2010 3 1 - homepage.fudan.edu.cnhomepage.fudan.edu.cn/jgying/files/2011/06/book2.pdf · r? u...

x(;1nfMSM2010 O 3 1

hH1Nu^ÆaGd RCxwwJUTh*? Merton k Scholes % 6wFkvQ39vE?\r\1\v L*B"^ÆaGwb9 860A+Æw4m?RL =NL*`&w^ÆaGKDh:wCw `1^ÆaGwL~jk bAP0w ?4mPB"^Æri Markov w kh+mw^ÆaGa1wSPQw7k:V_wM5GJ3*w^ÆaGw> ei 09^ÆIwL~>`H'a1w^b)4O^Æ^R?s$/^Æ&wk12B"b9L~w kj )H12B"r KkmUr?^ÆwbA_|Zh^Ær?9'iw awP(? 1<"`r GLw0|1+mB"r C Hww R?+1 %+mpdi)b Black-Schloes Merton w V/)9N4a4O^ÆaGw ka C8vM1 bA'5w 29w^!7ba?v6f Markov _ Ewp?5CUdeRw Markovwa C :V?_wr\8/.Kqfz?0b)^.0`1daJw?W)B" Markovw kL~ awSPb:?8 Hhw14+B""g_ OphH4UTw Markov #wF4m_$we1$JeiU?+m:V?dHwj^.E wRLFX Uw/uU.H? 4m_Ew8 86e7b [9]. ^ÆaG :?+w/Gb0w/ei7bk/wSP*^ÆaG83 [10], Billingsley[1], /E6w Kallenberg[6].

PJ i t ` 1

§1.1 o& my . . . . . . . . . . . . . . . . . . . . . . . . . 1

§1.2 bCe kEF . . . . . . . . . . . . . . . . . . . . . . . 3

§1.3 n/ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5 i( 9

§2.1 +mD^Æ . . . . . . . . . . . . . . . . . . . . . . . . . . . 9

§2.2 ^Æ Markov C . . . . . . . . . . . . . . . . . . . . . . . . . 16[ 2%K 27

§3.1 rIdw . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27

§3.2 1 % . . . . . . . . . . . . . . . . . . . . . . . . . . . 39g i(# 48

§4.1 9t* . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 48

§4.2 Kolmogorov, . . . . . . . . . . . . . . . . . . . . . . . . . . 54v Markov A 65

§5.1 Markov . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 65

§5.2 Galton-Watson $7aG . . . . . . . . . . . . . . . . . . . . . . . . 85

§5.3 eM Markov # . . . . . . . . . . . . . . . . . . . . . . 93F Levy # 103

§6.1 _'aG . . . . . . . . . . . . . . . . . . . . . . . . . . . 103

§6.2 Levy-Khinchin ); . . . . . . . . . . . . . . . . . . . . . . . . . . . 105

§6.3 Poisson aG . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 110U Brown 113

§7.1 Brown wM# . . . . . . . . . . . . . . . . . . . . . . . . . . . 113

ii

sq iii

§7.2 # . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 115

§7.3 rCF . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 118

§7.4 &; . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1204ry 124

u!a§1.1 'v9.;=8 w o&?1 9w Uw0b.?_skp RL7a ,poIW0[F[wdaEbwo&a #+Cw8 pPw8 %E9 w U& (Ω, F , P) ?bA8j*^R?U Ω ?#jn8 HD[j*F ? Ω w σ- iR P ?e9j* (Ω, F ) w89& ξ ?bA^Æ& B ∈ B(R), m

µ(B) = P(ξ ∈ B). (1.1.1)F* µ ? (R, B(R)) w89^R?U (R, B(R), µ) ?8j* µ D ξ w$/wweRF (x) = µ((−∞, x]) = P(ξ ≤ x), x ∈ R (1.1.2)D ξ w$/eR$/eR $/? (1.1.2) ,ybw"6H#B"w Dynkin tkmy ?8 H>wbAt a, 69H_EwG,Xw|my Dynkin H#: `wM [?] _i^D Dynkin t& F0 ? Ω w9^gEwn DyF* σ(F0) ); F0 *Ew σ- iRaxs Ω F0 wh6 σ- iRDbA^y? π-y`a*6' Euclid j*`kk? π- yDbA^y? Dynkin y λ- y`a ∅, Ω w-\ .6e+\'%? π- y σ- iR? Dynkin y9.jaQ kA Dynkin yw6? Dynkin yr\ Ω w l^y A , ba bA A wh6 Dynkin y δ(A ), ^yXzD A *Ew Dynkin y

Dynkin y σ- iRw;*R v/ ke+'r/.6e+' v61at G$89w=n1::69bAy? σ- iRqj&*a? Dynkin y Dynkin tR?>[1bA6 Dynkin y? σ- iRwbA+mwL$n/1

gz |dv 2< 1.1.1 (Dynkin) & F0 ?bA π-y% δ(F0)?bA σ-iRr\ σ(F0) =

δ(F0).JmFFZ6 δ(F0) *6\' A ∈ δ(F0), mκ[A] := B ∈ δ(F0) : A ∩ B ∈ δ(F0).#Z6 κ[A] ?bA Dynkin y=5E69 κ[A] (1) -\' (2).,6we+\' (1), B ∈ κ[A], % A, Ac, A∩B ∈ δ(F0), r\ A∩Bc = [Ac ∪ (A∩B)]c ∈

δ(F0), r\ Bc ∈ κ[A].69 (2), Bn ⊂ κ[A] ?.6%% A ∩ Bn ? δ(F0) H.6r\ A⋂

(⋃

n Bn) ∈ δ(F0), |Q ⋃

n Bn ∈ κ[A].r F0 ? π- yS A ∈ F0 [ κ[A] ⊃ F0 κ[A] ⊃ δ(F0). 0k[qA ∈ δ(F0) 3 κ[A] ⊃ F0. r\ κ[A] ⊃ δ(F0), δ(F0) HY*6\'my? Dynkin twbKx%HOpahuEteraXwlEeaEw?&*69nw89Ew> ^Æ&wRL ^E my0.a?bZ~w=y< 1.1.2 & L ? (Ω, F ) wb9e9eRgEw+Cj* F0 ?bA π-y(d(1) l A ∈ F0 1A ∈ L ;

(2) 1 ∈ L ;

(3) ` fn ? L HbCw#3eRwU n m% supn fn ∈ L .F* L σ(F0)- e9weRkProof. Dynkin tev L σ(F0) H knw;CeRv lbA σ(F0) w#3e9eRei); σ(F0) |'+meRw*L meRw'C|Q> k^1ua&lI:& P ? (Ω, F ) w89 ξ ?^Æ& ξ w$/? (R, B(R)) (d l a, b ∈ R, P(ξ ∈ (a, b)) = µ(a, b)wb89Tkt Borel σ- iR?*k*Ew*k?bAπ- yF*myfev R l Borel e9eR f

E[f(ξ)] =

∫

f(x)µ(dx).

gz |dv 3~bAEt Dynkin tw?w8PC?8 HbA#>:Vww8Pa?8 *9 wbA(pw8PhP?^A=/`9w=/ A, B ∈ F D,yw`P(A ∩ B) = P(A)P(B).w8P%?w^ÆEZ0A8Pw|ZAbA| AbAs^0/=y?wa1w>`.,y0 n A=/ A1, · · · , An ∈ F ,y?>dH kA=/s3*w8xB_*w8wF l

k ≥ 1 1 ≤ n1 < · · · < nk ≤ n P(An1 ∩ · · · ∩ Ank

) = P(An1 ) · · ·P(Ank).~^Æ& ξ1, · · · , ξn ?> l x1, · · · , xn ∈ Rd

P

(

n⋂

k=1

ξk ≤ xk)

=

n∏

k=1

P(ξk ≤ xk). 098Ptbhu1t=/ywC8P& A1, · · · , An ⊂ F ?=/yF* A1, · · · , An ,y?>^.A=/yHQbA=/ Ak ∈ Ak, 1 ≤ k ≤ n A1, · · · , An ,y_zyX+CiRHw+CUC*A=/yw,yC?>dH k*A?,ywF*^Æ&wCei=/ywC)b< 1.1.1 ^Æ& ξ1, · · · , ξn qwFq=/y σ(ξ1), · · · , σ(ξn),yProof. & n = 2, 1E&* l Borel B1, B2 ∈ B(R)

P(ξ1 ∈ B1, ξ2 ∈ B2) = P(ξ1 ∈ B1)P(ξ2 ∈ B2).w Dynkin te6^Æ3^?Ew^R?Ue ^Æ&wn9*wC35R?e a1*Ew σ- iR9*wC0[CR tbwÆU§1.2 "6+8# =` ξ eF* limn E(ξ; |ξ| ≥ n) = 0. `bbe^Æ&F*0A*bCFR?bCew8P

gz |dv 4 1.2.1 bAe^Æ&e ξi : i ∈ I D?bCewlim

N→∞sup

IE(|ξi|; |ξi| ≥ N) = 0..HZ6 ξi : i ∈ I bAe^Æ&_kE% ξi ?bCew"6CQbCewbAx%n/#69bAtU9e^Æ&?^Jw< 1.2.1 `^Æ& ξ e% l ǫ > 0,a δ > 0,8v l A ∈ F ,

P(A) < δ, E(|ξ|; A) < ǫ.

Proof. rE(|ξ|; A) = E(|ξ|; A, |ξ| ≥ N) + E(|ξ|; A, |ξ| < N)

≤ E(|ξ|; |ξ| ≥ N) + N · P(A),S l ǫ > 0, a N 8v E(|ξ|; |ξ| ≥ N) < 12ǫ, 0 < δ < 1

2N ǫ, r\P(A) < δ, R E(|ξ|; A) < ǫ.< 1.2.1 & ξi : i ∈ I ?^Æ&e%a?bCewLn/?(1) L1- supi∈I E|ξi| < ∞.

(2) bC^J l ǫ > 0, a δ > 0 8vq A ∈ F , P(A) < δ 3supi∈I E(|ξi|; A) < ǫ.

Proof. C k A ∈ F , N > 0, E(|ξi|; A) = E(|ξi|; A ∩ |ξi| ≥ N) + E(|ξi|; A ∩ |ξi| < N)

≤ E(|ξi|; |ξi| ≥ N) + NP(A).bCeC|Q ξi ?bC^Jw : A = Ω, vE|ξi| ≤ E(|ξi|; |ξi| ≥ N) + N,vt ξi w L1- CL$C& ξi ?bC^Jw L1- w Chebyshev .x:sup

iP(|ξi| ≥ N) ≤ 1

Nsup

iE|ξi|.

gz |dv 5r\ l ǫ > 0, a N 8v l i ∈ I, P(|ξi| ≥ N) < δ. v ξi wbC^JCvt l ǫ > 0, a N > 0, 8v E(|ξi|; |ξi| ≥ N) ≤ ǫ _i ∈ I EbCeC< 1.2.2 e^Æ&I ξn L1- F ξ wLn/? ξn ?bCeww ξn

p−→ξ.

Proof. CH# ξnp−→ξ ?%w ξn w L1- C^?%wEZ6 ξn wbC^JCRO k A ∈ F ,

E(|ξn|; A) ≤ E(|ξ|; A) + E(|ξn − ξ|). l ǫ > 0, a N , q n > N 3 E(|ξn − ξ|) < 12ǫ. ~r ξ, ξ1, · · · , ξN eSa δ > 0, P(A) < δ % E(|ξ|; A), _ E(|ξi|; A), 1 ≤ i ≤ N , 6 1

2ǫ. r\ P(A) < δ, R supn E(|ξn|; A) < ǫ.L$CH#69 ξ ?ewr 8FSa ^ ξkn wVVF ξ. Fatou t L1- C E|ξ| ≤ lim infn E|ξn| < ∞. r\ ξ e k ǫ > 0,bC^JCa δ > 0 P(A) < δR E(|ξn|; A) < ǫ l n Ew E(|ξ|; A) < ǫ E ξn

p−→ξ, a N q n > N 3RP(|ξn − ξ| > ǫ) < δ. r\

E|ξn − ξ| ≤ E(|ξn − ξ|; |ξn − ξ| < ǫ) + E(|ξn − ξ|; |ξn − ξ| ≥ ǫ)

≤ ǫ+E(|ξn|; |ξn − ξ| ≥ ǫ)+E(|ξ|; |ξn − ξ| ≥ ǫ) < 3ǫ.69 ξnL1

−→ξ.` ξn i L1 F ξ, F* limn Eξn = Eξ, $ *ei6U$*61r>8 Lebesgue kEF^ÆI ξn bAe^Æ& η kE3 ξn ?bCew5 l N > 0 E[|ξn|; |ξn| > N ] ≤ E[η; η > N ].r\ Lebesgue kEF?OwbAhKyp

§1.3 m1Tq'i8 (h*?r Markov ) %z.`n/ 0AL8Pn/ ?^1N8wn/88I:N4k(huw& (Ω, F , P)

gz |dv 6?8j* A, B ∈ F , mP(A|B) :=

P(A ∩ B)

P(B),D? B *wn/" B *w8/+mz A U B wn/8\mHq$& P(B) > 0. 8 HhLwI:9b?8I:&

Ωn : n ≥ 1 ? Ω we9$y% l A ∈ F , P(A) =

∑

n≥1

P(A|Ωn)P(Ωn).r=/ A eiaE^Æ& 1A, P(A) = E(1A). r\1ei8I:|Zt^Æ&ypEξ =

∑

n≥1

E(ξ|Ωn)P(Ωn), (1.3.1)dH ξ ?bAe^Æ& l=/ B, yXzm B w__E(ξ|B) :=

E(ξ; B)

P(B).Tk E(ξ; B) := E(ξ · 1B).!V: (1.3.1), `mbA;w^Æ&

ξ′ :=∑

n

E(ξ|Ωn)1Ωn,F*8I:ei:E Eξ = Eξ′. w ξ′ ? Kkmh:H ξ w^Æ&5 kCbAI xn, m η :=∑

n xn1Ωn, F*

E(ξ − η)2 = E(ξ − ξ′)2 + E(ξ′ − η)2,^R?U ξ′ L2 Yzwkm" _yX η 0[w^Æ&Hz ξ hHw ξ′ 0AhHwkm"? (wVV,xwkm") bw696x:wU.?E[(ξ − ξ′)(ξ′ − η)] = 0.\:Ewr? (1) ξ′ Ωn >R (2) l n,

E(ξ′; Ωn) = E(ξ; Ωn).

gz |dv 8

(2) -n/ wmE(E(ξ|A )) = E(E(ξ|A ); Ω) = E(ξ; Ω) = E(ξ).

(3) E η = 1B, B ∈ A , 69Rei 0m?%w(4) l A ∈ A , E(ξ; A) = Eξ1A = Eξ · P(A) = E(Eξ; A).

(5) r E(ξ|A1) ? A2 e9wS E(E(ξ|A1)|A2) = E(ξ|A1) ?%w69bAx:/? A1 e9wE69 l A ∈ A1, E(ξ; A) =

E(E(ξ|A2); A), 0 A ∈ A2 ?%w(6) m%"6U9n/ E(ξ|A ) ? A e9w^Æ&Hz ξ hHwFA< 1.3.2 & ξ ?_ e^Æ& A ? F w^ σ- F*

E(ξ − E(ξ|A ))2 = infE(ξ − ζ)2 : ζ ?U A e9w_ e^Æ&,n/ ?_U A e9w_ e^Æ&H ξ w_ YzhHwJr ζ E(ξ|A ) ?U A e9w_ e^Æ&S6wf vE(ξ − ζ)2 = E(ξ − E(ξ|A ))2 + E(E(ξ|A ) − ζ)2 + 2E[(ξ − E(ξ|A ))(E(ξ|A ) − ζ)]

= E(ξ − E(ξ|A ))2 + E(E(ξ|A ) − ζ)2,\aQ> ?%w^lw7ua Euclid j*wbAtdbA^j*whÆYz?0At0Aj*wuwYz_ e^Æ&k?bAqfw+Cj*U A e9w_ e^Æ&k?awbA+C^j*r\ E(ξ|A )R? ξ t0Aj*wu0^Cn/ bA:V?A^Æ& ξ aEbAE9w^Æ& A aEg8=F* A e9w^Æ&k? g8 A wri" ξ w9k E(ξ|A ) nf?dHz ξ YzhHwbA9/U?h 9r\Un/ E(ξ|A ) ?g8= A wn/"ξ wh 90[ 1.3.1 wCFRqfw?A (6) U9q,= A = ∅, Ω 3 ξ wh 9R?aw qdO= A = F3 ξ wh 9R? ξ; (4) U9q A ξ 3aww=9 ξ ,S\=k,=b[_zU σ- A k=?CQ Ω w$y ω ∈ Ω, A ∈ A , 1KU?* ω ∈ A.

j)Æ^ÆaG:V?^Æ'4H3*TIwRWawSP4?r>? ITw*A^Æ&h+mw^ÆaG?^Æa?>s$/^Æ&IgEw1$kI 5HG^L0bAs[LzeKCw,paw SPh 1708N_L/ Remond de Montmort._i^ÆwSPgq?w7 0b)H12.sw SP^ÆH#?L~wgn uf D+m^Æ4RD+m^Æ?^ÆHh+mwbya .AV3ik3weKC?b[w0K^Æei+mzAb+I5w|w :u5' 1$1B"t Markov Cv0A uSP+m^Æwb9j§2.1 0k*^Æ?h"RL!UTkSPw^Æ'4ra#>w>-AeibA|`956zbm6Lbm0[b:"R?bA+mw^Æq|?hbA^Æ*jwk1eiAs^"ed|2R`xBK ildHAHwqR3*TI"u pzbmr3Lbmq3"0[ pRQ'bnxx"wDbn[u"bn[u_09eKw[u?xeKQ'w05gLC1U+mD^Æw zMb X-Y_6w4R?gEwj*& m < n, a, b?4Rbn (m, a)t (n, b) w?u?>4R (sm, sm+1, . . . , sn) (d

(i) sm = a, sn = b;

(ii) m < k ≤ n, sk − sk−1 = 1 −1, sk im 1 33":+2d,w (k − 1, sk−1) (k, sk) :AEbn.+ n − m D?uw?%La_d (m, a)t (n, b)w?ubR ^ (0, 0)t (n−m, b−a)w?ubR?,sw"61 Nn,x ); (0, 0) t (n, x) w?ub9

gj xmi 10R%a : (0, 0) (n, x) w?uqwFqa p, q ∈ Z+ 8vn = p + q, x = p − q. 5 p k q );?uH3k3"w.+RRei 0[bn^ (0, 0) Qw?uLa (n, x) qwFqa p n3w.+r\TwV/v

Nn,x =

p + q

p

=

n

n+x2

<#?bA+maYw>` GwbÆ?Maxwellk Kelvin H#t+w< 2.1.1 & a, b > 0, m < n, %^ (m, a) t (n, b) ww x- O,w?ubR ^ (m, a) t (n,−b) w?ubR,sJ$* A, B );^ (m, a) t (n, b) ww x- O,w?uk ^(m, a) t (n,−b) w?uk A w lu (sm, . . . , sn), &hJt x-O? k 3fF* m < k < n. 2u k t n 1$ x- O#v?u

(sm, . . . , sk,−sk+1, . . . ,−sn).a? B Hw?ujZ6#(sm, . . . , sn) 7→ (sm, . . . , sk,−sk+1, . . . ,−sn)1 A t B wbbwr\> EEV 2.1.1 & b > a > 0, 69 (0, 0) t (n, a) .t y = b w?uR

Nn,a − Nn,2b−a.EV 2.1.2 (L0#) & a, b > 0,−b < c < a, 69 (0, 0) t (n, c) w.t−b w?uR

∑

k

(Nn,2k(a+b)+c − Nn,2k(a+b)+2a−c).

gj xmi 11?uCQbA8j*Kz4 l n ≥ 1, Wn ); (0, 0) Q? n w?uk Pn ?dwQ~8 B ⊂ Wn, F*Pn(B) =

|B||Wn|

=|B|2n

."6w[jw??#wwa Whitworth (1878) k Bertrand

(1887) iQ> 2.1.1 bu[HuK P,Q wv[$* m, n w m > n, F* 4Au[aGH P w[Rb:# () Q w[Rw8 (m − n)/(m + n). n Au[vP,Qw[R; sn. F*u[aG?bn (0, 0) t(m + n, m − n) w?u (s0, s1, . . . , sm+n), 0[wubR Nm+n,m−n. _u[aG?x8w?bAQ~8j (0, 0) t (m+n, m−n)w?uSh~.t x- Ow80A=/xs (1, 1) t (m + n, m−n) wu.t x- OdHwubRx Nm+n−1,m−n−1 , (1, 1) t (m + n, m− n)t x- OwubRw#v/? Nm+n−1,m−n+1. r\8

Nm+n−1,m−n−1 − Nm+n−1,m−n+1

Nm+n,m−n=

m − n

m + n.F*&*\ P w[Rb:.! Q w[Rw8IF\ (0, 0) Qt (m+n, m−n).Wt x-O"6wubR_d 1, 0x (0, 1)t

(m + n, m − n + 1) .t x- OwubR#0x Nm+n,m−n −Nm+n,m−n+2, r\8

Nm+n,m−n − Nm+n,m−n+2

Nm+n,m−n=

m + 1 − n

m + 1.' B"h+mw^Æ/UbD+m^ÆT8j*

(Ω, F , P) ds$/^Æ&I Xn : n ≥ 1, dHP(Xn = 1) = P(Xn = −1) =

1

2.

S0 = 0, Sn :=n∑

k=1

Xk, n ≥ 1,

gj xmi 12D 1 Dw+m^Æ ω ∈ Ω,

(S0(ω), S1(ω), . . . , Sn(ω), . . . )D? ω w[uawbn (0, 0) Qw?u.F\` (s0, . . . , sn) ∈ Wn, F* P(Si = si; 0 ≤ i ≤ n) = 12n , r\`

B ⊂ Wn, F* P((S0, . . . , Sn) ∈ B) = Pn(B). Z~Ub Sn B (*?) w=/w8kbr6_1w?uw8?bCwr\Dw+m^Æw8j*wjx%?uMEw8j*wjeiR?uw \"61u\^Æwb9ww$/H#f H3*w$/1#\?u x- O w8t,h u2n := P(S2n = 0),

n ≥ 0. F* u0 = 1,

u2n = N2n,0/22n =1

22n

(

2n

n

)

.< 2.1.1 n ≥ 1,

P(S1 > 0, S2 > 0, . . . , S2n > 0) =1

2u2n,

P(S1 ≥ 0, S2 ≥ 0, . . . , S2n ≥ 0) = u2n.J%P(S1 > 0, S2 > 0, . . . , S2n > 0)

=n∑

k=1

P(S1 > 0, S2 > 0, . . . , S2n−1 > 0, S2n = 2k).#(d S1 > 0, S2 > 0, . . . , S2n−1 > 0, S2n = 2k w?ubR?N2n−1,2k−1 − N2n−1,2k+1,r\

P(S1 > 0, S2 > 0, . . . , S2n > 0)

=1

22n

n∑

k=1

(N2n−1,2k−1 − N2n−1,2k+1)

gj xmi 13

=N2n−1,1

22n=

1

2u2n. l k ≥ 0,(d S1 ≥ 0, S2 ≥ 0, · · · , S2n = 2k w?uR? (0, 0)t (2n, 2k) :+ y = −1 .6w?u#bRx N2n,2k −N2n,2k+2. r\

P(S1 ≥ 0, S2 ≥ 0, · · · , S2n ≥ 0) =1

22n

n∑

k=0

(N2n,2k − N2n,2k+2) = N2n,0/22n.E69 τ );Hw3*τ := infn > 0 : Sn = 0.Tk τ =? ‘3*’, ?uweR^R?^Æ&w1TSa=r1.8uu?*bt 0 ~` τ *ab?SRv61L>t0[w^Æ&a+D 0 wHH3 (H;Hw3*).< 2.1.2 l n ≥ 1,

(1) P(τ = 2n) = u2n−2 − u2n = 12n−1u2n, n ≥ 1.

(2) u2n =∑n

r=1 P(τ = 2r) · u2n−2r.Jtfv8P(S1 6= 0, S2 6= 0, . . . , S2n 6= 0) = P(S2n = 0).S^Æ 2n 3fHw8

P(τ = 2n) = P(S1 6= 0, . . . , S2n−1 6= 0, S2n = 0)

= P(S1 6= 0, . . . , S2n−2 6= 0) − P(S1 6= 0, . . . , S2n−1 6= 0, S2n 6= 0)

= P(S2n−2 = 0) − P(S2n = 0).I: (2) 8I:|Qr>,h f2n := P(τ = 2n). aQ P(τ < ∞) = 1, 0U9^Æi8 1 *3*J Stirling I:u2n ∼ 1/√

πn, r\ Eτ = ∞,^R?U__3*?*w

gj xmi 14:[f eGkHra3*w$/An := maxS0, S1, . . . , Sn,

Tx := infn ≥ 0 : Sn = x,$*D Sn weG Hra x w3* (x wHH3). %` x ≥ 0,% An ≥ x qwFq Tx ≤ n. & x > 0, k ≤ x, #^ (0, 0) t (n, k) wt:+ X = x w?ubRx^ (0, 0) t (n, 2x − k) w?ubRr\P(Sn = k, An ≥ x) = P(Sn = 2x − k),F*

P(Sn = k, An = x) = P(Sn = k, An ≥ x) − P(Sn = k, An ≥ x + 1)

= P(Sn = 2x − k) − P(Sn = 2x + 2 − k).eGw$/P(An = x) =

∑

k≤x

P(Sn = k, An = x)

= P(Sn = x) + P(Sn = x + 1).% P(Sn = x) P(Sn = x + 1) bA#v\Hra3w$/Tx = n = S1 < x, . . . , Sn−1 < x, Sn = x.E\ (0, 0) t (n − 1, x − 1) w.t:+ X = x w?ubR#ax

Nn−1,x−1 − Nn−1,x+1.r\ P(Tx = n) = (Nn−1,x−1 − Nn−1,x+1)/2n.hvf >z3knwG3w$/ L2n );? 2n w?uhvt 0 w3*L2n := supk ≤ 2n : Sk = 0.D 0 w>z3a ?SR

gj xmi 15< 2.1.3 P(L2n = 2k) = u2k · u2n−2k, 0 ≤ k ≤ n.J S′j =

∑ji=1 X2k+i. %

L2n = 2k = S2k = 0, S2k+1 6= 0, S2k+2 6= 0, · · · , S2n 6= 0

= S2k = 0, S′1 6= 0, S′

2 6= 0, · · · , S′2n−2k 6= 0,r\Ckt 2.1.1,

P(L2n = 2k) = P(S2k = 0)P(S′1 6= 0, S′

2 6= 0, · · · , S′2n−2k 6= 0) = u2k · u2n−2k.E69% l n ≥ 0, / Sn−1 Sn ?#3w1U n A3?5w/ Sn−1 Sn ?#5wU n A3?3w σ2n ? 0 t 2n 35w3R

σ2n :=

2n∑

k=1

1Sk−1≥0,Sk≥0,D^Æ 5w3_ σ2n ^ ?SR< 2.1.4 P(σ2n = 2k) = u2k · u2n−2k, 0 ≤ k ≤ n.J n \Gq n = 1 3% P(σ2 = 0) = P(σ2 = 2) = 1/2, > E~t 2.1.1 8 P(σ2n = 0) = P(σ2n = 2n) = u2n, > ESE 0 < k < n 69e' & P(σ2m = 2k) = u2k · u2m−2k l m < n k0 < k < m E τ );H 0 w3*F*q 0 < k < n 38I:k\G$&

P(σ2n = 2k) =

n∑

r=1

P(σ2n = 2k|τ = 2r)P(τ = 2r)

=n∑

r=1

1

2(P(σ2n−2r = 2k − 2r) + P(σ2n−2r = 2k))P(τ = 2r)

=1

2

n∑

r=1

(u2k−2ru2n−2k + u2ku2n−2r−2k)P(τ = 2r)

=1

2(u2ku2n−2k + u2ku2n−2k),\|Q> hvbAxh 2.1.2(2) vt

gj xmi 16f(x) =

1

π√

x(1 − x), x ∈ (0, 1),F* Stirling I: u2k · u2n−2k ∼ 1

nf(k/n). ∑

k<xn

u2k · u2n−2k ∼∑

k/n<x

1

nf(k/n) −→

∫ x

0

f(y)dy =2

πarcsin

√x,\|QQ:w5$^R?U L2n/2n k σ2n/2n w$/eR0Hz?6w5$eRTkt4eR f wAW1' L2n σ2n $/9H Jfw?5$ ^ÆaG .eVowb'C

§2.2 k* Markov ~&8j* (Ω, F , P) dw Bernoulli IX1, X2, . . . , Xn, . . . ,dH P(Xn = 1) = p, P(Xn = −1) = q w p + q = 1. 0[ws$/^ÆIwa C| 2.3.1[9] |Qa^?v6B"wEbw Kolmogorov ,wh0|1CbA+mPxw69Tk0[w8j* ^ÆIwa C?,x:VwiC1>>a^?E69wbbAJ.wV|w^ÆEZa2<*0[bAI0[w^ÆID (EG8 p) w Bernoulli I< 2.2.1 & p, q ≥ 0, p + q = 1, %a 8j* (Ω, F , P) ds$/^Æ&I X1, X2, . . . , Xn, . . . , dH

P(Xn = 1) = p, P(Xn = −1) = q.J [0, 1] w Lebesgue 9j*k8j* (Ω, F , P). H#?q : p$* [0, 1] [0, q), [q, 1]; m X1 1A* −1, 2A* 1;v? q : p2`1$W [0, qq), [qq, q), [q, q+pq), [q+pq, 1];m X2 fR −1, SR 1; b:J0AaG. q : p 2.A*|$ n 32*|$ 2n A*m Xn fR −1, SR 1,' X1 = ǫ1, . . . , Xn = ǫn); n|$vwbA6*

gj xmi 17d?? qkpn−k, dH k = |0 ≤ i ≤ n : ǫi = −1|, r\jZ6 X1, X2, . . . ?(dws$/^Æ& x ∈ Z, Sx

0 := x, Sxn := x +

n∑

k=1

Xk, n ≥ 1.q x = 0 3+mz: Sn. r\ Sxn = x + Sn. Sx

n D? x Qw+m^Æa3dbAm w8? p, 3idbAm w8? q. qp > q 3UaY3w9Y3iw p = q ?r6f awD^Æ' 1at 8j* (Ω, F , P) "1be^ÆI (Sx

n : n ≥0) : x ∈ Z, ^R?UTw89".sw^ÆI"612~wVuas[wj^R?UsbA^ÆI .sw89"/ $/wkm",*vbKaE %k:V .8/\v0KV aN4wym1uaae!Nw$/eR1eiKDua$/eRw5'H#a 8j* (Ω, F , P),(d l$/ F , a ^Æ& ξ 8vaw$/n? F (- 2.1.3[9]), 0?UsbA8p.sw^Æ&5'_w$/~1êi,tbATwe9j* (Ω, F ) kbATw^Æ& ξ (d l$/ F , a 8µF , 8v ξ (Ω, F , µF ) "w$/eR? F , sbA^Æ&.sw85'_w$/w6U0Kj RL)B *H^?L>w );ZeiTYzUbI| 1$IêiUb63 I|^R?UeiYz3*);Zêi3*Yz);ZZK); :)BeiU63 I|VA^ /_qtmBwg31L>+mz 1v|0?U 30 miles/gallon, H_s[wj1> U+0?U 8 + / ÆI|' 1R/,^ÆaGw~bK); W );w4R=ukKzU?"#wk

w : Z+ −→ Z. w ∈ W , m ξn(w) := w(n), ? W w4R=eR);u w w n- l(B ); W 8v_w ξn : n ≥ 0 E^Æ&wh6=/F* ξn

gj xmi 18? W U=/ B w^Æ&I (+D^ÆI), W ξn r>D?~%[j* ~%^ÆI"61^ (Ω, F , P) Cj* (W, B) m8 lCw ω ∈ Ω, x ∈ Z, # n 7→ Sxn(ω), n ≥ 0 ?bA4R=u

W HwY φx(ω). S l x ∈ Z, ω 7→ φx(ω) ?[j* Ω t~%[j* W wbA#F*5 ξnφx = Sxn.< 2.2.1 l B ∈ B, φx ∈ B = ω ∈ Ω : φx(ω) ∈ B ∈ F .J#hKw B = ξn = y, F* φx ∈ B = ξnφx = y = Sx

n = y ∈ F . A );BH8vtHUEwYk% A ⊂ B. r ξn = y ∈ A ,_i.A ξn ?U A w^Æ&~eiZ6 A '^?bA=/r\|Q B ⊂ A , r B ?h6wE690[w#>D?^Æ#03 l x ∈ Z, m8 Px

Px(B) := P(ω : φxω ∈ B) = E(1Bφx), B ∈ B.

(W, B) w8 Px D? P # φx "w4% ξn P

x "w*n$/ Sxn P "w*n$/bC l n ≥ 0, x0, . . . , xn ∈ Z,

Px(ξi = xi : 0 ≤ i ≤ n) = P(Sx

n = xi : 0 ≤ i ≤ n). x, y ∈ Z, p(x, y) :=

q, y = x − 1;

p, y = x + 1;

0, d`F*1eivz:Q*$/< 2.2.2 l n ≥ 0, x0, . . . , xn ∈ Z, P

x(ξi = xi : 0 ≤ i ≤ n) = 1x=x0p(x, x1)p(x1, x2) · · · p(xn−1, xn).J::z\P

x(ξi = xi : 0 ≤ i ≤ n) = P(Sxn = xi : 0 ≤ i ≤ n)

gj xmi 19

= 1x=x0P(Si = xi − x : 1 ≤ i ≤ n)

= 1x=x0P(X1 = x1 − x, X2 = x2 − x1, . . . , Xn = xn − xn−1)

= 1x=x0P(X1 = x1 − x)P(X2 = x2 − x1) · · ·P(Xn = xn − xn−1)

= 1x=x0p(x, x1)p(x1, x2) · · · p(xn−1, xn).uj* (W, B) w8 D = Px : x ∈ Z D(Xw (~%) +m^Æ Px " ξn ? x Qw^Æs Sx

n 8 P 9"wB5 Px ?7C ?uw (-j) 0K(Xw b^ÆaGh*? MarkovaG0K ? E.B.Dynkin tw,

Sxn, P uU ξn, Px wfV?r/8?bA^ÆI?^ x &wv/1?VbATw^ÆIr/?bAN4[j*v/?Xkw[j*

Markov 6wten/8);P

x(ξn+1 = xn+1|ξn = xn, . . . , ξ0 = x0) = p(xn, xn+1) = Pxn(ξ1 = xn+1). (M1)0R? D w Markov Cg8' ξn = xn, 2u ξn+1 weK D a

ξ0, . . . , ξn−1 w D1n/ w u); MarkovC Bn ); n + 1- ^Æ&(ξ0, . . . , ξn) *Ew=/F* Bn ⊂ Bn+1. Tk Bn ?z=/ B b.?zw< 2.2.3 l n ≥ 0, x, y ∈ Z,

Px(ξn+1 = y|Bn) = p(ξn, y) = P

ξn(ξ1 = y), (M2)Tk# Pξn(ξ1 = y) ?# w 7→ ξn(w) x 7→ P

x(ξ1 = y) /w0nv6yXwhyX?J l x0, · · · , xn ∈ Z, `=/ sn = xn, . . . , s0 = x0 w8.x 0,F*a? Bn w^ (M1) 2.3.1[9] |Q (M2) E

gj xmi 201t W w|d\^ n ≥ 0, w ∈ W , mθnw(k) := w(k + n), n ≥ 0,r>w eiZ6 θn ? (W, B) t_'we9#+w

ξkθn = ξk+n, k, n ≥ 0.

θn <u w w`v n A;wu θnw ^uw n 3f`9r\D(i) |d\^|dvwuw k l(?uw n + k l(q θn ? W t_'w#st 2.2.1w69b[eZ6 l B ∈ B,

w ∈ W : θn(w) ∈ B ∈ B. 5 Gn ); ξn, ξn+1, . . . *Ew=/F*` B ∈ B, w ∈ W : θn(w) ∈ B ∈ Gn. ` Y ? W ^Æ&F* Y θn? Gn ^Æ&r> Bn i)a ξn i)' Gn i)2u0KÆU (M2) ei:EE

x(1ξ=yθn) = Pξn(ξ1 = y).Ebz< 2.2.2 n ≥ 0, x ∈ Z #3^Æ& Y , F*

Ex(Y θn|Bn) = E

ξn(Y ). (M3)Jr w 7→ Eξn(Y )? Bn ^Æ&SmEZ6 l^ A ∈ Bn,

Ex(Y θn; A) = E

x(Eξn(Y ); A) (2.2.1)` (2.2.1) ;CeR Y = 1B, B ∈ B EF*n/ wCFa#3+m^Æ&^E|'~ Y w+m^Æ& Yn, r\mF|Q (2.2.1) EH# Y = 1B, dH B = ξ0 = x0, . . . , ξm = xm A = ξ0 = y0, . . . , ξn =

yn, F*t 2.2.2,

Ex(1Bθn; A)

= Px(ξ0 = y0, . . . , ξn = yn, ξn = x0, ξn+1 = x1, . . . , ξn+m = xm)

gj xmi 21

= 1x=y0p(y0, y1) · · · p(yn−1, yn)1x0=ynp(x0, x1) · · · p(xm+1, xm)

= 1x=y0p(y0, y1) · · · p(yn−1, yn)Pyn(B)

= Ex(Pξn(B); A).0[r Bn ?zww6A:w A wk*ES (2.2.1) Y = 1B, dH B ?6wA: l A ∈ Bn E^6A:w 1B t#3 B e9w^Æ& Y a\wYzp5R?l_my ?? RdO 58v (2.2.2) Ew^Æ& Y k?bA+Cj* L , wmFE6A:w B k*6?'ww*E B, r\ L k B e9w#3/w^Æ&k69H_twj?8 HL>t_w^;CeRt#3^Æ&w ^?8 w(X v62tF31..!U9I: (M1), (M2), (M3) ? Markov Cw);A:X Markov T3* n ∈ Z+ Ew?_Æq3w^Æ3* W = Z+ ∪

+∞ w (Zm) ^Æ&D?q3` l n ≥ 0, T ≤ n ∈ Bn. vmBT := B ∈ B : B ∩ T ≤ n ∈ Bn, n ≥ 0.1uZ6 BT ^?bA=/=5% ∅, W ∈ BT , ~ B ∈ BT ,% Bc ∩ T ≤ n = T ≤ n − B ∩ T ≤ n ∈ Bn, S BT -\'hv

(∪Bn) ∩ T ≤ n = ∪(Bn ∩ T ≤ n), r\ BT e+\^'"6=5wZ6?+mwkj (q3w.!f e-W))(1) q T ?bAT3* n 3 T ?q3w BT = Bn.

(2) & T , S ?q3F* T ∧ S ^?(3) & T , S ?q3w S ≤ T , F* BS ⊂ BT .6wCF (1) U98,h BT ?nw> 2.2.1 y ∈ Z, m Ty := infn ≥ 0 : ξn = y, D? y wHra3%

T ≤ n = ∪0≤k≤nξk = y ∈ Bn,

gj xmi 22r\ Ty ?q3Hra3?* 3* n r*ei^V: 0 t n w?uv80?q3wF~ τy := infn ≥ 1 : ξn = y ^?q3D? y wHH3% τy ≥ 1 w τy Ty ? F9^ y Qwu* 09u Ty = 0. 1eiZ6"6wCF(1) =/ Ty ≥ n Ty = Tyθn + n Ep Tyθn + n ≥ Ty sE(2) =/ Ty ≥ Tx Ty = TyθTx

+ Tx E(3) =/ τy > n τy = τyθn + n E τy = n .EZ6 (1), r

Tyθn = infk ≥ 0 : ξkθn = y

= infk ≥ 0 : ξk+n = y = infk ≥ n : ξk = y − n,q Ty ≥ n 3 infk ≥ n : ξk = y = Ty, _i (1) E =/ T = n m ξT (w) := ξn(w), ξT :=∑

n≥0 ξn1T=n. Tk ξT T < ∞ a?^Æ q3 T 3fw D1U[j*weR X =/ A ? F ^Æ&?> l x ∈ R, X ≤ x ∩ A ∈ F .< 2.2.4 ` T ?q3F* ξT T < ∞ ?U=/ BT w^Æ&JEZ6 y ∈ Z, ξT = y ∩ T < ∞ ∈ BT , l n ≥ 0,

ξT = y ∩ T < ∞ ∩ T ≤ n = ∪0≤k≤nξk = y, T = k ∈ Bn.mq3w|d θT w(n) = w(n + Tw), θT = θn ` T = n. s[θT ^?q T < ∞ 34kma? (W, B) t (W, B) w^Æ#rB = ξn = x, w ∈ W : θT (w) ∈ B ∩ T < ∞ =

⋃

kξnθk = x ∩ T = k =⋃

kξn+k = x ∩ T = k ∈ B. "6?Uq3w Markov CD D wtMarkov C< 2.2.3 & T ?q3 Y ?#3^Æ&F* T < ∞

Ex(Y θT |BT ) = E

ξT (Y ),

gj xmi 23 l A ∈ BT E

x(Y θT ; A, T < ∞) = Ex(EξT (Y ); A, T < ∞). (M4)JyX6w691E Y = 1B, B ∈ B Z6 (M4) ERdO ~ (M3), r A ∩ T = n ∈ Bn, S

Ex(1BθT ; A, T < ∞)

=∑

n≥0

Ex(1BθT ; A ∩ T = n)

=∑

n≥0

Ex(1Bθn; A ∩ T = n)

=∑

n≥0

Ex(Pξn(B); A ∩ T = n)

=∑

n≥0

Ex(PξT (B); A ∩ T = n)

= Ex(PξT (B); A, T < ∞).E 697/Rzw^ÆbALwCFj*w_d.&Cj*gC^ x Qu" bAn B Jw8x^ x+y Qu" B+yJw8 Æ_dbAm 0?r p(x, y)_d.&Cp(x+z, y+z) = p(x, y), x, y, z ∈ Z.vz7OEt_d\^ l y ∈ Z, w ∈ W , γyw(n) := w(n)+y,

n ≥ 0. ry 24Au_d y m a? W w^Æ#w(d ξnγy = ξn +y,

γ−1y = γ−y.EV 2.2.1 69 Txγy = Tx−y, x, y ∈ Z.< 2.2.4 j*_d.&C W #3^Æ& Y E

xY γy = Ex+yY ,

x, y ∈ Z.Js[E Y = 1B, B = ξ0 = x0, . . . , ξn = xn, Z6RdO r p(x, y) = p(0, y − x), SE

x(1Bγy) = Px(ξ0γy = ξ0, . . . , ξnγy = xn)

gj xmi 24

= Px(ξ0 = x0 − y, . . . , ξn = xn − y)

= 1x=x0−yp(x0 − y, x1 − y) . . . p(xn−1 − y, xn − y)

= 1x+y=x0p(x0, x1) . . . p(xn−1, xn)

= Px+y(B).E 69r 1γy∈B = 1Bγy, :8);R? P

x(γy ∈ B) = Px+y(B), h*z P

x(B) = P0(γx ∈ B), _w$/ P

x P0 _dvtdn#_O&f0=H12w^Æwt MarkovCu\Hra3wAeR l x, y ∈ Z, φx,y ? Ty 8j* (W, B, Px) wAeR

φx,y(t) := ExtTy .H#j*_d.&C

φx,y(t) = ExtTy = E

0tTyγx = E0tTy−x = φ0,y−x(t)

φx,x(t) = φ0,0(t) = E0tT0 = 1.& x > 0, r^Æ.dbAm S^ 0 Q3ra x F#ra 1, Tx ≥ T1, 03 Tx = TxθT1 + T1, r\t Markov C

φ0,x(t) = E0tTx = E

0tT1+TxθT1

= E0tT1 · tTxθT1

= E0(tT1E

sT1 tTx)

= E0tT1 · E1tTx = φ0,1(t) · φ1,x(t). φ := φ0,1, F* φ0,x = φx.b 6^ 0 Q^ Tx ≥ 1, 03 Tx = Txθ1 + 1, Markov C

φ0,x(t) = E0tTx = E

0t1+Txθ1

= tE0(Es1tTx)

= t((E1tTx)p + (E−1tTx)q)

gj xmi 25

= tpφ0,x−1(t) + tqφ0,x+1(t). x = 1, 1vφ(t) = tp + tqφ(t)2,r φ(t) ≤ 1, S?v

φ(t) =1 −

√

1 − 4t2pq

2tq.SCv

φ0,−1(t) = E0tT−1 =

1 −√

1 − 4t2pq

2tp.\|Q 0 Q *0Jtb 1 w8

P0(T1 < +∞) = φ(1) =

1 − |p − q|2q

=

1, p ≥ q;

pq , p < q.q^ÆdY33a *3*Jtb 1. 03\d = h

E0T1 = φ′(1) =

+∞, p = q;

1p−q , p > q. \ 0 Q 0 wHH3 τ0 wAeRr^ 0 Q τ0 ≥ 2 > 1, S

τ0 = 1 + τ0θ1, Markov CE

0tτ0 = E0t1+τ0θ1 = tE0tτ0θ1

= tE0(Es1tτ0) = tpE1tτ0 + tqE−1tτ0

= tpφ0,−1(t) + tqφ0,1(t) = 1 −√

1 − 4t2pq.

0 Q *0J 0 w8P

0(τ0 < ∞) = 1 − |p − q| =

2q, p > q

1, p = q

2p, p < q,Tk D+m^Æ3\8 1.x l

gj xmi 26

1. & ξn ?+m^Æ` T ?q3 T < ∞, m ξ′n = ξn+T − ξT ,

n ≥ 0, 69 ξ′ ?bA^ 0 Qw+m^ÆA4zU^Æ^q3 T L;`92. bA^ −N,−N + 1, · · · , N − 1, N k+m^Æ (n ≥ 1), 3b0w8 p, 3ib0w8 q = 1 − p, −N N ?FWd&bA^^ 0 Q\a 0 rFw83. A, B bAs^,L L q`s^?SR%y.Cq`s^? 1, % A C B bq`s^? 3 5, % B C A bq b:tbLY>Q pk ); `93 A k qhvzw8:QU pk w;$ G4. bA^Æ3r?w8? p 3vbAw8? q = 1− p, a > 0,

π(a) );0A^Æ*0tb 0 w869π(a) = pπ(a + 2) + qπ(a − 1),q p ≤ 1

3 3 π(a) = 1.

\ 3 &Lr? 1<Ptwp1Ni Doob wFkC r15w*;' r?^Æ$HwH<Jk.e!wFX 0b)H12+mB"rd 'iC Hww§3.1 'E,W f r1En/ x8PE!w?le^Æ& ξ ^ σ- A wn/ E(ξ|A ) R?(d l A ∈ A , E(ξ; A) = E(η; A)w A e9w^Æ& η. ^Æ&n ηi : i ∈ I, σ(ηi : i ∈ I) );

η−1i (B) : i ∈ I, B ∈ B *Ew σ- Tka^?8v ηi : i ∈ I e9wh6 σ- E(ξ|ηi : i ∈ I) := E(ξ|σ(ηn : i ∈ I)). ^ÆaG Hr?bA?*w[:Vz4r?I_ wi:[D^Æ& ξn ?s$/^ÆI$/ P(ξn = 1) = p,

P(ξn = −1) = 1− p. ei?+mw,p Xn :=∑n

i=1 ξi, n T HbwLzR%q p = 1/2 3 ?I_w03E(Xn|Xn−1, · · · , X1) = P(ξn|Xn−1, · · · , X1) + Xn−1 = Xn−1,^R?U`1ig8wr n− 1 T w>`u9"bTLz ?

0. & (Ω, F , P) ?bA8j*3* I ?4RnwbA^.!&a?4RwbA*d50.?qL`.?*1.! n + 1 ? I H4R n w"bA4R & Xn : n ∈ I ?^ÆI % l n ∈ I, mFn := σ(Xi : i ∈ I, i ≤ n),?3* n rw^Æ&_*Ew σ- q Fn U n ?|'wn Fn :

n ∈ I D?^ÆIw_ 3.1.1 ew5=^ÆI Xn : n ∈ I DrI` l i, j ∈ Iw i < j, E(Xj |Fi) = Xi.

27

gu nhr 28~ ≥ hE3D"r ≤ E3Drrwmx% l n ∈ I, E(Xn+1|Fn) = Xn. bArw n Ur EXn = E(E(Xn+1|Fn)) = EXn+1. "rw ? n w|'I> 3.1.1 ^ÆIswr& I = N, ξn ?ew^ÆIwEξn = 0, F* ξ1 + · · · + ξnn≥1 ?rI` ξn ≥ 0, w Eξn = 1, F*ξ1 · ξ2 · · · ξnn≥1 ?rI> 3.1.2 (Wald r) & ξn ?b=Hmw+m^ÆF*dHw σ- Bn ?|'w λ > 0,

Yn := λξn , n ≥ 0.r ξn ? Bn e9wS Yn ^?E

x(Yn+1|Bn) = Ex(λξn+Xn+1 |Bn) = λξn · Ex(λXn+1 |Bn)

= Yn · (λp +q

λ).r\ λξn(λp + q

λ )−n?rIDWaldrq p 6= q3 λ = qp , λp + q

λ = 1,r\ (

qp

)ξn

?rI> 3.1.3 (Doob r) & ξ ? (Ω, F , P) e^Æ& Fn : n ∈ I ? F wbAU n |'w^ σ- wnξn := E(ξ|Fn),F* ξn ?bAU (Fn) wr+mz& I ?z3* 0, 1, 2, · · · , pL;$h>`%J4Rwn^?Ew& (Ω, F , P) ?8j* (Fn : n ∈ I) ?bA^ÆI Hn : n ≥ 0 D?eÆw` H0 ? F0 e9ww l n ≥ 1, Hn ?

Fn−1 e9w& X ?waG Hn ?eÆaGm(H•X)0 := H0X0,

(H•X)n := (H•X)n−1 + Hn(Xn − Xn−1), n ≥ 1.D?aG H U X w^Æ$a?b^Æ$wzA: ( ,h*F ^Æ$S# E1:F3b.)

gu nhr 29< 3.1.1 & X ?bAwaG H ?eÆaG` X ?rF*aG H•X ?r` X ?"rw H #3F* H•X ?"rJ% (H•X)n ?ew+w Fn e9ww n ≥ 1,

E[(H•X)n − (H•X)n−1|Fn−1] = E(Hn(Xn − Xn−1)|Fn−1)

= HnE(Xn − Xn−1|Fn−1),` X ?r%#? H•X ?r` X ?"rw H #3%#^#3 H•X ?"r6w?^Æ$H#>Fw>`oÆNu1b?/ Z'_~w8LZU90?wwi^ (kz?A 0ALZ^nA [9] 30 _H_tw Feller w~& Xn ? n ,v A w_\% Xn − Xn−1 ? A n ,HLzwRCbA B A wkHn ?F^^R? B w8p B ^.eK8"bT A wLz Hn KDW X0, X1, · · · , Xn−1 w>` Hn ? Fn−1e9w (^0Akm?AeÆ^GwD.eÆ) >Q B wk.eK A Ef^.eKE A3*q?,?bK+m81tq3w8Pa?8 HhLw8P9b 3.1.2 bA= I w^Æ& T D? (, (Fn : n ∈ I) w) q3` l n ∈ I, T = n ∈ Fn.~wq3?HH3` A ? Borel m T ?I Xn : n ∈ I Ht A w3* T := infn ∈ I : Xn ∈ A, F* T ?q3?T = n = Xn ∈ A ∩ Xn−1 6∈ A ∩ · · · ∩ X0 6∈ A ∈ Fn.r Fn U n |'S T ?q3x% l n ∈ I, T ≤ n ∈ Fn. ` T S ?q3F* T ∧ S ≤ n = T ≤ n ∪ S ≤ n, S"6t< 3.1.1 ` T S ?q3F* T ∧ S T ∨ S ^?q3^ÆI Xn : n ∈ I,_z n T = nm XT := Xn, n ∈ I,D X q3 T Vw Dm T - q?I

XTn (ω) := Xn∧T (ω), n ≥ 0,

gu nhr 305aei:XT

n =

n−1∑

k=0

Xk1T=k + Xn1T≥n

=

n−1∑

k=0

Xk(1T≥k − 1T≥k+1) + Xn1T≥n

= X0 +

n∑

k=1

1T≥k(Xk − Xk−1). T ≥ n = T < nc ∈ Fn−1, w 3.1.1 vtq?< 3.1.2 (Doob) (1) ` Xn : n ∈ I ?r T ?q3F*rwq?IXT

n : n ∈ I ^?rGb0` T ?wF* EXT = EX0. (2) & X ?"r S, T ?q3w S ≤ T , % XTn − XS

n : n ∈ I ?"rr\ EXSn ≤ EXT

n .JE69 (2). 6 Y Tn w)b:kn/ S ≤ T ,

Y Tn − Y S

n =

n∑

k=1

1T≥k\S≥k(Yk − Yk−1).|Q> H (1) wA> r>D Doob q??#>wpj?q3b.?w_iSP4*yp" EXT = EX0 E?#>kmwjH#VA^U9> b?.w> 3.1.4 & Xn : n ≥ 0 ?:+ 0 Qww+mD^Æa?rm T ? 1 wHH3F* XT = 1, _i EXT = 1 6= 0 = EX0.p"6wU9 ^Æwyp" T weCK6x:E< 3.1.3 (Wald) & ξn : n ≥ 1?es$/^ÆIT ?eq3% E∑T

n=1 ξn = ET · Eξ1.J& µ = Eξ1. m Xn :=∑n

i=1(ξi − µ), F* Xn : n ≥ 1 ?r Doobq3 l n, EXT∧n = 0. r\` T > E"6169q T e3 XT e=5E

T∧n∑

i=1

|ξi| = E(T ∧ n) · E|ξ1|,

gu nhr 31r\mFvE|XT | ≤ E

T∑

i=1

|ξi| = limn→∞

E(T ∧ n) · E|ξ1| = ET · E|ξ1| < ∞. E(XT∧n) = E(XT ; T ≤ n) + E(Xn; T > n). H#kEF8#b1 limn E(XT ; T ≤ n) = EXT . b 6 E(|Xn|; T > n) ≤ E(∑T

i=1 |ξi|; T >

n), r ∑Ti=1 |ξi| eSkEF|Q E(|Xn|; T > n) −→ 0. r\

E(∑T

i=1 ξn − Tµ) = EXT = 0."61269rqfwFCFQ? Doob w`.x:CN , .! I = n ∈ Z : 0 ≤ n ≤ N, X = Xn : n ∈ I ?5=w^ÆI−∞ < a < b < ∞, m

τ1 := infn ≥ 0 : Xn ≤ a;

τ2 := infn ≥ τ1 : Xn ≥ b;

· · · · · ·

τ2k+1 := infn ≥ τ2k : Xn ≤ a;

τ2k+2 := infn ≥ τ2k+1 : Xn ≥ b;

· · · · · ·Xzb?$&jw"? +∞,% τn : n ≥ 1?bA|'wq3IeiaQ τn+1 ≥ n, q n > N 3 τn+1 ≡ +∞. mUX

I [a, b] := maxk : τ2k < +∞,^Æ& UXI [a, b] ^ÆI X 3* I J^ a "oB b w`R"6?Q:w Doob `.x:< 3.1.4 (Doob) ` X ?bA"r%

EUXI [a, b] ≤ 1

b − a[E(XN − a)+ − E(X0 − a)+].J Yn := (Xn−a)+, Jensen.x:Y = (Yn)^?bA"r τ1, τ2, · · ·?2 0, b−a, Y $*i a, b, X vmwq3_ UX

I [a, b] = UYI [0, b−a]. k 8 2k − 1 ≥ N , F* τ2k ≥ 2k − 1 ≥ N , r\

YN − Y0 =

∞∑

n=1

(Yτn∧N − Yτn−1∧N )

gu nhr 32

=∑

n≥1

(Yτ2n∧N − Yτ2n−1∧N ) +

k−1∑

n≥0

(Yτ2n+1∧N − Yτ2n∧N )

≥ (b − a)UYI [0, b − a] +

∑

n≥0

(Yτ2n+1∧N − Yτ2n∧N ),dHwk?*kh N A 3.1.2, EYτ2n+1∧N ≥ EYτ2n∧N , r\EYN − EY0 ≥ (b − a)EUX

I [a, b] +

k−1∑

n=0

(EYτ2n+1∧N − EYτ2n∧N )

≥ (b − a)EUXI [a, b].

Doob `.x:?69_wr"rFw FX< 3.1.5 & Xn : n ≥ 0 ?"rw K = supn E|Xn| < ∞. % Xn → X a.s.,dH X ?bAe^Æ&~ Xn ?bAbCer% XnL1

−→X wXn = E(X |Fn).J& X∗, X∗ $*? n → +∞ 3 Xn w* "*%

X∗ > X∗ =⋃

a,b∈Q

X∗ < a < b < X∗.#?e+`.x:EUX

0,N [a, b] ≤ 1

b − a(E|XN | + a) ≤ K + a

b − a.mF E limN UX

0,N [a, b] < +∞. r\ limN UX0,N [a, b] < +∞ a.s.

X∗ < a < b < X∗ ⊂ limN

UX0,N [a, b] = +∞,S P(X∗ < a < b < X∗) = 0, ee"C|Q X∗ = X∗ a.s. *weC Fatou tvt` Xn ?bCe% 2.4.6[9], Xn

L1

−→X wXn = limm E(Xm|Fn) = E(X |Fn).6f w?r"r3Fwj n −→ +∞ 3wF"6U9"r3iwFjv n −→ −∞ 3wF< 3.1.6 & X = (Xn : n ≤ 0)?U (Fn : n ≤ 0)w"rw infn EXn > −∞.% X ?bCewwq n → −∞ 3 Xn wVVw L1 FbAe^Æ& X−∞.

gu nhr 33J& n ≤ 0, r EXn ≥ EXn−1, S infn EXn > −∞ [ x = limn→−∞ EXna w*C ǫ > 0, k 8v EXk − x < ǫ, F*q n ≤ k 3E(|Xn| : |Xn| > λ) = E(Xn : Xn > λ) − E(Xn : Xn < −λ)

= E(Xn : Xn > λ) + E(Xn : Xn ≥ −λ) − EXn

≤ E(Xk : Xn > λ) + E(Xk : Xn ≥ −λ) − EXk + ǫ

≤ E(Xk : Xn > λ) + E(−Xk : Xn < −λ) + ǫ

≤ E(|Xk| : |Xn| > λ) + ǫ,~P(|Xn| > λ) ≤ 1

λE|Xn| =

1

λE(2X+

n − Xn)

=1

λ(2EX+

n − EXn) ≤ 1

λ(2EX+

0 − x),\|Q X ?bCew(2) yX 3.1.5 w69& X∗, X∗ $*?q n → −∞ 3 Xn w* "*%

X∗ > X∗ =⋃

a,b∈Q

X∗ < a < b < X∗.34R N , 2`.x:"r XN , XN+1, · · · , X0,

EUXN,0[a, b] ≤ 1

b − a(E|X0| + a).r\s[ limN→−∞ UX

N,0[a, b] < +∞ a.s. p?X∗ < a < b < X∗ ⊂ lim

N→−∞UX

N,0[a, b] = +∞,S P(X∗ < a < b < X∗) = 0, |Q X∗ = X∗ a.s. limn→−∞ Xn(ω) wVVa * X−∞. Xn : n ≥ 0 wbCeC Xn ^? L1 FX−∞, r\ X−∞ ?ewk3>(rwbAw1u69 KolmogorovteRa|Z Borel teR (7b 2.4.4[9]).

gu nhr 34< 3.1.7 (Kolmogorov) & ξn ?s$/e^ÆI%1

n(ξ1 + ξ2 + · · · + ξn)wVVw L1 F Eξ1.J n ≥ 1 m X−n := 1

n (ξ1 + ξ2 + · · · + ξn), F−n := σ(X−k : k ≥ n). F*(X−n, F−n : n ≥ 1) ?r5 l n ≥ 1, n/ E(ξk|

∑ni=1 ξi) U

1 ≤ k ≤ n ?.&wr\E(ξ1|

n∑

i=1

ξi) =1

n

n∑

i=1

ξi = X−n.^Æ&e ∑ki=1 ξi : k ≥ n ∑n

i=1 ξi, ξn+1, ξn+2, · · · ) eiy,+C);S F−n = σ(∑n

i=1 ξi, ξn+1, ξn+2, · · · ), Cw 2.3.2(6)[9] 8uE(X−1|F−n) = E(ξ1|F−n) = E(ξ1|

n∑

i=1

ξi) = X−n, X−n, F−n : n ≥ 1 ?reCk6w 3.1.6 |Q X−n a.s. w L1 FbAe^Æ&X . "669 X = EX a.s. .!& Eξ1 = 0, % EX = 0. φ ); ξ1 wh3eRF* φ 0 ew φ′(0) = 0. r\

EeitX = limn

EeitX−n = limn

(

φ(t

n)

)n

= 1. X = 0 a.s."6B"wr.x:^?L~ww#>< 3.1.8 (Doob) & Xn : n ≥ 0 ?r(1) l λ > 0 54R N ,

λP( max0≤n≤N

|Xn| ≥ λ) ≤ E|XN |.

(2) l p > 1 54R N ,

E max0≤n≤N

|Xn|p ≤(

p

p − 1

)p

E|XN |p.h*w p = 2 ?h>w=nE max

0≤n≤NX2

n ≤ 4EX2N .

gu nhr 35Jr |X | 5?#3"r_i"6.!& X ?#3"r(1) τ := min0 ≤ n ≤ N : Xn ≥ λ, % τ ?bAq3w τ ≤ N , S

EXN ≥ EXτ

= E(Xτ ; max0≤n≤N

Xn ≥ λ) + E(Xτ ; max0≤n≤N

Xn < λ)

≥ λP( max0≤n≤N

Xn ≥ λ) + E(XN ; max0≤n≤N

Xn < λ),r\λP( max

0≤n≤NXn ≥ λ) ≤ E(XN ; max

0≤n≤NXn ≥ λ) ≤ EXN .

(2) ξ := XN , η := maxn≤N Xn, q := pp−1 . % (1) w69H8u tP(η ≥

t) ≤ E(ξ; η ≥ t), >n Fubini k Holder .x:Eηp = E

∫ η

0

ptp−1dt =

∫ ∞

0

ptp−1P(η ≥ t)dt

≤∫ ∞

0

ptp−2E(ξ; η ≥ t)dt

≤ pEξ

∫ η

0

tp−2dt =p

p − 1Eξηp−1

≤ q(Eξp)1p (Eη(p−1)q)

1q

= q(Eξp)1p (Eηp)

1q .#sS (Eηp)

1q v"6A^%;r ?L~j3ww> 3.1.5 e"ul"61tr6f w+m^Æ D. a ∈ Z,

a > 0, T := T0 ∧ Ta, Hra 0 a dbw3* 0, a wG3^b=w> 8 l x ∈ Z, 0 ≤ x ≤ a, Px(T0 < ∞) P

x(Ta < ∞) B!bA? 1, S Px(T < ∞) = 1. % T < ∞ = T0 < Ta ∪ T0 > Ta, 1

qx := Px(T0 < Ta),^ x Qw^Ætb 0 tb a 9rw8A4z0,q

A, B Bh x, a−x +|7"bA+m^ÆA:w, BdHLY_w|3>Qr\ qx r>^DLY8 T ? HJ3* 0D?bAXF! 0, a w+m^Æ

gu nhr 36` p = q = 12 , F* ξn ?bArF* ξT

n ^?rSE

xξn∧T = Exξ0 = x,

Exξn∧T = E

x(ξn∧T ; T ≥ n) + Ex(ξn∧T ; T < n)

= Ex(ξn; T ≥ n) + E

x(ξT ; T < n).q n ≥ T 3 ξn ≤ a, SE

x(ξn; T ≥ n) ≤ aPx(T ≥ n) −→ 0,mF

Ex(ξT ; T < n) ↑ E

x(ξT ; T < ∞) = ExξT .

ExξT = E

x(ξT ; T0 < Ta) + Ex(ξT ; T0 > Ta) = aP

x(T0 > Ta),r\qx = 1 − P(T0 > Ta) = 1 − x

a=

a − x

a.` p 6= q, .!& p > q, F* ( q

p )ξn ?bAwr^ ( qp )ξn∧T ^?yXzmF1

(q

p)x = E

x(q

p)ξn∧T

= Ex[(

q

p)ξn ; n ≤ T ] + E

x[(q

p)ξT ; n > T ].w

Ex[(

q

p)ξn ; n ≤ T ] ≤ P

x(n ≤ T ) −→ 0,

Ex[(

q

p)ξT ; n > T ] ↑ E

x(q

p)ξT

= Ex[(

q

p)ξT ; T0 < Ta] + E

x[(q

p)ξT ; T0 > Ta]

= qx + (q

p)a · Px(T0 > Ta)

gu nhr 37

= qx[1 − (q

p)a] + (

q

p)a,r\

qx =( q

p )x − ( qp )a

1 − ( qp )a

.> 3.1.6 _/\HJ3* T wAeR1EbA T wr^ 3.1.2 8u l λ > 0,

λξn(λp +q

λ)−n?bArq p ≥ q 3 λ 6∈ ( q

p , 1), |λp + qλ | ≥ 1. ~dq?I?rwCFv

λx = Exλξn∧T (λp +

q

λ)−n∧T

= Ex[λξn∧T (λp +

q

λ)−n∧T ; n ≤ T ] + E

x[λξn∧T (λp +q

λ)−n∧T ; n > T ]

Ex[λξn∧T (λp +

q

λ)−n∧T ; n ≤ T ] ≤ λa

Px(n ≤ T ) −→ 0,r\1

λx = ExλξT (λp +

q

λ)−T

= Ex[(λp +

q

λ)−T ; T0 < Ta] + λa

Ex[(λp +

q

λ)−T ; T0 > Ta]. µ = q

λp , F* e = µp + qµ . i

(q

µp)x = E

x[(µp +q

µ)−T ; T0 < Ta] + (

q

µp)a

Ex[(µp +

q

µ)−T ; T0 > Ta],2 µ : λ vA G

λx = Ex[(λp + q

λ)−T ; T0 < Ta] + λaE

x[(λp + qλ)−T ; T0 > Ta],

( qλp )x = E

x[(λp + qλ )−T ; T0 < Ta] + ( q

λp )aE

x[(λp + qλ )−T ; T0 > Ta].?Q

Ex(λp +

q

λ)−T = E

x[(λp +q

λ)−T ; T0 < Ta] + E

x[(λp +q

λ)−T ; T0 > Ta]

gu nhr 38

=λx−a( q

p )a − λx − λa−x( qp )x + λ−x( q

p )x

( qp )aλ−a − λa

. |t| ≤ 1, e = 1t , F* λ AD

λ1(t) =1 +

√

1 − 4t2pq

2pt, λ2(t) =

1 −√

1 − 4t2pq

2pt.% λ1 = q

λ2p , r\E

xtT =λ2(t)

x(λ1(t)a − 1) − λ1(t)

x(λ2(t)a − 1)

λ1(t)a − λ2(t)a.1au\ T wRL

Dx := ExT = lim

t↑1

1 − ExtT

1 − t.& p > q, F* t ↑ 1 3 λ1 −→ 1, λ2 −→ q

p , w t = λ1

λ21p+q

, r\Dx = lim

t↑1

1 − λx2 (λa

1−1)−λx1 (λa

2−1)λa1−λa

2

1 − λ1

λ21p+q

= limt↑1

λx1(λa−x

1 − 1) + λa2(λx

1 − 1) + λx2(1 − λa

1)

(λ1p − q)(λ1 − 1)(λa1 − λa

2)(λ2

1p + q)

=−(a − x) + x( q

p )a + a( qp )x

(p − q)(1 − ( qp )a)

=a

p − q·1 − ( q

p )x

1 − ( qp )a

− x

p − q.` p = q = 1

2 , F* λ2 = λ−11 , r\ λ ); λ2 ( λ1), 1v

ExtT =

λx(λ−a − 1) − λ−x(λa − 1)

λ−a − λa

=λx(1 − λa) − λa−x(λa − 1)

1 − λ2a

=λx + λa−x

1 + λa.yXzHJ3*w

Dx = limt↑1

1 − λx+λa−x

1+λa

1 − 2λ1+λ2

gu nhr 39

= limt↑1

(1 − λx)(1 − λa−x)

(1 − λ)2· 1 + λ2

1 + λa= x(a − x).

§3.2 |TY-=M =whv12B"r C Hwwrw CB=h*? % ww?\w_kZ8vCRLwSPEHNuwbA wY9 Scholes, Merton v Nobel LL3 (bPY9 Black \3ga<).1 \bw?+mCB= 1 %p?dV/e_BzwJ3*wC H#uaR[B=18z3* 0, 1, 2, · · · , n, Sk );R[ k 3fw%??bA^Æ& k + 1 3f%?$*i8 p, q +B uSk k"B dSk, q u, d ?>R(d 0 < d < 1 < u, p, q > 0 wp + q = 1. vzU ξ1, · · · , ξn ?8j* (Ω, F , P) ws$/^Æ&w$/

P(ξk = u) = p, P(ξk = d) = q.

S0 ?bA5>R Sk := S0 · ξ1 · · · ξk, k = 1, 2, · · · , n. aG S0, S1, · · · , Sn7O R[%?w&D?bAR[B=a5>R u, d, p, q dH+ "w.CU9R[B=?)&wB= lei&'R[ k 3f&bRR[ "b3f'Q~ Sk+1 − Sk, 9 k 3f'bRR[ "b3f&~ Sk − Sk+1. TkAReK?3w Fk ); ξ1, · · · , ξk*Ew=/1 ≤ k ≤ n, F0 := ∅, Ω. r Sk ξkeiy,);S Fk = σ(S0, S1, · · · , Sk), 03 l Fk e9w^Æ& X ei); ξ1, · · · , ξk weR>+mz:EX = X(ξ1, · · · , ξk).& P ? (Ω, F ) ~bA8` l n A ∈ Fn, P(A) > 0 qwFq

P(A) > 0, 1U P x% P.$&a bAB=1DB=R?U leiiTw~ r (3sB) an (^sB) jnasB (wzjQ) w 1 m

gu nhr 40 w "b3fei (wz\) 1 + r m B=?.a )&wa>R r 0[67Ow)&wR[B=k)&wB=gE 1_Æw+mCB=anjn&'R[?B=wr>u\Bq0?bAd+w=.diYq15wR[B=pu dHwV/?(fw15wB=ik' lei 0AB=ra&'R[iajnubtVqwCww~~C?>`bih6wi%Vhe~bAu\8?>_r AB=wu\^R? .A3f k ~wR[R ∆k, 0 ≤ k ≤ n − 1, +-\CasB.O33sBjn 0A3fu\_w=? 0 t k 3fR[w%? >S ∆k ? Fk e9w/U? ξ1, · · · , ξk weR 3.2.1 ^ÆaG ∆ = ∆0, ∆1, · · · , ∆n−1 D?bAu\8 (portfolio),` ∆k ? Fk e9w 0 ≤ k ≤ n − 1. bAu\8?eGw` l ∆k1UbAu\?_\ :w`SP9u\ X0 ~ v6w6jaGH.^B=~v\C^.^HNQ\C Xk );u\ k 3f~w54 ("R[%=), %X1 = ∆0S1 + (1 + r)(X0 − ∆0S0),

X2 = ∆1S2 + (1 + r)(X1 − ∆1S1),

· · · · · · · · · · · ·

Xk+1 = ∆kSk+1 + (1 + r)(Xk − ∆kSk),

· · · · · · · · · · · ·aG X = Xk : 0 ≤ k ≤ n D?P9u\ X0, u\8 ∆ w_\ :"w54aG+D54aG6)b:ei:E(1 + r)−k−1Xk+1 − (1 + r)−kXk = ∆k((1 + r)−k−1Sk+1 − (1 + r)−kSk).r\.' w54aG?u\8U.' wR[%?w^Æ$1UbAB=g~Æ?>a bAu\88ei)&z ( R[B= l&) ^b_Vtq/1UB=,g~ÆU`bAsBwan~=bAsBwjn~F*0|Ra g~

gu nhr 41Æ ($&.b6jE), rLei^vbAsBjnarbAsBv)&w~;nw$&?bAENwB=,g~Æ5B=3wg~Æp1w~~C8B=w lg~ÆPZ5/' 1Cg~0A8PbAR?wm 3.2.2 bAB=a g~?>a bAeGwu\8 ∆ 8vdP9u\X0 = 0, _\ :"w54aG X (d Xn ≥ 0 w P(Xn > 0) > 0.H#169$&B=g~%

d < 1 + r < u.=5` 1 + r ≥ u, ^R?U~a=F*1ei#'Q (&j) 1 RR[v\C S0,asB "b3fQan&R[~ (1+r)S0−S1,r 1 + r ≥ u > d, _i\^Æ&?#3wwi58e 0, r\B=a g~s[1ei69`~ay 1 + r ≤ d, F*1eijn&R[v)&FnSB=^g~ÆCB=Ha ewX*6^DC<a1wa ez CB=X*6? 5R[%?wn1i uU9R: ?> w3feiiw%? (.W\3fR[w15%?) uN&R[wb&n ?A~p.?m (.4 wn F<B). %bA 3f m i%? K N&R[wR: 3f m w%=Vm := (Sm − K)+,^R?U`tw3f m, R[%? Sm =%? K, F*<Bni% K N&B=% Sm wR[~ Sm − K, `\3R[%? Sm y%? K 8vn~ezF*K$"l<Bn~ 0. Vm ? SmweR? Fm e9w^Æ&B=g~|Q .K?w %jR?0A~ 3f 0 %=lw6'!q U9jwF1& m = 1. Tkt wN&/wÆ? _R[%? >wU^R?UN& vwFneiraB=r>u\Bv031U ei0E (replicate), Q' w^ÆQ' w)&eiraB=r>u\Bu["031U eiM

(hedge). 1R 0AV/u 0 3fw%= V0.

gu nhr 42$&\C V0, 1K GB= (1) N& (2) r>u\N& v 3f 1 w%= V1, `r>u\1EN& (') !R[v-w\CasB (.Ow~^sBjn), $&N& ∆0 R (\R3k['QR[), l2\C V0 − ∆0S0 asB (\R3k[3sBjn). F* V0, ∆0 ?lwr>u\ : 3f 1 w%=∆0S1 + (1 + r)(V0 − ∆0S0),b1$?R[Fn1$~e!,=qK ∆0 ei8v ~ r>u\~bC

V1 = ∆0S1 + (1 + r)(V0 − ∆0S0).Tk V1, S1 ?^Æ&_i\:::?Q ∆0 ?,wr 3f 0 N&R[Rw v3f 1 wR[%??e7wp?1ei:$KypbR[+/: V u1 := V1|ξ1=u = (uS0 − K)+, V d

1 := V1|ξ1=d =

(dS0 − K)+, F*vtV u

1 = ∆0uS0 + (1 + r)(V0 − ∆0S0), (3.2.1)

V d1 = ∆0dS0 + (1 + r)(V0 − ∆0S0), (3.2.2),,v

V u1 − V d

1 = ∆0S0(u − d), ∆0 =V u

1 − V d1

uS0 − dS0.i G (3.2.1), ?v V0

V0 =1

1 + r

(

1 + r − d

u − dV u

1 +u − (1 + r)

u − dV d

1

)

.s31^,t M0A v8\C V0 `9N& ∆0 RR[2-w\CasB~ei69` w%?.? V0 F*bg~Æa =5&B=eii%? W0 N&0[b& I%ayW0 < V0, %ei^b_'Q ∆0 RR[_vw\C ∆0S0 N&b& duB= "b3f' ∆0 RF*w~

V1 + (1 + r)(∆0S0 − W0) − ∆0S1 = (1 + r)(V0 − W0) > 0,

gu nhr 43B=g~Æs[ei69 W0 > V0,B=^a g~Æ R?UbAg~wB=H V1 3f 0 w%?b?6w V0.:tJfw? V0 F v u, d, r, K, S0, B=HhY<wk'B=)&w8 p, q Ur\1U V0 ?)&HCw' p := 1+r−du−d , q :=

u−(1+r)u−d , % p + q = 1, B=g~w$&" p, q > 0. r\ \$&" j* (Ω, Fn) a 8 P 8v ξ1, · · · , ξn ?s$/wp?$/

P(ξ1 = u) = p, P(ξ1 = d) = q,F*1V0 =

1

1 + rEV1,R 1

1+r ?54w.'B=w)&~"bA3fm 1w54 '3fw%= 11+r . 0[:ei:Vz?A ;w8" '3fw%=x"b3fd %=w.'8 P^DB=w)&HC8?bA#>w85?a w%4*I< 3.2.1 8 P "La.'wR[%? (1 + r)−kSk : k = 0, 1, 2, · · · , n?bArJ69?+mwr ξk+1 Fk S

E((1 + r)−k−1Sk+1|Fk) = (1 + r)−kSkE(1

1 + rξk+1)

= (1 + r)−kSk · 1

1 + r

(

1 + r − d

u − du +

u − (1 + r)

u − dd

)

= (1 + r)−kSk.E 69DW 3.1.1, 1f"6w> < 3.2.2 bAeGu\8w_\ :"w.'vw54aG (1+r)−kXk :

0 ≤ k ≤ n )&HC89"?bAr"6169 3.2.1 wM^< 3.2.3 B=g~qwFqa bAx% P w8 P 8v.'vwR[%? (1 + r)−kSk P 9"?bAr

gu nhr 44JE6L$C`.'vwR[%? P 9"?r%d l_\ :"w54aG X ?rr\ EXn = (1 + r)nEX0. P9u\ X0 = 0,% EXn = 0, r\ Xn ≥ 0 P(Xn > 0) = 0. r P P x%S|Q

P(Xn > 0) = 0. r\B=g~ 3.2.3 bA 3f m t w+mR:X*6?>bA Fm e9w^Æ&bA 3f m t w+mR:X*6 Vm DeD (/U0E) w?>a bA>R V0 bAu\8 ∆ 8d_\ :"w54aG X (dX0 = V0, Xm = Vm. bAB=?w?> l+mR:X*6eM6_f wR: ?~w+mR:X*61wCw?69 l+mR:X*6eMw1eiXkM#QMw 1H#$& Vm ?bAeMw+mR:X*6ma bA>RV0 bAu\8 ∆ 8d_\ :"w54aG X (d X0 = V0, Xm = Vm.&B=g~F* )&HCw8 P 9" (1 + r)−kXk ?r

(1 + r)−kXk = E((1 + r)−mXm|Fk)

= E((1 + r)−mVm|Fk),

Xk = (1 + r)kE((1 + r)−mVm|Fk)._\CU9

Xk+1 − (1 + r)Xk = ∆k(Sk+1 − (1 + r)Sk).EvzXk+1(ξ1, · · · , ξk, ξk+1) − (1 + r)Xk(ξ1, · · · , ξk)

= ∆k(ξ1, · · · , ξk)(Skξk+1 − (1 + r)Sk), ξk+1 $*= u, d, %thXu

k+1 := Xk+1|ξk+1=u = Xk+1(ξ1, ξ2, · · · , ξk, u),

Xdk+1 := Xk+1|ξk+1=d = Xk+1(ξ1, ξ2, · · · , ξk, d),a1? Fk e9wF*1vXu

k+1 − (1 + r)Xk = ∆k(uSk − (1 + r)Sk),

gu nhr 45

Xdk+1 − (1 + r)Xk = ∆k(dSk − (1 + r)Sk),?Q

∆k =Xu

k+1 − Xdk+1

(u − d)Sk.069 Vm eM%P9u\ V0 = (1 + r)−m

EVm, u\8 ∆ 6wf R:+mX*6 6iH bAM#MwV& Vm ?bAR:+mX*6#mVk := (1 + r)k

E((1 + r)−mVm|Fk), k = 0, 1, · · · , m − 1.q Vk ? ξ1, · · · , ξk weRyXzV u

k+1 := Vk+1|ξk+1=u = Vk+1(ξ1, ξ2, · · · , ξk, u),

V dk+1 := Vk+1|ξk+1=d = Vk+1(ξ1, ξ2, · · · , ξk, d),m∆k :=

V uk+1 − V d

k+1

(u − d)Sk, k = 0, 1, · · · , m − 1. (3.2.3)% ∆ = ∆0, ∆1, · · · , ∆m−1 ?bAu\8"6169P9u\ V0 u\8 ∆ w_\ :"w54aG X0, X1, · · · , Xm (d

Xk = Vk, k = 0, 1, 2, · · · , m,nfM Vm.=5\G% V0 = X0. & Xk = Vk, 169 Xk+1 = Vk+1. H#r (1 + r)−kVk ? P- rSVk =

1

1 + rE(Vk+1|Fk) =

1

1 + r(pV u

k+1 + qV dk+1),54aGwm

Xk+1 = (1 + r)Xk + ∆k(Sk+1 − (1 + r)Sk)

= (1 + r)Vk +V u

k+1 − V dk+1

(u − d)Sk(Skξk+1 − (1 + r)Sk)

gu nhr 46

= pV uk+1 + qV d

k+1 +V u

k+1 − V dk+1

(u − d)(ξk+1 − (1 + r))

= (p +ξk+1 − (1 + r)

u − d)V u

k+1 + (q − ξk+1 − (1 + r)

u − d)V d

k+1

=ξk+1 − d

u − dV u

k+1 +u − ξk+1

u − dV d

k+1

= 1ξk+1=uVuk+1 + 1ξk+1=dV

dk+1 = Vk+1.0[169 "6w< 3.2.4 bAg~B=?wwbA m 3ft w+mR:X*6

Vm 0 3fw%=?V0 =

1

(1 + r)mEVm,aeiiP9u\ V0 u\8 ∆ w_\ :uMdH ∆k (3.2.3)mJ1E69` Vm 0 3fwI%.? V0, F*B=Rg~Æ$&I% W0 > V0, %'Qb&X*6v\C W0, vdHw V0 `9iCQwu\8F* B=l&hJbeiM'QwX*6r\vb&)&wFnW0 −V0 > 0, SB=g~yXei69qI% W0 < V0 3B=^g~ÆE6961+mzB" zCB=w ?w/e7b,U(w6Tkw?0?CB=wbK$&C=aeiS?ACB=w9'4Yz15wCB=ww;vqx l

1. bAk^H 3f 0bAt| bA |^Æz^k^HbA|v2a"+"bA,sVw|*zL0\aGXn n vkH |R b|R969 Xn ?r2. 69_=w_ er Xn ?56'aG l n1 < n2 ≤

n3 < n4 E(Xn4 − Xn3)(Xn2 − Xn1) = 0.

3. & Xn : n ≥ 0 ? (Fn) wwe^ÆI(dE(Xn+1|Fn) = αXn + βXn−1, n ≥ 1,

gu nhr 47dH α > 0, β > 0, α+β = 1. al=3I Y0 := X0, Yn := aXn+Xn−1? (Fn) r4. & Yn ?s$/5^Æ&I8v EYn = 1. Xn := Y1Y2 · · ·Yn.

(a) 69 (Xn) ?rwwVVFbA^Æ& X ;

(b) & Yn i8 12 $*= 1

2 32 . Z6 X = 0 a.s. r\ E

∏

n≥1 Yn 6=∏

n≥1 EYn.

5. & Xn : n ≥ 1 ?bA_= 0 w_ ewrZ6 E[(Xn+k − Xn)2] =∑k

i=1 E[(Xn+i − Xn+i−1)2]. 69` ∑

n E[(Xn+1 − Xn)2] < ∞, F* Xni8 1 F6. & Xn ?_ erw supn EX2

n < ∞, 69 Xn _ F7. 69 Wald & ξn ?s$/^ÆI Eξn = 0, Eξ2

n = 1, Xn :=

∑ni=1 ξi, T ? Xn weq3% EX2

T = ET .

8. & Xn ?rw |X0(ω)|, |Xn(ω)− Xn−1(ω)| bA ω n Uw>RkE` τ ?bAX*_=wq369 Xτ ew EXτ = EX0.

9. & Xn : n ≥ 1 ?rm ξn = Xn − Xn−1, ξ1 = X1, 69 DXn =∑n

i=1 Dξi.

10. 69HJ3*wAeR ExtT ? t weR

11. ::z69LY8 qx (d;$ G

qx = pqx+1 + qqx−1, 0 < x < a

q0 = 1, qa = 0.

12. ::z69HJ3* T w Dx (d;$ G

Dx = 1 + pDx+1 + qDx−1, 0 < x < a,

D0 = Da = 0.

13. 69R[%?aG Sn ?bA Markov aG14. & u = 2, d = 1

2 , S0 = 4, r = 14 . b&R: (S2 − 5)+ 0 3f%=!b&+mR:X*6 max0≤k≤2(Sk − 5)+ %=!

h j)$ §4.1 :=H#B"9M# %1L08j*wm 4.1.1 & Ω ?#jnbA^y F D? Ω wbA σ- ( σ- iR=/x), `

(1) ∅, Ω ∈ F ;

(2) A ∈ F Ac ∈ F ;

(3) An ∈ F , n ≥ 1 ⋃

n An ∈ F .D (Ω, F ) ?bAe9j* F w#3eR P D?9`(1) P(∅) = 0;

(2) ly.,w An : n ≥ 1 ⊂ F , P(⋃

n≥1

An) =∑

n≥1

P(An).03 (Ω, F , P) D?bA9j*Gb0` P(Ω) = 1, F*1U P ?89+D889#?bKhKw9gr6q89HO+69w>`+,t P(Ω) = 1, r\a1bw9^?wEw3u12h*z>QQ~8_$w8j*r>?+mw1)Bei[j*VQupbw8j*?,q0wVd σ- k89w?.eKw0|sC0wPr?e\k9wee"Cw_i1#tbA+mw^ybA^y A D? Ω w`a jkΩ, -\'w*+\' σ- wma;*+.ebA*+'bAe+'p5w>M σ- w>M+mGr\1wV? M#bA1_ w89vt*t*Ew σ- 0FR?5&eRhGHB"w_Æ Caratheodoryw9M# 9^t*t σ- wbAu?~9~9.?9p?aG

48

gw xmlefk~ 49 4.1.2 m Ω w5 (/_^) w#3eR µ D? Ω wbA~9`(1) mC l A, B ⊂ Ω, ` A ⊂ B, F* µ(A) ≤ µ(B);

(2) ee" l An ⊂ Ω, n ≥ 1, µ(⋃

n≥1 An) ≤∑n≥1 µ(An).~9wbA?aqj<*"6tU9 lbA^yw#3eReisbA~9< 4.1.1 & A ? Ω wbA^y µ ? A w#3eR l A ⊂ Ω,mµ∗(A) := inf

∑

n≥1

µ(An) : An ⊂ A ,⋃

n≥1

An ⊃ A

, inf ∅ := ∞,% µ∗ ?bA~9D? (µ, A ) sw~9J#3C?%wmCu_"wmU.?69ee"C kC^ Bn, 69 µ∗(⋃

n≥1 Bn) ≤∑n≥1 µ∗(Bn). `#bA?F*.x:%E_i1.!&#.bA?*w l ǫ > 0, a An,k ∈ A 8v ⋃

k≥1 An,k ⊃ Bn w∑

k≥1

µ(An,k) < µ∗(Bn) + ǫ/2n.F* ⋃

n,k An,k ⊃ ⋃n≥1 Bn, r\ ∑

n,k µ(An,k) ≥ µ∗(⋃

n Bn). \|Q∑

n

µ∗(Bn) + ǫ ≥ µ∗(⋃

n

Bn),hv ǫ w kCd[ee"~9wA?ab?eivtbA9& µ∗ ?~9CA ⊂ Ω. ` l E ⊂ Ω

µ∗(E) = µ∗(E ∩ A) + µ∗(E ∩ Ac),F*1U A ? µ∗- e9Tk~9we"C6wxhx%≥ h M ∗ ); µ∗- e9k< 4.1.2 ` µ∗ ?~9F* (Ω, M ∗, µ∗) ?9j*

gw xmlefk~ 50JH#69 M ∗ ? σ- q%a jk Ω +w-\'U.?Z6e+\'#Z6 M ∗ *+\' A, B ∈ M ∗, %E ⊂ Ω,

µ∗(E) = µ∗(A ∩ E) + µ∗(Ac ∩ E)

= µ∗(A ∩ E) + [µ∗(B ∩ Ac ∩ E) + µ∗(Bc ∩ Ac ∩ E)]

= [µ∗(A ∩ (A ∪ B) ∩ E) + µ∗((A ∪ B) ∩ Ac ∩ E)]

+ µ∗((B ∪ A)c ∩ E)

= µ∗((A ∪ B) ∩ E) + µ∗((B ∪ A)c ∩ E),|Q A ∪ B ∈ M ∗. 0d[ M ∗ *6\^'S"61FFZ6M ∗ .,6we+\'& An ? M ∗ H.,6

Bn :=

n⋃

i=1

Ai, A :=

∞⋃

n=1

An,~9wmCvµ∗(E) = µ∗(E ∩ Bn) + µ∗(E ∩ Bc

n)

≥ µ∗(E ∩ Bn) + µ∗(E ∩ Ac)

= µ∗(E ∩ Bn ∩ Bn−1) + µ∗(E ∩ Bn ∩ Bcn−1) + µ∗(E ∩ Ac)

= µ∗(E ∩ Bn−1) + µ∗(E ∩ An) + µ∗(E ∩ Ac)

= · · · · · · =n∑

i=1

µ∗(E ∩ Ai) + µ∗(E ∩ Ac). n w kC µ∗ wee"C|Qµ∗(E) ≥

∞∑

i=1

µ∗(E ∩ Ai) + µ∗(E ∩ Ac)

≥ µ∗(E ∩ A) + µ∗(E ∩ Ac),r\ A ∈ M ∗. .F\^6w69aGHeiaQ l E ⊂ Ω,

µ∗(

E ∩ (

n⋃

i=1

Ai)

)

=

n∑

i=1

µ∗(E ∩ Ai),

gw xmlefk~ 51S|Q µ∗ *e"Cµ∗(

n⋃

i=1

Ai

)

=n∑

i=1

µ∗(Ai).>n µ∗ wee"C|Q µ∗ M ∗ ee"Cr\ µ∗ ? (Ω, M ∗)w9^y A w#3eR µ s~9 µ∗, pbzµ∗ k µ A ?.,swF*jR?4*3ua1?,sw^R?U4*3u µ∗ ? µ wt* ?ee"C 4.1.3 1Dm A w#3eR µ ?e"w`a(d (1)

µ(∅) = 0; (2) lA HyKw A, B µ(A∪B) = µ(A)+µ(B). `a(d l A Hy.,w+ A Hwee"C An : n ≥ 1 ⊂ A.,w ⋃

n An ∈ A , µ(⋃

n

An) =∑

n

µ(An),F*Da?9% σ- w9R?9~bA#3e"eRE9qwFqa ∅ VJ< 4.1.3 & µ ? A w9F* (1) µ∗|A = µ; (2) A ⊂ M ∗.J#69 l A ∈ A , µ∗(A) = µ(A). m µ∗(A) ≤ µ(A) ?%w6.x:b#1& µ∗(A) < ∞, F* l ǫ > 0, a .9 A wAn ⊂ A 8v

µ∗(A) + ǫ ≥∑

n≥1

µ(An) ≥ µ(A).|Q> Tk0t µ A wmCkee"C,t9w1"669 A ⊂ M ∗. & A ∈ A , l E ⊂ Ω, .!& µ∗(E) < ∞, % l ǫ > 0, a ^ An ⊂ A (d ⋃

n An ⊃ E w ∑

n µ(An) < µ∗(E) + ǫ, rµ∗(E ∩ A) + µ∗(E ∩ Ac)

≤ µ∗((⋃

n

An) ∩ A) + µ∗((⋃

n

An) ∩ Ac)

gw xmlefk~ 52

= µ∗(⋃

n

(An ∩ A)) + µ∗(⋃

n

(An ∩ Ac))

≤∑

n

[µ∗(An ∩ A) + µ∗(An ∩ Ac)]

≤∑

n

[µ(An ∩ A) + µ(An ∩ Ac)]

=∑

n

µ(An) < µ∗(E) + ǫ,dHhvwx:u_ µ A w*e"C\|Q µ∗(E∩A)+µ∗(E∩Ac) ≤µ∗(E), A ∈ M ∗.' 1#mt*w8P& A ?bA^yµ ? A bAeR` µ∗ ? σ(A ) w9w µ∗|A = µ, F*1U µ∗ ? µ wbAt*' eiuHO Caratheodaoryw9t*6wtj|QM ∗ ?bA A w σ- r\~9 µ∗ ? µ wbAt*D Caratheodoryt*< 4.1.1 (Caratheodory) & µ ? A w9F* µ bAt*_z1^4*3ut*?bw"6U89wt*b?bw< 4.1.2 ` µ ? A w9w µ(Ω) = 1, F* µ wt*?bwJ` µ At* µ1, µ2, m

F := A ⊂ σ(A ) : µ1(A) = µ2(A).F* F ⊃ A . ~ kZ6 F ? Dynkin y A 6'r\ Dynkin v F ⊃ σ(A ). ^R?U µ1 = µ2.0A?U σ- w9M#k9wM#?x%wp?1v6ktw>M+mGr\9wM#^+mv"61aa [0, 1] w Lebesgue 9wa CwR?69& Ω = [0, 1]. U I ⊂ Ω ?bAi`^*?>a bA R wi`* I ′ 8v I = I ′ ∩ Ω. 0|x% I ? R wi`*/? 0 w* A0 ); Ω wi`^*kA , F $*); A0 *Ewkσ- F* A ?*A.6i`^*+wk (0EZ60[w^y?RdO ), F R? [0, 1] w Borel ' A bA_we

gw xmlefk~ 53RR?m*?wk∣

∣

∣

∣

∣

n⋃

i=1

Ii

∣

∣

∣

∣

∣

:=

n∑

i=1

|Ii|,dH Ii : 1 ≤ i ≤ n ⊂ A0 wy.,6"6wU.?690[mweR?A9., An ⊂ A w ⋃

n An = A ∈ A , 169∑

n |An| = |A|. eiÆ An, A ?* An = (an, bn], A = (a, b]. r l n,⋃n

i=1 Ai ⊂A, S ∑

i≥1 |Ai| ≤ |A| ?%wau ǫ > 0, r ⋃

n(an, bn + ǫ/2n) ⊃[a + ǫ, b], *.9a n 8v ⋃n

i=1(ai, bi + ǫ/2i) ⊃ [a + ǫ, b], r\∞∑

i=1

(bi + ǫ/2i − ai) ≥n∑

i=1

(bi + ǫ/2i − ai) ≥ b − a − ǫ.|Q ∑∞i=1 |Ai| ≥ |A| − 2ǫ, ǫ w kCv8 ∑∞

i=1 |Ai| ≥ |A|.1^KO69$/eR?ei5'w1byAk^E=/bwyAyXH#4*eReiD?bA$/eRI#$&bA8j* (Ω, F , P) k^Æ& (X, Y ). F (x, y) := P(X ≤ x, Y ≤ y), x, y ∈ R.F*a(d

(1) a?Jw(2) l (x, y) ∈ R2, a, b ≥ 0, m

(x+a,y+b)(x,y) F := F (x + a, y + b) + F (x, y) − F (x + a, y) − F (x, y + b),F* (x+a,y+b)(x,y) F ≥ 0.

(3)

limx or y→−∞

F (x, y) = 0, limx,y→+∞

F (x, y) = 1;dH (2) ?Fwa F $m|'tawE?rP ((X, Y ) ∈ (x, x + a] × (y, y + b]) =

(x+a,y+b)(x,y) F.1(d6weR F D$/eR

gw xmlefk~ 54CbA$/eR F . A0 ); R2 UA (x, x + a] × (y, y + b] wkmµ((x, x + a] × (y, y + b]) :=

(x+a,y+b)(x,y) F. A , B2 $*); A0 *Ewk σ- s[ A HwYei);*A.6UAw+S µ wmqjTt A yXzt*t B2 wU.?690[w µ ?8969k [0, 1] wyX., An ⊂ A w ⋃

n An = A ∈ A , 169 ∑

n µ(An) = µ(A). eiÆ An, A ?UAAn = (xn, xn + an] × (yn, yn + bn], A = (x, x + a] × (y, y + b].r l n,

⋃ni=1 Ai ⊂ A,S∑

i≥1 µ(Ai) ≤ µ(A)?%wau ǫ > 0,JC l n, a δn > 0 8vF (xn + an + δn, yn + bn + δn) − F (xn + an, yn + bn) < ǫ/2n,wa δ > 0 8v F (x + δ, y + δ) − F (x, y) < ǫ. 0d[

0 ≤ (xn+an+δn,yn+bn+δn)(xn,yn) F − µ(An) < ǫ/2n;

0 ≤ µ(A) − (x+a,y+b)(x+δ,y+δ)F < ǫ.r (xn, xn+an+δn)×(yn, yn+bn+δn) : n ≥ 1.9 [x+δ, x+a]×[y+δ, y+b],*.9a n8v (xi, xi+ai+δi)×(yi, yi+bi+δi) : 1 ≤ i ≤ ndi.9r\ (xi, xi+ai+δi]×(yi, yi+bi+δi] : 1 ≤ i ≤ n.9 (x+δ, x+a]×(y+δ, y+b],e"C|Q

∞∑

i=1

µ((xi, xi + ai + δi] × (yi, yi + bi + δi]) ≥ µ((x + δ, x + a] × (y + δ, y + b]).6wJCv∑∞i=1 µ(Ai) ≥ µ(A)−2ǫ, ǫw kCv8∑∞

i=1 µ(Ai) ≥µ(A).

§4.2 Kolmogorov Z=6wf U9bA* Euclidj*w$/eRb?ei5'w' 12f ^ÆaGwb9 ji 1 bA'5w U9

gw xmlefk~ 55H#V:^Æ&/^Æ3a15?(d9n/w Ω t Euclid j*w#g b95wjH1EbEbwj*0[w|Z%,FwrH"6wm^?_w 4.2.1 & (Ω, F ) k (E, E ) ?e9j*bA# ξ : Ω −→ E De9#` ξ−1(E ) ⊂ F . ÈF*1>9? F/E - e9#^ÆaGwmq+ma?bA8j* (Ω, F , P)tbAe9j* (E, E )wbeITwe9# ξt : t ∈ T, dH T ?bAI03uE D^ÆaGwWdj* (,j*), T D?^ÆaGw3*>(q T ?Z w^3r>D^ÆIr6f w^Ækr?^ÆI? bAC8j*(dbn/w^ÆI bAg8w^ÆIQeivt;w^ÆI> 4.2.1 & F ? R $/eRF*a 8j* (Ω, F , P) k^ÆIξn wa1w$/eR? F . Gb0zm S1 = ξ1, · · · , Sn :=

∑ni=1 ξi, F* Sn ^?^ÆIa^D R wbA^Æ` F w x 0,% Sn ?bAr^Æwa C v §2.3 H69w^ÆIwa CpGj.K0[+mz~^ÆIu?1#b:"6w=> 4.2.2 %f3J& G = (V, E) ?bA3z V , E $*?k#wn$&a?T1*w.A*AS' bA%fÂQ G [%?0[w (1) UbQX`* (2) \0.K954=UbQ2AL3WMY9F :N ?.?bA8j*ei0Aj7OxTU ξn ); n 3f (1b?%fvtAw3*D3f) %fw D1E0[bA8j* (Ω, F , P) u"^ÆI ξn, 8va(d

P(ξn+1 = y|ξn = x) =

1c(x) ; ` y ? x wS;

0, *%,0| c(x) ); x wSAR> 4.2.3 ^Æh &bA6p$h 1, 2, 3, · · · wAZÂZêilAg6pLb?l ,gwZ^Hh$h6wZ^q

gw xmlefk~ 56"Aguw3u6p0[w[%h (1)BObQ(/0[O1_aP(2) BO(/0K9_a17G<_aPZ[1OS+)D1?*ei^ÆaGu7O0A= ξn ); n Ag_htwZ^h$s[z?*bA8j* (Ω, F , P) 8v ξ1 = 1,

P(ξn+1 = i|ξ1, · · · , ξn) =

|k:ξk=i|n+1 ; i ≤ max(ξ1, · · · , ξn);

1n+1 ; i = max(ξ1, · · · , ξn) + 1;

0; da.6w^ xTzU9 "61jw=yjyX §2.1 H_wbA$/?*ei5'^Hw=?\:V81e0[w8j*b?a wpkRLwbA1$1ER?z69a\#B"?AbA^ÆaG_Ew ÆUH#t49w8Paq+m5yXr6w$/eR` (Ω, F )C bA89 P, F*~e9# ξ, qj (E, E ) mbAeR µξ "µξ(A) := P(ξ−1(A)) = P(ω ∈ Ω : ξ(ω) ∈ A), A ∈ E .~M4wCFjZ6 µξ ?bA89aD P ξ "w49/ ξ w$/^ ξP / Pξ−1, '4zU ξ P st (E, E ) .HaQ` η ? E tbAe9j* F we9#F*

P(ηξ)−1 = (Pξ−1)η−1.h*z& ξ ?^Æ&F* µξ ? (R, B(R)) w89? ξ w$/B"Fe9j*w8P& (E1, E1), (E1, E2) ?e9j* A1 ⊂ E1,

A2 ⊂ E2, A1 × A2 := (x1, x2) : x1 ∈ A1, x2 ∈ A2.a?Fj* E1 × E2 wUA` A1 ∈ E1, A2 ∈ E2, 0AUADe9UA%e9UAk6'? π yp.? σ- F*1_zie9UA*Ew σ- E1 × E2, k E1 × E2 w_w σ- j*

gw xmlefk~ 57

(E1 × E2, E1 × E2) DF (e9) j*m#πi(x1, x2) = xi, i = 1, 2,a1$*D?Fj*t E1, E2 wuF σ- ei"f< 4.2.1 σ- E1 × E2 ? E1 × E2 8vAue9wh6 σ- h*z1bj*w_FH#jme9j* (E, E ) w n _Fe9j*1a (En, E n). j*wbAr>: (x1, · · · , xn), ^R?l();1^eiaaE?n [n] = 1, 2, · · · , n t E wbA#

x : i 7→ xi. aubA0[w#^ei:EFj*wbAZ~UEn k [n] t E w#k?bbww_i1.$Fj*wbAka_wwbA#F σ- n?8v_ue9wh6 σ- K.s7asbAjwfV?v/eiqj|Ztwyp6wn [n]+#Fwaei lbAYAR nwnlira.a?u(6j*Hwl(TIw_i kbAIn (5I+.? Fw) T, ,h ET ); T t E w_#wn^D E wTFj*h*z En = E1,··· ,n. q T?*ET DFj*h+mwFj*? EN, a?= E wk_iFj*w8P+.?*&*uma6w σ- I?e9UAw /Uuw )v615?b T ? N wypp HO"6wKolmogorov ,3/,FwH;*_i1?bIT OwEw~/Raqk N u?# x : T → E, t ∈ T , ,h x(t) xt ? x 3f t Vw=l( kC T w*^ I, (v61: I ⊂ T 3> I ?*^) muπI : ET −→ EI ,

πI(x) := (x(t) : t ∈ I), x ∈ ET.+mz πt := πt. m E T ET 8v_ πt : t ∈ T e9wh6 σ- 0AmyXr6Aj*FwyA1E E T wbAx%f l A ⊂ EI , a u πI "wM2 π−1I (A) D? A wR0A:Dw?_6wbAzA j*duwM2?bARk (e9) Rk?>

ET

0 :=⋃

I

π−1I (E I),

gw xmlefk~ 58dH I ' T w*^"6tU9 E T R?R*Ew σ- < 4.2.2 E T = σ(E T

0 ).0[1vt bAFe9j* (ET, E T) w πt : t ∈ T ?i E Wdj*we9#' $&C^ÆaG8j* (Ω, F , P)i (E, E )Wdj*w^ÆaG ξt : t ∈ T. mξ(ω) := (ξt(ω) : t ∈ T), ω ∈ Ω.% ξ(ω) ∈ ET, aD? ω w ([) u0A ξ ? Ω tj* ET w#"6wtU9 ξ ?e9#< 4.2.3 6mw ξ : Ω −→ ET ?e9# µξ ); P ξ "w489a8v (ET, E T, µξ) E8j* 0A8j*au πt : t ∈ T ?Wdj* E w^ÆaGD~%aG5R?^ÆaGw$/ra uaGwUR4r>w^Æ& $/wUb[/E4b=HB"w^Æk?uwU* Euclid j*w$/eR?ei5'wXkl5' E|5'+.L 0, 1 w_$/eiV|5'ei<|5'eiNp5'^eiAs^5'6wM#U9$/eR5ei Euclid j*_'5'w$/eRwk?)b_EwCFwp15K)b=w.? kw Borel ?+mw*b Borel wCF?N4w9wmuYqw^R?U*?)b=w-B_z^ÆaG1w6 ~%j* (ET, E T) uM#85'F*Xk4*-Bu)b1_EwCFIFR?Rwnk^ÆaG dwu"6#69Rwn?< 4.2.4 ` I, J ? T w*^w I ⊂ J , F* π−1

I (E I) ⊂ π−1J (E J), S

E T

0 ?bAC8j* (Ω, F , P) w^ÆaG ξt : t ∈ T k T w*^ I (1b?$&dHYTIT), m µI (ξt : t ∈ I) w$/µI(A) = P((ξi : i ∈ I) ∈ A), A ∈ E

I ,^R? P # πIξ "w49 µξ πI "w49µI = P(πIξ)

−1 = (Pξ−1)π−1I = µξπ−1

I .

gw xmlefk~ 59a? (EI , E I)w89D PwbA*$// P EI wuak' aGw1$=8j*wn (EI , E I , µI) : I ⊂ T D?^ÆaG ξt : t ∈ T w*$/e*$/eR?^ÆaGw$/ µξ RE T

0 w)',q Lebesgue 9 *wk'_i^R?^Æ&w$/eRw7A4zU^ÆaGw$/R?b+ ew b*/)bQu*$/eR4b**u)bsb+w/b[s$/eR|'JCb[*$/^bALwCF,C:VzU.s/wFK,C4K\Eb+4w& I kJ ? T w*^w I ⊂ J , m# πJ

I : EJ −→ EI πJ

I (xt : t ∈ J) := (xt : t ∈ I).a?j* EJ t EI wu% πJIπJ = πI , r\

µJ(πJI )−1 = µξπ−1

J(πJ

I )−1 = µξπ−1I = µI . µI x µJ EI wu0U9` I ? J w^ µI ? µJ w0R?"6wt< 4.2.5 ^ÆIw*$/e(d,C& I k J ? T w*^w I ⊂ J , F* µJ (πJ

I )−1 = µI .,C?bAL8P1CabAm 4.2.2 C3* T ke9j* (E, E ), ` T w l*^ I, µI ?(EI , E I) w8F*D8j*wn (EI , E I , µI) : I ⊂ T ?j* E k3* T wbA*$/e~`a1(dt 4.2.5 Hw,CF*D?,w*$/ea_O8j* (Ω, F , P) i E Wdj*w^ÆaG ξt : t ∈ T bA,w*$/e (EI , E I , µI) : I ? T w*^ . ' eiUxTU.wj $&CbA,w*$/e?*a 8j*(Ω, F , P) i E Wdj*w^ÆI ξt : t ∈ T 8vaw*$/eR?CwFAI`RU0A,w*$/e?ei5'w ?iw` E ?bAe$j*Tkq1e bA`j*3`,daU9_z E ?dw Borel σ-

gw xmlefk~ 60< 4.2.1 (Kolmogorov) ` E ?e$j*F*a6w,8j*?ei5'w0?'i8 HwhPw9b 6wXFk1;.ei690A ;wR?$wbAL1 5|CQd69< 4.2.2 & E ?e$j*% E l89 µ ?5%w l A ∈ E , µ(A) = supµ(K) : K ⊂ A, K E.6C 4.2.1 1J1d\ e9j* (ET, E T) M#Q_Ew9 µ, a(d I ⊂ T (*^"6yX), µπ−1

I = µI . ^R?U l H ∈ E I ,

µ(π−1I (H)) = µI(H).0f>[1 µ Rw E T

0 w6&*m B ∈ E T

0 , F*a I ⊂ Tk H ∈ E I 8v B = π−1I (H). vm

µ(B) := µI(H).p0mB!a R B w);Ur\ mwnC F69µ(B) . v B w);Z~U`~w I ′ ⊂ T k H ′ ∈ E I′ 8vB = π−1

I′ (H ′), F* µI′

(H ′) = µI(H). =5r π−1I (E I) U I ?|'wS.!& I ⊂ I ′, r πI = πI′

IπI′ , S

π−1I′ ((πI′

I )−1(H)) = π−1I (H) = π−1

I′ (H ′),wrub?(#|Q (πI′

I )−1(H) = H ′. v,C6µI(H) = µI′

((πI′

I )−1(H)) = µI′

(H ′).0U9 µ ? E T

0 mw#3eRa(d(1) µ(ET) = 1; =5 l t, ET = π−1

t (E), S µ(ET) = µt(E) = 1.

(2) ` A, B ∈ E T

1 w A ∩ B = ∅, F* µ(A ∪ B) = µ(A) + µ(B). =5a I ⊂ T 8v A, B ∈ π−1I (E I), H, K ∈ E I , yK8v A = π−1

I (H),

B = π−1I (K). r\ µ(A ∪ B) = µ(π−1

I (H ∪ K)) = µI(H ∪ K) = µI(H) +

µI(K) = µ(A) + µ(B).

gw xmlefk~ 61' 4.1.1,1E69 µ jVJCRO 169` An ⊂ E T

0 w An ↓ ∅, % µ(An) ↓ 0.q µ(An) ?|,w&a ǫ > 0 8v µ(An) > ǫ. H#a In ⊂ T,

Hn ∈ E In 8v An = π−1In

(Hn). ?r π−1I (E I) U I |'_i.!Æ In U n |' ' An = π−1

In(Hn) dH Hn ∈ E In . F* µ(An) = µIn(Hn).re$j*wF?e$j*Sa E Kn ⊂ Hn8v µIn(Hn \Kn) < ǫ/2n. Bn = π−1

In(Kn), a? An HwRp U n |,r\ Cn :=

⋂ni=1 Bi, a? An HwRwU n |,'

µ(An \ Cn) = µ

(

n⋃

i=1

(An \ Bi)

)

≤n∑

i=1

µ(Ai \ Bi) =

n∑

i=1

µIi(Hi \ Ki) < ǫ.0d[ l n, µ(Cn) > 0, |Q Cn #j x(n) ∈ Cn gE ET wr Cn |,S l m ≤ n, Cn ⊂ Cm ⊂ Bm = π−1Im

(Km), 0U9(x(n)(t) : t ∈ Im) : n ≥ m ⊂ Km. r Km ESa Km HFw^r\1eivt E wbA x(t) : t ∈ ⋃n In 8v l m, (x(t) : t ∈ Im)? (x(n)(t) : t ∈ Im) : n ≥ m wbA*|Q (x(t) : t ∈ Im) ∈ Km, ^k e ∈ E, t 6∈ T \ ⋃n In, m x(t) = e, F* x = (x(t) : t ∈ T) ∈ ET, 6w> U9 πIm

(x) ∈ Km /U x ∈ π−1Im

(Km) ⊂ Am, 0sC)rx ∈ ⋂n≥1 An.% Euclid j*?e$j*~/iz`we^?e$j*p?z`w.eR.?e$w&*XkuCbA,w*$/eIH# 3*z T = N wyp"=y5+mv E N

1 :=⋃

n π−1[n] (E

[n]). % E N1 ⊂ E N

0 +w E N = σ(E N1 ), R E N

1 RdO*E E N ~q*$/e (EI , E I , µI) : I ⊂ N ,3aaw^ (E[n], E [n], µ[n]) : n ∈ N (+mzDa*$/ewP^) r l*^ I b?A [n] w^8j* (E[n], E [n], µ[n]) : n ∈ N D,w` l n ∈ N,

µ[n+1](π[n+1][n] )−1 = µ[n], EXkzU l A1, · · · , An ∈ E

µ[n+1](A1 × A2 × · · · × An × E) = µ[n](A1 × A2 × · · · × An).q%bA,w*$/eHwP^b?,wau^bA,w8j*eibM#bA,w*$/e=5 l*^

gw xmlefk~ 62 n L$e8v I ⊂ [n], m µI := µ[n]π[n]I , ,C|Qm n wUjZ60[mw µI MEbA,w*$/e^R?Uq3*z3,w*$/eeiEjM#w,8j*x%);$& E ?zeCbA^ÆI ξn : n ≥ 0, FI:|8

P(ξ0 = x0, · · · , ξn−1 = xn−1, ξn = xn)

= P(ξ0 = x0)

n∏

i=1

P(ξi = xi|ξ0 = x0, · · · , ξi−1 = xi−1),0U9n$/? ξ0 w$/kn/8w:Vz1aG h93f E|q8ur nA3faGwBvaG"bf&* XzbA^ÆIwP93fr> 0,_i1"6$& N = n : n ≥ 0. au`18uaG 0 3fi8 µ0(x) x (x ∈ E);`g8aGwn 3frwB ξi = xi, 0 ≤ i < n, F*a "b0i8 pn((x0, · · · , xn−1), x)dt x (x ∈ E), F*0[w6dei7ObA^ÆI 1:"+690A> "6Uw$/b?>bAkx 1 w#3RoK 4.2.1 & E ?zeµ0(x) : x ∈ E? E $/ lw n ≥ 1 xi : 0 ≤ i < n ⊂ E, pn((x0, · · · , xn−1), x) : x ∈ E ? E $/F*a 8j* (Ω, F , P) d^ÆI ξn : n ∈ N 8v P(ξ0 = x) = µ0(x) w(q"6n/8km3)

P(ξn = x|ξ0 = x0, · · · , ξn−1 = xn−1) = pn((x0, · · · , xn−1), x),a1$/DaGwP9$/kUd8Jm µ[0] = µ0, l n ≥ 1, xi ∈ E, 0 ≤ i ≤ n, \Gmµ[n](x0, · · · , xn−1, xn) := µ[n−1](x0, · · · , xn−1)pn((x0, · · · , xn−1), xn).jZ6 µ[n] ?89w

∑

x∈E

µ[n](x0, · · · , xn−1, x) = µn−1(x0, · · · , xn−1).r\ µ[n] : n ∈ N?%89 4.2.1|Qa 8j* (Ω, F , P)d^ÆI ξn : n ∈ N (dP(ξ0 = x0, · · · , ξn−1 = xn−1, ξn = xn) = µ[n](x0, · · · , xn−1, xn).

gw xmlefk~ 63\j|Q6wn/8a 4.2.2 k 4.2.3, ^ÆaGwa C?%w 4.2.2, kTa ∈ E, m µ0(x) = 1x=a,

pn((x0, · · · , xn−1), x) =1x∼xn−1c(xn−1)

,dH ∼ );/, 4.2.3, 3*^ 1 `9m µ1(x) = 1x=1,

pn+1((x1, · · · , xn), x) =

1n+1

∑ni=1 1xi=x, x ≤ max(x1, · · · , xn);

1n+1 , x = max(x1, · · · , xn) + 1;

0, x > max(x1, · · · , xn) + 1.jZ6/(d| wn/I 4.2.1. 4.2.2 "6A|Q/E ?j*wbACFj*w^?,EwqwFqa?w0A> e$8 M`vei,t< 4.2.3 & µ ?j* E w89% l A ∈ B(E) k ǫ > 0, a ` G k F 8v F ⊂ A ⊂ G w µ(G \ F ) < ǫ.J& d ? E ` A F* F = A, Gn = x : d(x, A) < n−1.% Gn `w Gn ↓ A. r\ µ(Gn \ F ) ↓ 0, S> E' 1E69(dn/w A ?bA σ- RO 0.HZ6kj< 4.2.4 & µ ?e$j* E w8% l ǫ > 0, a EK ⊂ E, 8v µ(K) > 1 − ǫ.Je$C l n, a N 1/n weA| An,k : k ≥ 1 .9 E.F*

limi→∞

µ

⋃

i≥k≥1

An,k

= 1,Sa in 8v µ(

⋃in

k=1 An,k

)

> 1 − ǫ/2n. A =⋂

n≥1

⋃in

k=1 An,k, F* A ?wµ(Ac) ≤

∑

n≥1

µ

(

in⋃

k=1

An,k

)

< ǫ, K ); A wF* K ?Ew µ(K) ≥ µ(A) > 1 − ǫ.

gw xmlefk~ 64x l1. & φ ?n E′ t E w(#69 l A, B ⊂ E, φ−1(A) = φ−1(B) d[ A = B.

2. 69bA#3e"eRE9qwFqa ∅ VJ3. & E1, E2 ?e$j*69 B(E1 ×E2) = B(E1)×B(E2), Borel

σ- wF σ- R?Fj*w Borel σ- `j*,e$C> ?*E4. 69` I, J ?*^w I ⊂ J , F* π−1

I (E I) ⊂ π−1J (E J).

w Markov B§5.1 Markov C& E ?bA*/e_zE w`?z` σ-R E wk^& Xn : n ≥ 0 ?8j* (Ω, F , P) w^ÆIF* b=H1Ua8ua B3f n9r.A3fw D"b0w DbA$/R?n/$/ P(Xn+1 = y|Xi = xi, 0 ≤ i ≤ n) : y ∈ E.-3*wI`1bA3f n D' F*6 n wDae

n wD2uZ~U2uwB[%ak' Lsp G^Æ=H8u' w D"b0w[%R 4.2.2 H%f"b0&* a' _Vw DU ar6&* t' w DU ( 4.2.3 ,0[wCF) §3.1 Hf w^Æ%^0[wCF0ACFH# Markov N4QuSD MarkovC 5.1.1 ^ÆI Xn (d Markov C?> l3f n k y ∈ E,

P(Xn+1 = y|X0, · · · , Xn) = P(Xn+1 = x|Xn).n/ wCF0x% l x0, · · · , xn ∈ E P(Xn+1 = y|Xi = xi, 0 ≤ i ≤ n) = P(Xn+1 = y|Xn = xn).^R?Un/$/ v' w D xn. n/8 P(Xn+1 = y|Xn = x) D? (3f n w D x t"3fw D y) wUd80AUd.F DU^ 3*U&*? 3*U0AjI$&#7"bA+mwA|, n TwLz? n qF*# n TvGHw\ Xn (d

Markov Cp?# "bTL/zwqR?kTRUw`[%?.bTwLz?TwqRF*# "bTLzwqRkTRU +j1$&n/8 P(Xn+1 = y|Xn = x) (km3) 3* n U0A$&wCFD3*gC 5.1.2 bAWdj* E w(d MarkovCk3*gCw^ÆID (Wdj* E w) Markov 65

gy Markov o 66bA Markov w3*D %4R54R/#34R"6r>8#34Rk3*' & Xn : n ≥ 0 ? Markov Xn ?3f n w D X0 ?P9 Dpx,y := P(Xn+1 = y|Xn = x), x, y ∈ E, %3^: p(x, y), aD? Markov wUd8%FI: Markov w*$/ei:E

P(Xi = xi, 0 ≤ i ≤ n) = P(X0 = x0)p(x0, x1) · · · p(xn−1, xn),^R?U*$/Ud8k X0 w$/ (P9$/) _jZ6Ud8 p : E × E −→ R (d (1) px,y ≥ 0; (2)∑

y∈E px,y = 1.au E × E weR (x, y) 7→ px,y D? E wUdeR`a(d (1) px,y ≥ 0; (2)∑

y∈E px,y = 1. E3 px,y e: p(x, y). ~1^2 px,yakbAU2 P D (x, y) wYP := (px,y : x, y ∈ E),F* P D? E wUdU20[UdeR UdU2?bbwwTka?bAeKeB wU2p?awY?#3ww.Bwk?

1. qjZ6UdU2w n 5 Pn ?UdU2 D (x, y) wYp(n)x,y, D? n 0Ud8' 1HO+69 Markov wa C< 5.1.1 C E bAUdeR (px,y : x, y ∈ E), l E w$/

µ = (µx : x ∈ E), a bAe9j* (Ω, F ), dwbAe9I Xn : n ≥ 0 8 Pµ, (d

(1) l x ∈ E, Pµ(X0 = x) = µx;

(2) l n ≥ 0, x0, x1, · · · , x, y ∈ E,

Pµ(Xn+1 = y|Xn = x, · · · , X1 = x1, X0 = x0) = px,y.J.! Ω = EN, F = E N, Xi = πi. k x ∈ E, m8

µ[n](x0, · · · , xn) := µx0p(x0, x1) · · · p(xn−1, xn), xi ∈ E, 0 ≤ i ≤ n.F*UdeRwCF|Q (E[n], E [n], µ[n]x ) ?,w8j*r\

4.2.1 8 (Ω, F ) a 89 Pµ 8va E[n] wu? µ[n],

Pµ(X0 = x0, · · · , Xn = xn) = µx0p(x0, x1) · · · p(xn−1, xn).

gy Markov o 67vqjZ6 Pµ (dHwACFHw (1) U9P9$/? µ, (2) ?d[%. v µ. 1vt be8 Pµ : µ ? E $/ . h*z P

x );P9$/? x wm$/ww8 Px(X0 = x) = 1. F* 8 P

x auMarkov Xn ?^x Qw+w

Pµ(A) =

∑

x∈E

µ(x)Px(A), A ∈ EN, P

µ Px : x ∈ E 0[1(Ω, F , Xn : n ≥ 0, Px : x ∈ E)D?[UdeR (px,y) w Markov +m X . kC n ≥ 1, x = x0, x1, · · · , xn ∈ E, CF (2) vn$/

Px(Xn = xn, · · · , X1 = x1, X0 = x0) = p(x, x1) · · · · · p(xn−1, xn), (5.1.1)i#?A X W[+ x, x1, · · · , xn w80n$/?bA#>wI:H# xn = y, x1, · · · , xn−1 ' E kv

Px(Xn = y) =

∑

x1,··· ,xn−1∈E

p(x, x1) · · · · · p(xn−1, y) = p(n)(x, y),?A p(n)x,y lD? n 0Ud8~ n ≥ 1,

Pz(Xn+1 = y|Xn = x) =

Pz(Xn+1 = y, Xn = x)

Pz(Xn = x)= px,y,# n U^R?U^ x Udt y w8 Udw3*U0ACFD Markov w3*gC Fn ); X0, X1, · · · , Xn *Ew=/%a?zww.!Æ

F = σ(Xn : n ≥ 1). F*CF (2) k 2.3.1[9],

Px(Xn+1 = y|Fn) = pXn,y = P

Xn(X1 = y).t Ω w|d\^ θn, n ≥ 0, (d l k ≥ 0, Xkθn = Xk+n.< 5.1.2 l^Æ& Y , n ≥ 0, x ∈ E,

Ex(Y θn|Fn) = E

Xn(Y ).

gy Markov o 68Jr ω 7→ EXn(ω)(Y ) ? Fn ^Æ&SmEZ6 l^

A ∈ Fn,

Ex(Y θn; A) = E

x(EXn(Y ); A) (5.1.2)` (5.1.2) ;CeR Y = 1B, B ∈ F EF*n/ wCFa#3+m^Æ&^E|'~ Y w+m^Æ& Yn, r\mF|Q (5.1.2) EH# Y = 1B, dH B = X0 = x0, . . . , Xm = xm A = X0 =

y0, . . . , Xn = yn, F* (5.1.1),

Ex(1Bθn; A) = P

x(X0 = y0, . . . , Xn = yn, Xn = x0, Xn+1 = x1, . . . , Xn+m = xm)

= 1x=y0p(y0, y1) · · · p(yn−1, yn)1x0=ynp(x0, x1) · · · p(xm+1, xm)

= 1x=y0p(y0, y1) · · · p(yn−1, yn)Pyn(B)

= Ex(PXn(B); A).0[r Fn ?zww6A:w A wk*ES (5.1.2) Y = 1B, dH B ?6wA: l A ∈ Fn E169 l B ∈ F , (5.1.2) Y = 1B E H );6A:w

B wk A );8 (5.1.2) Ew Y = 1B w=/ B ∈ F wkq169 H ⊂ A . `1KO69 A ^? σ- F*\ UeU9 lXn ? A ^Æ& F ?h6wS F = A , RE69 ::Z6 A ?=/qrHpeijzaQ (1) H ? π yU*6\?'w8wCF|Q (2) A ?bA Dynkin (a) ∅, W ∈ A ;

(b) B ∈ A Bc ∈ A ; (c) ` Bn ∈ A wyK% ⋃

n Bn ∈ A . w5 2.3.1 H Dynkin aU A H *Ew σ- A ⊃ F .6w3* n eiby^Æ3*il^R?t MarkovCEbA# τ : Ω −→ [0,∞] D?bAq3` l n ≥ 0, τ ≤ n ∈ Fn. 0x% l n ≥ 0, τ = n ∈ Fn. Tk τ ei +∞ =wb? F∞ (Zm) ^Æ&jZ6T3* n ?bAq3m Fτ ? F∞ H(d ln ≥ 0, A ∩ τ ≤ n ∈ Fn wY A wkF* (1) Fτ ?bA σ- iR (2) qτ ≡ n 3 Fτ = Fn. (z/_BZ6) mq? D Xτ := Xn q τ = n3 n ≥ 0. 0[ Xτ n τ < ∞ mw \n? Fτ e9wyXm θτ := θn q τ = n 3s[ θτ ^ n τ < ∞ m X "

gy Markov o 696wt Markov C& τ ?bAq3% l B ∈ F∞, x ∈ E E

x(1Bθτ ; τ < +∞|Fτ ) = EXτ (1B)1τ<+∞.69?+mw A ∈ Fτ , F* l n ≥ 0, A ∩ τ = n ∈ Fn, MarkovC

Ex(1Bθτ ;A ∩ τ < ∞) =

∞∑

n=0

Ex(1Bθτ ; A ∩ τ = n)

=

∞∑

n=0

Ex(1Bθn; A ∩ τ = n)

=∞∑

n=0

Ex(EXn(1B); A ∩ τ = n)

= Ex(EXτ (1B); A ∩ τ < ∞).0R69 t Markov Cr\ Markov wyA" Markov C t MarkovC?x%wpt Markov C?bAqwFX"612Pf UdeRwCFkWdw$y> 5.1.1 (Bernoulli-Laplace t=) & A, B -HB r A|dH r A r Ap X0 ?`93 A -H |ARvB b| Xn ?L n v A -H |AR X = (Xn : n ≥ 1) ?^ÆIWdj*?

E = 0, 1, 2, · · · , n. jZ6 X ? Markov wpx,x−1 =

(x

r

)2

, px,x = 2i(r − x)

r2, px,x+1 =

(

r − x

r

)2

.0?bAUKaknw8=> 5.1.2 (*E^Æ) s$/^ÆI ξn, (d P(ξn = 1) = p,

P(ξn = −1) = q = 1 − p, X0 = x, Xn = X0 +∑n

i=1 ξi, % X = (Xn : n ≥ 0) ?bA^ x Qw Markov UdU2P :=

· · · · · · · · · · · · · · · · · · · · ·· · · q 0 p 0 0 · · ·· · · 0 q 0 p 0 · · ·· · · 0 0 q 0 p · · ·· · · · · · · · · · · · · · · · · · · · ·

.

gy Markov o 70Ebz& E ?ez" ( Zd), ξn ?Wdj* E ws$/^ÆIm X0 = x, Xn = X0 +∑n

i=1 ξi, % X = Xn :≥ 0 ?^ x QwMarkov & ξi w$/ P(ξi = x) = px, x ∈ E. r P(X1 = y|X0 = x) =

P(ξ1 = y − x) = py−x, S Markov wUd8 px,y = py−x, x, y ∈ E.> 5.1.3 (XF!w^Æ) & Markov X Wdj* E = 0, 1, · · · , r,UdU2P :=

1 0 0 0 · · · 0 0 0 0

q 0 p 0 · · · 0 0 0 0

0 q 0 p · · · 0 0 0 0

· · · · · · · · · · · · · · · · · · · · · · · · · · ·0 0 0 0 · · · q 0 p 0

0 0 0 0 · · · 0 q 0 p

0 0 0 0 · · · 0 0 0 1

,

p + q = 1. :Vzq Xn Wd 1 r − 1 9*3%"b03iw8q, 3w8 p, q Xn bt# 0 r 3a2.z`> 5.1.4 A, B A zw8? p, Lw8? q = 1 − p. (1) `[%?t.Q=z A zw8 (2) `[%?Jz=z A zw8[% (1). Xn );t n T>Q3 A wOzTR% Xn ?AXF! Markov Wd -2,-1,0,1,2, dUdU2

P :=

1 0 0 0 0

q 0 p 0 0

0 q 0 p 0

0 0 q 0 p

0 0 0 0 1

. pA(x) );^Wd x `9hJ A zw8F* pA(−2) = 0, pA(2) = 1. 8I:pA(−1) = p · pA(0)

pA(0) = q · pA(−1) + p · pA(1)

gy Markov o 71

pA(1) = q · pA(0) + peijz|Q A hJzw8? pA(0) = p2

1−2pq .[% (2). 0A Markov .K?j1 Xn = A ( Xn = B));t n T A z (B z). Yn = (Xn−1, Xn), n ≥ 2. % Yn WAWd AA,AB,BA,BB. UdU2

P :=

1 0 0 0

0 0 p q

p q 0 0

0 0 0 1

.% pA(AA) = 1, pA(BB) = 0, pA(AB) = 0+p·pA(BA), pA(BA) = p+q·pA(AB).#V:vT8I:pA2 = p2 + pq · pA(AB) + qp · pA(BA) =

p2(1 + q)

1 − pq.> 5.1.5 (D^Æ) & E ? Rd w4R? k x ∈ E 2d AS ( x wYzx 1), ` y ? x wSm px,y := 1

2d , *%m px,y := 0.% P = (px,y : x, y ∈ E) ?bAUdU2bAX\UdU2w MarkovD?D^Æ> 5.1.6 ( zww+m^Æ) & E ?bA+mzwn lx, y ∈ E, x, y +:::% S(x, y) = 1 *% S(x, y) = 0. % S ?Dw&

px,y =S(x, y)

∑

z∈E S(x, z),F* (px,y : x, y ∈ E) m E wbAUdeRww MarkovD?zw+m^ÆHwD^ÆR?4R?zw+m^Æ' & (px,y : x, y ∈ E) ?UdeR X ?ww Markov & x, y ∈ E,D x eb y, `a n ≥ 0, 8 p

(n)x,y > 0; D x, y yb` x eb y w y eb

x.EV 5.1.1 69 x eb y qwFqa n ≥ 1, xi ∈ E, 0 ≤ i ≤ n 8vpxi−1,xi

> 0, 1 ≤ i ≤ n w x0 = x, xn = y.

gy Markov o 72 Chapman-Kolmogorov GWd*ybU?x%UD X ` E w lAWdeyb y ∈ E, τy := infn ≥ 1 : Xn = y ( inf ∅ = ∞), jZ6 τy ?bAq3aD?Wd y wHb30?bA#> wq3GCFkw n ≥ 0, f(n)x,y := P

x(τy = n), %f

(n)x,y ≤ p

(n)x,y. r τy = k ∈ Fk, Xτy

= y, S n ≥ 1,

p(n)x,y = P

x(Xn = y) = Px(Xn = y, τy ≤ n)

=

n∑

k=1

Px(Xn−kθk = y, τy = k)

=

n∑

k=1

Ex(PXk (Xn−k = y); τy = k)

=

n∑

k=1

Px(τy = k)Py(Xn−k = y)

=

n∑

k=1

f (k)x,yp(n−k)

y,y .mfx,y := P

x(τy < ∞) =∑

n≥1

f (n)x,y ,0?aG^Wd x L*0tb y w8eiaQq x 6= y 3 x eb y qwFq fx,y > 0.EV 5.1.2 69AI: (1) l x, y ∈ E,

fx,y

∞∑

n=0

p(n)y,y =

∞∑

n=1

p(n)x,y.

(2) f(n+1)x,y =

∑

z 6=y px,zf(n)z,y .bAWd x ∈ E D?>w`^ x Qi8 1 t x, fx,x = 1,*% x D!w Markov X D?>w`d_Wd>D!`d_Wd!EV 5.1.3 69 y ∈ E ?>wqwFq l k ≥ 1 P

y(τyθk < ∞) = 1.< 5.1.3 1"X%(1) Wd x ?>wx% P

x(lim supXn = x) = 1 ^x% ∑

n p(n)x,x = ∞;

gy Markov o 73

(2) Wd x ?!wx% Px(lim supXn = x) = 0 ^x% ∑

n p(n)x,x < ∞.J y ∈ E,

τ (0) := 0, τ := τy,

τ (k) := τ (k−1) + τθτ (k−1) , k ≥ 1.F* τ (k) 5? X k t y w3fa^?bAq3m Ak(y) :=

τ (k) < ∞, );=/ X B! k tb y. % Ak(y) : k ≥ 1 ?m5wlim sup

nXn = y =

⋂

k

Ak(y),t Markov Cwr Xτ (k−1) = y,

Px(Ak(y)) = P

x(τ (k) < ∞) = Px(τ (k−1) < ∞, τθτ (k−1) < ∞)

= Ex(PX(τ (k−1))(τ < ∞); τ (k−1) < ∞)

= fy,yPx(τ (k−1) < ∞)

= · · · = fx,y(fy,y)k−1.r\q y = x 3P

x(lim supn

Xn = x) = limk

(fx,x)k =

0, fx,x < 1,

1, fx,x = 1. Borel-Cantelli t ∑

n p(n)x,x < ∞ [

Px(lim supXn = x) = 0,SEw691FE69 fx,x < 1 [∑

n p(n)x,x < ∞. P (t), F (t)$*); (p

(n)x,x : n ≥ 0), (f

(n)x,x : n ≥ 0) wAeR6|swI:

P (t) = 1 +

∞∑

n=1

p(n)x,xtn

= 1 +

∞∑

n=1

n∑

k=1

f (k)x,xp(n−k)

x,x tn

= 1 +

∞∑

k=1

f (k)x,xtk

∞∑

n=k

p(n−k)x,x tn−k

gy Markov o 74

= 1 + F (t)P (t),r\P (t) =

1

1 − F (t),|Q fx,x < 1 qwFq ∑

n p(n)x,x < ∞.")9 bAybx%yH/_Wd?>w/_Wd?!w< 5.1.4 ` x, y ∈ E ?ybw%"/9bE

(1) x, y ?>w Px(lim supXn = y) = 1 w ∑

n p(n)x,y = ∞.

(2) x, y ?!w Px(lim supXn = y) = 0 w ∑

n p(n)x,y < ∞.Jr x, y yba i, j ≥ 1 8v p

(i)x,y · p(j)

y,x > 0, l n ≥ 1,

p(i+n+j)x,x ≥ p(i)

x,yp(n)y,yp(j)

y,x, p(i+n+j)y,y ≥ p(i)

y,xp(n)x,xp(j)

x,y,r\ 5.1.3 |Q x, y ?>?!t Markov CP

x(lim supXn = y) = Px(lim supXn = y, τy < ∞)

= Ex(1lim supXn=yθτy

; τy < ∞)

= Ex(Py(lim supXn = y); τy < ∞)

= fx,yPy(lim supXn = y),r\ x, y ?!w% P

x(lim supXn = y) = 0 w(∑

n

p(n)x,y)p

(j)y,x ≤

∑

n

p(n+j)x,x ≤

∑

n

p(n)x,x < ∞,|Q ∑

n p(n)x,y < ∞. x, y ?>w%,X e|Q ∑

n p(n)x,y = ∞

Px(lim supXn = y) = fx,y,S1FEZ6 fx,y = 1 e=5& y > lw k ≥ 1, P

y(lim supXn = y) = 1 [P

y(⋃

n>kXn = y) = 1 0 [ Py(τyθk < ∞) = 1, S Markov Cv

p(k)y,x = P

y(Xk = x)

gy Markov o 75

= Py(Xk = x, τyθk < ∞)

= Ey(Py(Xk = x, τyθk < ∞|Fk))

= Ey(PXk(τy < ∞); Xk = x)

= fx,yp(k)y,x. y eb x |Q fx,y = 1.EV 5.1.4 69 (1) ` y ∈ E !% ∑∞

n=1 p(n)x,y < ∞, x ∈ E. (2) *Wd Markov B!bAWd?>wr\*Wd Markov ?>w^ C ⊂ E D?w` l x ∈ C,

∑

y∈C px,y = 1, x%z l x ∈ C, y 6∈ C, px,y = 0, C JwWd.etb C ~wWdF* X qwFq E ,#_^669whv569 `>Wd x ebWd y, F* y ^eb x r\|Q y ^>r\_ X w>Wdk?A^EV 5.1.5 69 (1) ` C ?wF* l n ≥ 1, x ∈ C, y 6∈ C p(n)x,y = 0; (2) bA Markov qwFqa,1^ (3) _>Wdk? (4) AWd x, y Drw` x eb y y eb x. r$7?>bArx%y69bAr$7?w> 5.1.7 1b 5.1.2 Hw*E^Æ%_ x, ^ x QL 2n 0t x w8 p

(2n)x,x =

(

2nn

)

pnqn, q p(2n+1)x,x = 0. Stirling I:

n! ∼ nn

en

√2πn, v (

2nn

)

∼ 4n

√πn

, Sp(n)

x,x ∼ 4n

√πn

(pq)n.r\q p = q = 12 3 ∑

n p(n)x,x = ∞, _Wd> X >*%_Wd! X !51ei\ f0,0. (`:

1√1 − 4t

=∞∑

n=0

(

2n

n

)

tn|Q p(n)0,0 wAeR

P (t) =1

√

1 − 4pqt2.

gy Markov o 76r\ f (n)0,0 wAeR

F (t) = 1 −√

1 − 4pqt2.|Q P0(τ0 < ∞) = f0,0 = F (1) = 1 − |p − q|. 5.1.3HwF!w^ÆjZ6^ lWd 0 < i < r Q

X b *0Jbt 0 r, v.z`r\Wd 0, r >d`Wd!< 5.1.5 (Polya) Rd wD^Æq d = 1, 2 3?>wq d ≥ 3 3?!wJ% n w8 R d U WdU q(d)n . d = 1 wypg 5.1.7 H69q d = 2 3s[fR0w8?SR 2n ? k 03 k 03" n − k 03i n − k 03r\

q(2)2n =

1

42n

n∑

k=0

(2n)!

k!k!(n − k)!(n − k)!

=1

42n

(

2n

n

) n∑

k=0

(n

k

)

(

n

n − k

)

=1

42n

(

2n

n

)2

∼ 1

πn.r\ ∑

n q(2)n = ∞.q d = 3 3s[w$v

q(3)2n =

1

62n

∑

k1+k2+k3=n

(2n)!

k1!k1!k2!k2!k3!k3!

=1

62n

(

2n

n

)

∑

k1+k2+k3=n

[

n!

k1!k2!k3!

]2

=1

22n

(

2n

n

)

∑

k1+k2+k3=n

[

1

3n

n!

k1!k2!k3!

]2

.HshH?bA1$/dk 1. r\∑

k1+k2+k3=n

[

1

3n

n!

k1!k2!k3!

]2

≤ 1

3n· max n!

k1!k2!k3!: k1 + k2 + k3 = n.

gy Markov o 77#he k1 = k2 = k3 h:HVbt Stirling I:he= 3n · n−1sr\ q(3)2n n− 3

2 sS ∑

n q(3)n < ∞, 3 D^Æ?!wq d ≥ 4 369yX x ∈ E, m xwN d(x), n n ≥ 1 : p

(n)x,x > 0wheIrR

(n?j d(x) := 0.) d(x) = 1, D x ?#N wD Markov?#N w`d_Wd?#N w< 5.1.1 (1)` x, yyb% d(x) = d(y). (2)` X ?#N wMarkov% x, y ∈ E, a n0, 8q n > n0 3 p(n)x,y > 0.J(1) a s, t ∈ T , 8v p

(s)x,yp

(t)y,x > 0, n ≥ 1, ` p

(n)x,x > 0, % p

(t+n+s)y,y ≥

p(t)y,xp

(n)x,xp

(s)x,y > 0, S d(y) 4S s+n+ t,03 p

(n)x,x > 0, S d(y) ^4S s+2n+ t,r\ d(y) 4S n, d(y) ≤ d(x), s d(x) ≤ d(y).

(2) & Gy := n ≥ 1 : p(n)y,y > 0, % Gy "'wr Gy wheIrR? 1, Sa m0, 8 Gy m0 ew__R (0?bAR j

Gy wheIr^? 1 [dHa *AY ni : 1 ≤ i ≤ k, a1yYr\a 4R αi : 1 ≤ i ≤ k 8v ∑

i αini = 1, Sa n 8v n, n + 1 ∈ Gy,v~"'wCF69> ) ~a s ∈ T , 8 p(s)x,y > 0, S l

n > n0 := m0 + s, p(n)x,y > 0.

E wbA8$/ (πy : y ∈ E) D?UdU2 P w_$/`∑

x∈E

πxpx,y = πy , y ∈ E.< 5.1.6 `bA#N w Markov X a _$/ (πx : x ∈ E),F*(1) X ?>w(2) l x, y ∈ E, limn p

(n)x,y = πy, _$/?bw

(3) l x ∈ E, πx > 0.J(1) _$/wm l n ≥ 1,

∑

x∈E

πxp(n)x,y = πy , y ∈ E.$& X ?!w% limn p

(n)x,y = 0, kEF l y ∈ E, πy = 0,

(πy) ?8$/)Tk0|.E X ?#N wn/

gy Markov o 78

(2) 1H#M# E × E wUdU2 (x, y), (u, v) ∈ E × E, mp((x, y), (u, v)) := px,upy,v,jZ6 (p((x, y), (u, v)) : (x, y), (u, v) ∈ E×E) ? E ×E wUdU2D?

P wqnU2a i E × E Wdj*w Markov ((Yn, Zn) : n ∈ T ) (ACwP9$/), iqnU2UdU2Ddqn% (Yn) (Zn)?Aw Markov r\p(n)((x, y), (u, v)) = p(n)

x,up(n)y,v ,t 5.1.1, qn^?#N ww π(x,y) := πxπy, x, y ∈ E × E ?qnw_$/Sqn^?>wT z ∈ E,

τ := infn ≥ 1 : (Yn, Zn) = (z, z)H;H (z, z) w3f% m ≤ n,

P(x,y)((Yn, Zn) = (u, v), τ = m)

= P(x,y)((Yn, Zn) = (u, v)|τ = m)P(x,y)(τ = m)

= P(z,z)((Yn−m, Zn−m) = (u, v))P(x,y)(τ = m)

= p(n−m)z,u p(n−m)

z,v P(x,y)(τ = m),\|Q

P(x,y)(Yn = u, τ = m) = P

(x,y)(Zn = u, τ = m).r\p(n)

x,u = P(x,y)(Yn = u) ≤ P

(x,y)(Yn = u, τ ≤ n) + P(x,y)(τ > n)

= P(x,y)(Zn = u, τ ≤ n) + P

(x,y)(τ > n)

≤ P(x,y)(Zn = u) + P

(x,y)(τ > n)

= p(n)y,u + P

(x,y)(τ > n),se6p(n)

y,u ≤ p(n)x,u + P

x,y)(τ > n),

gy Markov o 79S |p(n)x,u − p

(n)y,u| ≤ P

(x,y)(τ > n), qn?>w [ P(x,y)(τ < ∞) = 1, r\|Q

lim |p(n)x,z − p(n)

y,z | = 0, x, y, z ∈ E.b 6πk − p(n)

x,z =∑

y∈E

πyp(n)y,z − p(n)

x,z

=∑

y∈E

πy(p(n)y,z − p(n)

x,z),kEF limn p(n)x,z = πk, i, z ∈ E.

(3) r x, y yba i, j ≥ 1 8v p(i)x,y · p(j)

y,x > 0, l n ∈ T ,

p(i+n+j)x,x ≥ p(i)

x,yp(n)y,yp(j)

y,x, n → ∞, πx ≥ p(i)x,yp

(j)y,xπy , r\`A πx ?%_d`?S

πx : x ∈ E b?R?5w#N wn/?Fw1 E = 0, 1, p0,1 = 1, p1,0 = 1. %jZ6a?>_$/ π0 = π1 = 12 , p p

(n)x,y *.a < 5.1.7 `bA#N w Markov X ,_$/% l x, y ∈

E, limn p(n)x,y = 0.J1FEb X ?>wyp03` 669HM#wqn?!w% 5.1.4

∑

n

(p(n)x,y)

2 =∑

n

p(n)((x, x), (y, y)) < ∞,S limn p(n)x,y = 0.' &qn?>w$&> .r E ?BewSa bA^ (nu), 8 limu p

(nu)x,y _ x, y ∈ E a w.6w69He8*= x U αy := limu p

(nu)x,y . E w k*^ M ,

∑

y∈M

αy = limu

∑

y∈M

p(nu)x,y ≤ 1,S α :=

∑

y∈E αy ≤ 1. ~∑

z∈M

p(nu)x,z pz,y ≤ p(nu+1)

x,y =∑

z∈E

px,zp(nu)z,y ,

gy Markov o 80kEF∑

z∈M

αzpz,y ≤∑

z∈E

px,zαy = αy,0|Q ∑

z∈E αzpz,y ≤ αy, `dHbAR?6%α =

∑

y∈E

αy >∑

y∈E

∑

z∈E

αzpz,y =∑

z∈E

αz = α,sC)S l y ∈ E,∑

z∈E αzpz,y = αy, r α > 0, πy :=αy

α , F*(πy : y ∈ E) ? (px,y : x, y ∈ E) wbA_$/ n/) Gb0*>wy1t5> >w8P&Wd y ?>wDa?5 (positive) >w`__U3**

µy := Ey(τy) =

∑

n≥1

nf (n)y,y < ∞,*%D y ? (null) >wEV 5.1.6 69bA*Wd Markov,>Wdr\*Wd

Markov ?5>w< 5.1.8 & y ?>ww limn p(n)y,y a uy, % uy > 0 qwFq µy < ∞,03 uy = 1

µy.JR (p(n)y,y) (f

(n)y,y ) wAeR P (t) F (t) (d P (t)(1 − F (t)) = 1. r

limn p(n)y,y a Abel (- §1.5 j) d| |Q

limn

p(n)y,y = lim

t↑1

1 − t

1 − F (t)=

1

F ′(1).r\w> ?%w^i>`eiaQbA#N Markov e$Kyp

(1) ?!w03 ∑

n p(n)y,y < ∞, y ∈ E, .a _$/

(2) ?>p?>03 ∑

n p(n)y,y = ∞, p limn p

(n)y,y = 0, .a _$/

(3) ?>w?5>w03 ∑

n p(n)y,y = ∞, w limn p

(n)y,y = πy > 0, a _$/

gy Markov o 81> 5.1.8 & E ?#34RP :=

q0 p0 0 0 · · ·q1 0 p1 0 · · ·q2 0 0 p2 · · ·· · · · · · · · · · · · · · ·

dH px, qx ?5ww px + qx = 1, r\ P ?bAUdU2& X ?bAi PUdU2w Markov jZ6 X ?#N w n ≥ 1,∑n

k=1 f(k)0,0 =

1 − P0(τ0 > n) = 1 − p0p1 · · · pn−1, q n → ∞ 3F p0p1 · · · pn−1 *a a, % a ∈ [0, 1], w X ?>wqwFq a = 0.? G ∑

x∈E πxpx,y = πy , y ∈ E v πx = π0p0p1 · · · px−1, x ≥ 1. r\OMarkov _$/qwFqR

∑

x≥1

p0p1 · · · px−1Fhv1+mB" Markovw.&9wa bCr E e_iE w9 µ d mw9 µx = µ(x), x ∈ E σ- *Ck[_ µx ?*w E wbA σ- *9 (µx : x ∈ E) DUdeR(px,y) w Markov w.&9` l y ∈ E, µy =

∑

x∈E µxpx,y. %03 l n ≥ 1, µy =

∑

x∈E

µxp(n)x,y,wbA.&w89R?_9$& Markov X %d.&9`bA5F*k?5w_i"6U.&9a ?>5w.&9a bC?> lA.sw.&9,;bA>R"669>w Markov w.&9a wbTk.E#N wn/5#N wn/ 6_C^R?UUd8wFC< 5.1.9 (1)>wMarkovw.&9a wb(2)Markov?5>wqwFqa_$/wd_$/ πx = 1/E

xτx, dH τx ? xwHH3

gy Markov o 82J(1) & X ?>w Markov #69a C 0 ∈ E, τ := infn ≥ 1 : Xn = 0, l x ∈ E,

µx := E0

τ∑

n=1

1Xn=x.r X >S Xτ = 0. r\ µ0 = 1 w X ^ 0 Q%τ∑

n=1

1Xn=x =

τ∑

n=1

1Xn−1=x.~ τ ≥ n = τ < nc ∈ Fn−1, ' Markov Cq n ≥ 2 3E

01Xn=y,n≤τ = P0(X1θn−1 = y; n ≤ τ)

= E0(PXn−1(X1 = y); n ≤ τ)

=∑

x∈E

(E01Xn−1=x,n≤τ) · px,y

=∑

x∈E,x 6=0

(E01Xn−1=x,n≤τ) · px,y,# n ≥ 1 kv µy =∑

x∈E µxpx,y. 69 (µx) ? X w.&9"669bC& (νx : x ∈ E) ^? X w.&9m (yX Doobw h- &)

qx,y :=νy

νxpy,x, x, y ∈ E..&9wCF|Q Q = (qx,y : x, y ∈ E) ^?UdeRw(d l

n ≥ 1,

q(n)x,y =

νy

νxp(n)

y,x,w q(n)x,x = p

(n)x,x. r\ Q ww Markov ^?>w g

(n)x,y ); Y ^ xQ 3f n Htb y w8F* g

(n+1)x,y =

∑

z 6=y qx,zg(n)z,y /

g(n+1)x,y νx =

∑

z 6=y

νzg(n)z,y · pz,x. r

(n)x = P

0(Xn = x, n ≤ τ), F*6 69a Cw3u69 q n ≥ 1 3r(n+1)x =

∑

z∈E,z 6=0

r(n)z · pz,x.

gy Markov o 83q n = 1 3 g(1)x,y = qx,y, r

(1)x = p0,x, r\ g

(1)x,0νx = ν0r

(1)x , \aQ g(n)

x,0νx : n ≥1 k ν0r

(n)x : n ≥ 1 (ds[w\ Gw,swP=S l n ≥ 1,

g(n)x,0νx = ν0r

(n)x . ' # n k>C|Qi#x νx, #mx

ν0 · µx, ^ νx = ν0 · µx l x ∈ E E(2) & X ?5>w Markov F*abw.&9

µx = E0

τ0∑

n=1

1Xn=x, x ∈ E,r\ ∑

x∈E µx = E0τ0 < ∞, S πx := µx/

∑

x µx ?_$/w π0 = 1/E0τ0. r 0 ? kww_$/?bwS πx = 1/E

xτx l x ∈ E Eau` Markov _$/ πx, F* 5.1.6(1) >[1 X?>wbCa >R c 8vE

0τ0∑

n=1

1Xn=x = cπx, x ∈ E,r\ E0τ0 < ∞, X ?5>wEV 5.1.7 & X ?UdeR px,y w Markov l x ∈ E, m

Mn :=

n∑

i=1

1Xi=y −∑

x∈E

(

n∑

i=1

1Xi−1=x

)

· px,y,69 (1) l 0 ∈ E, (Mn : n ≥ 1) ? P0 r (2) 2 Doob q?w (Mn) 69>w X .&9x l

1. 69 5.1.3 HXF!w^Æ *0btWd 0 r.

2. & E = 0, 1.(a) Z6 p

(n)0,0 = p1,0 + (p0,0 − p1,0)p

(n−1)0,0 ;

(b) :Q p(n)0,0 w)b:+Q*

(c) Q p(n)x,y d*

3. #kH 3 Ap|hkH 3 A |.^kHB b|6"bkH pn );6 nv#kHw|VbCw8 limn pn.

gy Markov o 84

4. & X = (Xn) ?8j* (Ω, F , P ) i E Wdj*w Markov Yn := (Xn, Xn+1). F := (x, y) ∈ E × E : px,y > 0.(a) 69 (Yn) ?bAi F Wdj*w Markov (b) :QUd8+69` X ? #N % Y ^?(c) 69` πx ? X w_$/% πxpx,y : (x, y) ∈ F ? Y w_$/

5. r " !/ICu! IC9*qwFqm"wG#3hb tbv"""w8x p.

(a) Xn );`Q (!/IC) 03G#wwRCU9a?bAMarkov :QdUd8a?*_$/d_$/

(b) \`0w8w*(c) 69 l p, 5 ei6`i 95% iw8.0

6. xeKz M Ao^|"| ξn );" n A|vjowAR69 ξn ? Markov +:QawUdU27. b 5.1.2 Hez"w^Æ X , awUdeR? px,y = py−x,dH (px : x ∈ E) ?8$/ (1) (px) (d4*n/3X ?w

(1) 69` E *F* X .eK_$/r\a.eK?5>w8. &9 (µx) (d l y ∈ E,

∑

x∈E

µxpx,y ≤ µy.

(1) ` X 69 (µx) /sx 0, /sx/1?*5R (2) (dOn/w σ- *9D Markovwa$969` E *F*a$9?.&9 (3) & (µx) ?a$9mµ(n)

y :=∑

x∈E

µxp(n)x,y, y ∈ E.69 µ(n)

y : n ≥ 1 |, µiy := limn µ

(n)y , y ∈ E. 69 (µi

y : y ∈ E)?.&99. & X ? E > Markov * A ⊂ E, m σ ? A wHH

gy Markov o 853 σn ? n ;H A w3*^R?Uσn = σn−1 + σθσn−1 . Yn := X(σn). 69 (1) Y ?i A Wdj*w> Markov (=5? X wbA3*&D? X A w). (2) ` (µx : x ∈ E)? X w.&969a A w*E (µx : x ∈ A) ? Y w.&9

10. V+m^Æw^U9!w Markov^eK.&9p b§5.2 Galton-Watson %`."*E*Kw'??>R:wbA>R'?wh+mw^Æ=?_Æw Galton-Watson $7aG+mpaLw %=

Galton-Watson $7aG+D$7aGw7O?:Vw&^Æ&X w=?#34Ri)K*KAk;viRC& X w$/

pn = P(X = n), n ≥ 0,∑

n

pn = 1. f ); X wAeRf(s) = E[sX ] =

∑

n≥0

pnsn, s ∈ [0, 1].AeR?SP$7aGhLwFXq X ?>R3D_yAr03w'?=:?w"6b?$&#_wyA$&Kw.AAkis[w=:wdaAku;viRLÆU& (Ω, F , P) ?bA8j* X(k)n : n, k ≥ 1 ?ds$/^Æ& X ,s$/5X

(k)n +mD k iH n AAk_;wviRC

Z(j)0 = j, Z

(j)n+1 =

Z(j)n∑

i=1

X(n+1)i , n ≥ 0, (5.2.1) Z

(j)n );bAh9AkR j wKw n iviwRC Zn = Z

(1)n , 5 Z(j)

n ? Znw j A0Ewk~:Vza C Zn = j w

gy Markov o 86n/" Zn+k : k ≥ 0 ? Gn w Z(j)n : n ≥ 0 s$/wr\ Zn? Markov < 5.2.1 Z = Z(j)

n : n ≥ 0 ?bA Markov Proof. Fn = σ(Z1, Z2, · · · , Z − n), Gn = σ(X(k)

i : i ≥ 1, 1 ≤ k ≤ n), %Fn ⊂ Gn. H# l n ≥ 1, #34R k, j, jn−1, · · · , j1, r Zn = j

P(Zn+1 = k|Zn) = P(Zn+1 = k|Zn = j) = P(

j∑

i=1

Xn+1i = k),S

P(Zn+1 = k, Zn = j, Zn−1 = jn−1, · · · , Z1 = j1)

= P(

j∑

i=1

Xn+1i = k)P(Zn = j, Zn−1 = jn−1, · · · , Z1 = j1)

= E(P(

j∑

i=1

Xn+1i = k); Zn = j, Zn−1 = jn−1, · · · , Z1 = j1)

= E(P(Zn+1 = k|Zn); Zn = j, Zn−1 = jn−1, · · · , Z1 = j1), Markov CP(Zn+1 = k|Fn) = E(Zn+1 = k|Zn).3*gC?%wr

P(Zn+1 = k|Zn = j) = P(

j∑

i=1

Xn+1i = k) = p∗j

k ,dH p∗j = p∗jk : k ≥ 0 ? pk : p ≥ 0 w j- L[a? X w j A0Ewkw$/ p∗0 R? 0 w Dirac 9r\ Zn ?^ 1 Qw (wz

Z(j)n : n ≥ 0 ?^ j Qw) i pj,k = p∗j

k , j, k ≥ 0 UdeRw (3g) MarkovEV 5.2.1 Z w#WdybqwFq p0 > 0, p1 > 0, p0 + p1 < 1.' 1u\ Zn wAeRawAeR f(n), f w n- L0neR fn, f0 ?sx#0[fn+1(s) = E

[

sZn+1]

= E

[

sPZn

i=1 Xn+1i

]

gy Markov o 87

=∑

j≥0

E

[

sPj

i=1 Xn+1i

]

· P(Zn = j)

=∑

j≥0

(f(s))jP(Zn = j) = E

[

f(s)Zn]

= f(n)(f(s)).0|Q f(n) = fn, Zn wAeRR? X wAeRw n- L0n m, σ2 $*); X w ;F*m = f ′(1), σ2 = f ′′(1) + f ′(1) − f ′(1)2. (5.2.2)EV 5.2.2 69E[Zn] = mn, var(Zn) = σ2mn−1

n−1∑

i=0

mi.Tk X ?#_w kZ6"6wCF(1) f [0, 1] bA|'wx(2) f(0) = p0, f(1) = 1;

(3) m ≥ 1, % l s ∈ [0, 1) f(s) > s;

(4) m > 1, % f(s) = s [0, 1) bwDEV 5.2.3 69OCF^R?U f [0, 1] s 1 JhAD6wFA q.< 5.2.1 (1) m ≥ 1, % q = 1. m > 1, % q < 1;

(2) s ∈ [0, q), % fn(s) ↑ q;

(3) s ∈ (q, 1], % fn(s) ↓ q.

Proof. (1) ?%w (2) bA|'C xCq s ∈ [0, q) 3 q = f(q) >

f(s) > s. \G|Q s < f(s) < f2(s) < · · · < fn(s) < q, fn(s) |'* x, % f wJCv x = limn fn(s) = f(x). r x ≤ q, Sh6C x = q.

(3) v69yX> 5.2.1 & 0 < p < 1, 0 < b < 1 − p, pk = bpk−1, k ≥ 1;

p0 = 1 −∑

k≥1

pk = (1 − p − b)(1 − p)−1.

gy Markov o 88& X w$/ pk : k ≥ 0. %f(s) = 1 − b

1 − p+

bs

1 − ps, m =

b

(1 − p)2.Xkz\ Zn wAeR fn. 0|t$:eRwbACFjZ6 l5R

u, v,

f(s) − f(u)

f(s) − f(v)=

bs1−ps − bu

1−pu

bs1−ps − bv

1−pv

=s − u

s − v· 1 − pv

1 − pu.q m 6= 1, f(s) = s A.swD s0 1. u = s0, v = 1, v

f(s) − s0

f(s) − 1=

s − s0

s − 1· 1 − p

1 − ps0.#sF f(s) − 1, v s ↑ 1, e8 1−p

1−ps0= 1

m . r\f(s) − s0

f(s) − 1=

1

m· s − s0

s − 1,i

fn(s) − s0

fn(s) − 1=

1

m

fn−1(s) − s0

fn−1(s) − 1= · · · = m−n s − s0

s − 1,\?Q fn v

fn(s) =s0 − m−n · s−s0

s−1

1 − m−n · s−s0

s−1

.q m = 1 3 f(s) = p−(2p−1)s1−ps , \G.Hvt fn w)b: τ = infn : Zn = 0. % τ D? Z w8^3* P(τ < ∞) D?$7aG Z w8^8r τ < ∞ =

⋃

nZn = 0, 0 ?A)oS Zn ?|'wrP(τ < ∞) = lim

nP(Zn = 0) = P(lim

nZn = 0), Zn wAeR? fn, F* P(Zn = 0) = fn(0), r\

P(τ < ∞) = limn

fn(0) = q."< 5.2.2 $7aG Z w8^8AeR f [0, 1]6wD q. r\q m ≤ 13 q = 1, 8^?w wq m > 1 3 q < 1.

gy Markov o 89EV 5.2.4 69 (1) l#Wd?!wrP(Zn+i 6= k, i ≥ 1|Zn = k) ≥

pk,0, p0 > 0,

1 − pk,k, p0 = 0;

(2) 8 1 Zn → 0 Zn → ∞."61uf Zn w'?Z` m ≤ 1, %gL8u P(Zn → 0) = 1.< 5.2.2 Wn = Zn · m−n. % Wn ?bArProof. Markov C

E(Zn+1|Zn = j, Zn−1 = jn−1, · · · , Z1 = j1)

= E(Zn+1|Zn = j) = E

[

j∑

i=1

Xn+1i

]

= jm,\e8E(Zn+1 · m−(n+1)|Zn, Zn−1, · · · , Z1) = Zn · m−n, Wn ?rr Wn?#3rrFa e#3^Æ&W 8vWn

a.s.−→W .< 5.2.3 m > 1, σ2 < ∞, %(1) limn E(Wn − W )2 = 0;

(2)

EW = 1, varW =σ2

m2 − m;

(3) P(W = 0) = q.

Proof. (1), (2), rw Doob .x:E69 EW 2n RdO

EW 2n =

EZ2n

m2n=

σ2(1 − m−n)

m2 − m+ 1,\

sup EW 2n =

σ2

m2 − m+ 1 < ∞.

(3) r = P(W = 0), r EW = 1, S r < 1. ~Tktr6UawqZ1 = k 3 Zn + 1 : n ≥ 0 Zn w k A0Ewk?s$/w

r = P(W = 0) =∑

k≥0

P(W = 0|Z1 = k)P(Z1 = k)

gy Markov o 90

=∑

k≥0

P(W = 0)kP(Z1 = k) = f(r).\|Q r = q.hv1f $7aGw#Wdkm"w.&9j kw

Markov X = (Xn),Wdj* E,UdeR px,y . E w^ F ,m qx,y = px,y,

x, y ∈ F , k ∆ 6∈ F , m q(x, ∆) = 1 − q(x, F ), x ∈ F ; q(∆, ∆) = 1. F* qx,y : x, y ∈ F ∪ ∆? F ∪ ∆wUdeRww Markov X∆.% ∆ ? X∆ w)o(dπy =

∑

x ∈ Fπxpx,y, y ∈ F`w9 πx : x ∈ F D X∆ w.&9' $7aG1baw#Wdw Markov w.&9a bCj< 5.2.3 & p1 > 0, % k j ≥ 1, p(n)1,j /p

(n)1,1 |'

Proof. p1 > 0 [ p(n)1,1 b?5wAeRwCF

p(n+1)1,j

p(n+1)1,1

=1

j!

f(j)n+1(0)

f ′n+1(0)

.s\f

(1)n+1(s) = f ′(fn(s)) · f ′

n(s);

f(2)n+1(s) = f ′′(fn(s)) · (f ′

n(s))2 + f ′(fn(s)) · f (2)n (s)

· · · · · ·

f(j)n+1(s) = an,j(s) + f ′(fn(s)) · f (j)

n (s),dH an,j ?bAR#3wBRr\p(n+1)i,j

p(n+1)1,1

=1

j!· an,j(0) + f ′(fn(0))f

(j)n (0)

f ′(fn(0))f ′n(0)

≥ 1

j!· f

(j)n (0)

f ′n(0)

=p(n)1,j

p(n)1,1

.E69 πj = limnp(n)1,j

p(n)1,1

, j ≥ 1; P(s)? πj : j ≥ 1wAeRγ = f ′(q), q m = 13 γ = 1, *% γ < 1.

gy Markov o 91< 5.2.4 πj : j ≥ 1 (di".&Cγπj =

∑

k≥1

πkpk,j , j ≥ 1. (5.2.3)r\ p0, p1 > 0 3% k j ≥ 1, πj < ∞.

Proof. rp(n+1)1,1 = f ′

n+1(0) = f ′(fn(0)) · p(n)1,1 ,w fn(0) → q, S

p(n+1)1,1

p(n)1,1

= f ′(fn(0)) ↑ γ.a C-K Gp(n+1)1,j =

∑

k≥0

p(n)1,kpk,j ,#sS p

(n)1,1 v

p(n+1)1,j

p(n+1)1,1

·p(n+1)1,1

p(n)1,1

=∑

k≥0

p(n)1,k

p(n)1,1

· pk,j , n → ∞, mF|Qw> r π1 = 1, S 1 ≥ γ = γπ1 =∑

k≥1 πkpk,1, pk,1 = pk−10 p1 > 0, |Q

πk < ∞.EV 5.2.5 k ≥ 1, wa n 8v p(n)1,k > 0, 69 πk > 0.' & p0, p1 > 0. (5.2.3) w#F sj , v j ≥ 1 kv

γP(s) =∑

k≥1

πk

∑

j≥1

pk,jsj =

∑

k≥1

πk

∑

j≥0

pk,jsj − pk,0

=∑

k≥1

πkE

(

sPk

i≥1 X(1)i

)

−∑

k≥1

πkpk0

=∑

k≥1

πk(f(s))k − P(p0),w (5.2.3)

γ = γπ1 =∑

k≥1

πkpk,1 =∑

k≥0

πkkpk−10 p1 ≥ p1

p0P(p0),

gy Markov o 92r\ P(p0) < ∞, "6wx:P(f(s)) = γP(s) + P(p0). (5.2.4) (5.2.4), e8 k n ≥ 1, P(fn(p0)) < ∞. ^ k s ∈ [0, q),

P(s) < ∞, ^R?Ux: (5.2.4) _ 0 ≤ s < q E (5.2.4) GBivP(fn(s)) = γn

P(s) + (γn−1 + · · · γ + 1)P(p0). (5.2.5)q m ≤ 1 3 q = 1 w fn(s) ↑ 1. r\q m = 1 3 γ = f ′(1) = m = 1, ^∑

k≥1 πk = P(1) = ∞, πk ?9q m < 1 3 γ = f ′(1) = m < 1,F* ∑

k≥1 πk = P(1) < ∞, πk ?*9< 5.2.5 πj : j ≥ 1 k G (5.2.3) w? >Rwkm"bProof. E69k (5.2.4)w?#3Rw5R P >Rkm"?bw& R ?~bA?H# (5.2.4), P(0) = R(0) = 0, _ia1,swP9=~ (5.2.5) se8a1(d"6 G

P′(fn(s))f ′

n(s) = γnP

′(s). (5.2.6) l s ∈ [0, q), a k ≥ 0 8v fk(0) ≤ s ≤ fk+1(0), r\R′(s)

P ′(s)=

R′(fn(s))

P ′(fn(s))≤ R′(fn+k+1(0))

P ′(fn+k(0))

=R′(fn+k+1(0))

P ′(fn+k+1(0))

P ′(fn+k+1(0))

P ′(fn+k(0)).#^bAxh|Q

R′(0)

P ′(0)=

R′(fn(0))

P ′(fn(0)),# n Ur\w (5.2.6) v

R′(s)

P ′(s)≤ R′(0)

P ′(0)· P ′(fn+k+1(0))

P ′(fn+k(0))

=R′(0)

P ′(0)· f ′

n+k(0)

f ′n+k+1(0)

· γ =R′(0)

P ′(0)· γ

f(f ′n+k(0))

. n → ∞, vR′(s)

P ′(s)≤ R′(0)

P ′(0).

gy Markov o 93yXzwR′(s)

P ′(s)=

R′(fn(s))

P ′(fn(s))≥ R′(fn+k(0))

P ′(fn+k+1(0)),e|v

R′(s)

P ′(s)≥ R′(0)

P ′(0).sR R′ ?sR P ′ w>R R P wP=?S R ? P w>Rv6

§5.3 6Q Markov CÆHpN& X = Xn : n ≥ 0 ?Wdj* S wbA Markov l x, y ∈ S, a u x0 = x, x1, · · · , xn−1, xn = y, 8v l 0 ≤ i ≤ n − 1,

p(xi, xi+1) > 0. Markov CP

x(Xi = xi, 1 ≤ i ≤ n) = Πni=1p(xi−1, xi),0A8D? X WOu^ x Udt y w8`a S wbA#3w#eR π, 8v l x, y ∈ S, π(x)px,y =

π(y)py,x, F*D X ? S eM Markov (π(x) : x ∈ S) D?D9Tk1bei$& l x ∈ S, π(x) > 0. ?` π(x) = 0, F* l y ∈ S, py,x = 0, x ^ S HSv X ?eMw Markov & x =

x0, x1, · · ·xn−1, xn = x ? x t x wNw l 1 ≤ i ≤ n, p(xi−1, xi) > 0.F*π(xi−1)

π(xi)=

p(xi, xi−1)

p(xi−1, xi),r\|Q

Πni=1p(xi−1, xi) = Π1

i=np(xi, xi−1),^R?U X WUdw8 3Uau` X WUdw8 3U (^U X (d Kolmogorovw9E), F* X ?eMw=5 x ∈ S, π(x) > 0, r X wC k y ∈ S, bnwu^ x Udt y, r\1mπ(y) :=

Πni=1p(xi−1, xi)

Π1i=np(xi, xi−1)

π(x).

gy Markov o 94H# X WUdw8 3U0ACF6Om NUdeMC?%w069 "6< 5.3.1 Markov X eMqwFq X (d Kolmogorov 9E' 1a#& G = (S, E) ?bArw*zS ?=wn E ⊂ (x, y) : x, y ∈ S ?#wnU x, y ∈ S ,` (x, y) ∈ E. $&z?3w (x, y) ∈ E qwFq (y, x) ∈ E. Tk0[sbn# E HQ'& r ?m E wbA5DeR l# (x, y) ∈ E, r(x, y) =

r(y, x) > 0. D r(x, y) D?# (x, y) wff r D? G wfeRm#wssfwrR c(x, y) := r(x, y)−1, (x, y) ∈ E. _z` (x, y) 6∈ E, x 6= y, F*.!Æ r(x, y) = +∞, c(x, y) = 0. qfks?y,bwz G keR r ( c) bhD?bA#5# F :v4?15w# K$:w=UC0[bA#mc(x) :=

∑

y∈S

c(x, y), p(x, y) =c(x, y)

c(x), x, y ∈ S.F* (p(x, y) : x, y ∈ S) ? S UdeRw c(x)p(x, y) = c(x, y) = c(y, x) =

c(y)p(y, x), r\_is bA S weM Markov au& X ?*n S bAeM MarkovUdeR p(x, y), D9? π(x), x ∈ S.F*m E := (x, y) : p(x, y) > 0, sc(x, y) := π(x)p(x, y), x, y ∈ S.F*eMCc ? E D5eRz GdfeRseRbh (G, r) (G, c) DbA#r\# eM Markov ?bbww:Vz?AMarkov Rs #Wwia_zEj3sCEfwSUd"61R& (G, r)?bA#X ?wweMMarkov"61Pf /wU,h x ∼ y); x, y, (x, y) ∈ E. & H ⊂ S,U y

H ,` y H Hw, y ∼ H . m H := H∪y ∈ S : y ∼ H,D? H wa? H H ,wwkrz?rww H ?S w#j1^F* H b^?d H w1^

gy Markov o 95 5.3.1 UWdj* S weR f x ∈ S Vk`f(x) =

∑

y:y∼x

p(x, y)f(y).U f n H ⊂ S k`a H w lk% f x Vkx% f(x) = Exf(X1). keRwmA_Q~$HUkeRwmr\0|keR^yXwCF< 5.3.2 (1) & H ? S wbAr^ f : G → R H k` f H bthe= max f := maxx∈G f(x), F* f H ?>R (2) & H ?

G w1^` G weR f, g H kw G\H ,xF* f = g.

(3) & H ? S w1^ f0 ? G\H weRF*a G weR f , a H kw f |G\H = f0.J(1) ` f x kw f(x) = max f , F*kwm f(x) =∑

y:y∼x p(x, y)f(y) |Qq y ∼ x 3 f(y) = max f . r H ?rwS f H x max f . (2) h = f − g. F* h H kw G\H $& h H wV5% h H w x0 Vbthe= K ); x0 H Hwr$7 (1) |Q h K ?5>R' r G rw K 6= G,S K\K #jr K ? H wr$7S K\K ⊂ G\H , r\sQ)|Q h ≤ 0, psei|Q h ≥ 0, S h = 0. (3) & τ := τG\H ?n G\H wG3F*1uZ6 f(x) := E

xf0(Xτ ) ,nH# Xτ ∈ G\H , Smkm~ x 6∈ H , % Px(τ = 0) = 1, S f(x) = f0(x); x ∈ H , % MarkovC P

x(τ ≥ 1) = 1, S f(x) = Ex(f0(Xτ )θ1) = E

xf0(X1), f x Vk' #j^ A, B ⊂ S, A ∩B = ∅. "bA: A, B 8v A .AwQ? 1, B .AwQ? 0. wV:a QeR^R? SwbAeR v 8v v|A = 1, v|B = 0, eR i, a?# E weRkfbh(d"6w Kirchhoff k Ohm (1) Kirchhoff =.:: Iw x, x 6∈ A ∪ B,

∑

y∼x

i(x, y) = 0,Qwkxwk

gy Markov o 96

(2) Kirchhoff ` x = x0 ∼ x1 ∼ · · · ∼ xn = x ?bAF*n∑

i=1

i(xi−1, xi)r(xi−1, xi) = 0.

(3) Ohm ` x ∼ y, F* v(x) − v(y) = i(x, y)r(x, y).

Ohm [ i ? E wDeR i(x, y) = −i(y, x). ~ Ohm x% Kirchhoff =5 Ohm % [ Kirchhoff au` Kirchhoff E b ∈ B, l x 6∈ A ∪ B, a x t bwN b = x0 ∼ x1 ∼ · · · ∼ xn = x, mv(x) :=

n∑

i=1

i(xi, xi−1)r(xi, xi−1). Kirchhoff v(x) K$wNUr v|B = 0, a^ b wK$Uj|Q v, i, r (d Ohm "6U9Qei Markov u);< 5.3.3 (1)QeR v (A∪B)c k (2) l x ∈ S, v(x) = Px(τA <

τB).J(1) x 6∈ A ∪ B, = ∑

y∼x i(x, y) = 0, Ohm i(x, y) = (v(x) − v(y))c(x, y), S ∑

y∼x(v(x) − v(y))c(x, y) = 0. r\ v(x)c(x) =∑

y∼x c(x, y)v(y), |Q v x Vk (2) jZ6 x 7→ Px(τA < τB) ^

(A ∪ B)c w A ∪ B v b[bC8/sxr i DS∑

x,y∈S:x∼y

i(x, y) =∑

(x,y)∈E

i(x, y) = 0. Kirchhoff =∑

x∈A

∑

y∼x

i(x, y) =∑

x∈B

∑

y∼x

i(y, x) = 0,i#? A Qw#? B w`1?,xwa i(A, B).4AwQ? 1, _i A, B *wf? i(A, B)−1

, /Us? i(A, B), $*D? A, B *w8fk8s Reff(A, B), Ceff(A, B). A, B *

gy Markov o 978sx A, B "m Qvw8fx^ A m 3 A, B wQ;8s (f) eirai"Kx%&w :u\z/_B Ohm k Kirchhoff Z61. XAXwf r1, r2 wwfxbA r1 + r2;

2. +A+wf r1, r2 wwfx r1r2

r1+r2;

3. >A& a1, a2, a3 b?z:w>AF*aei& a1, a2, a37A:wc(a1, a2) =

γ

c(b, a3), c(a2, a3) =

γ

c(b, a1), c(a3, a1) =

γ

c(b, a2),dH

γ =Π3

i=1c(b, ai)∑3

i=1 c(b, ai).' & A = a, a 6∈ B. hH:: a il A. "6w1 8s 8wU< 5.3.4 & τ+

a := infn ≥ 1 : Xn = a. a Qt a rtb B w8P

a(τB < τ+a ) = Ceff (a,B)

c(a) .J"I8v v(a) = 1, v|B = 0. H#6w v(x) = Px(τa < τB).% τ+

a = τaθ1 + 1 ≥ 1, ^ a Q3 τB ≥ 1. r\P

a(τB < τ+a ) = E

a(1τB<τaθ1)

=∑

x∼a

Px(τB < τa)p(a, x)

=∑

x∼a

(1 − v(x))p(a, x)

=1

c(a)

∑

x∼a

(v(a) − v(x))c(a, x)

=1

c(a)

∑

x∼a

i(a, x) =Ceff(a, B)

c(a)."6w^U9&[0Au\8> 5.3.1 & a1, a2, a3, a4 ?5 AwWA.#ws? 1. F*, a1, a2 *w8s Ceff(a1, a2) = 1 + 1

3 = 43 . r\ a1 Qt a1 rtb

gy Markov o 98

a2 w8 Pa1(τa2 < τ+

a1) = Ceff (a1,a2)

c(a1)= 2

3 . ,w a1, a3 *w8sCeff(a1, a3) = 1

2 + 12 = 1, ww8 1

2 . 0A^9+m6w8e::\> 5.3.2 &bA5 k#.#ws ? 1. a, b ?,rabwx%& Ceff(a, b) = 127 , a Qt a rtb b w8

Pa(τb < τ+

a ) =Ceff(a, b)

c(a)=

4

7.0A^0A8w::\w?.eKw' & S ?ez (S, E) ?rwT1*w.AwS* c ? E wD5eRF*wbA S weM Markov X . Sn ?|'t S w*^ Bn := Sc

n, a ∈ S1. 6wEP

a(τBn< τ+

a ) =Ceff(a, Bn)

c(a).q n → +∞ 3 τBn

↑ +∞. r\ Pa(τ+

a = ∞) = 1c(a) limn Ceff(a, Bn). D* limn Ceff(a, Bn) ? a t ∞ ws Ceff(a,∞). r X >qwFq

Pa(τ+

a < ∞) = 1, S|Q X >qwFq Ceff(a,∞) = 0, a ws> 5.3.3 & X(1) ?:+4R?w+m^Æww#?4R?#.#ws? 1. Sn := −n,−n + 1, · · · , n − 1, n, Bn := Scn. F*

Ceff(0, Bn) = 2n , r\ Ceff(0,∞) = 0. S X(1) > 5.3.2 CbAz G = (S, E). & A, B ? S w#j^w A ∩ B = ∅. EwbADeR j D? A t B w` l x 6∈ A ∪ B,

∑

(x,y)∈E

j(x, y) = 0.yXzZ6∑

x∈A

∑

y∼x

j(x, y) =∑

x∈B

∑

y∼x

j(y, x) = 0,r\ j(A, B) :=∑

x∈A

∑

y∼x j(x, y) D? A t B wU j ? A t B wbAm ?> j(A, B) = 1. Cz G wf r, l A t B w j

gy Markov o 99m j <*wKE (j) = EA,B(j) :=

1

2

∑

(x,y)∈E

j(x, y)2r(x, y)."6169 Thompson < 5.3.5 (Thompson) & i? A"Q v > 0, B :z3<*wm F* _ A t B wm H i <*wKh6J& j ? A t B wm i′ := j − i. r i, j ?,swS i′? A t B wbAw\KE (j) =

1

2

∑

(x,y)∈E

j(x, y)2r(x, y)

= E (i) + E (i′) +∑

(x,y)∈E

i(x, y)i′(x, y)r(x, y).hvb1?r i ? Ohm wm v A ?>R v, B ?x=5v∑

(x,y)∈E

i(x, y)i′(x, y)r(x, y) =∑

(x,y)∈E

(v(x) − v(y))i′(x, y)

=∑

(x,y)∈E

v(x)i′(x, y) −∑

(x,y)∈E

v(y))i′(x, y)

= 2∑

(x,y)∈E

v(x)i′(x, y)

= 2∑

x∈S

v(x)∑

y∼x

i′(x, y)

= 2∑

x∈A∪B

v(x)∑

y∼x

i′(x, y)

= 2v · i′(A, B) = 0,r\ E (j) = E (i) + E (i′) ≥ E (i), <*wKh66w69aGeiaQ` i ?0[wbAm F*E (i) = v · i(A, B) = Reff(A, B),m <*wKx A, B *w8fvzU"6> E

gy Markov o 100oK 5.3.1 ` i ? A t B wm F*Reff(A, B) =

1

2

∑

(x,y)∈E

i(x, y)2r(x, y).\jz|Q"6w Rayleigh m< 5.3.6 (Rayleigh) 8fUfeR?|'wJ& r, r′ ? G AfeR r′ ≥ r ?> l (x, y) ∈ E, r′(x, y) ≥r(x, y). Reff(A, B) R′

eff(A, B) $*); A, B *wfeR r r′ w8f i i′ );wwm E E ′ wwKF* ThompsonR′

eff(A, B) = E′(i′) ≥ E (i′) ≥ E (i) = Reff(A, B),dHbA ≥ h?r r′ ≥ r, Axh?r# (G, r), i ? i′ ? G l x, y w+D?bA, G

x, y *:bAfw+D?bAÆ.!aEf r(x, y) ↑ +∞,r\feR?'"wÆ.!aE r(x, y) ↓ 0, r\feR,!0[ Rayleighm|Qv8f'"Æv8f,!r Markov X !qwFq8s Ceff(a,∞) > 0, x%8fReff(a,∞) < ∞, %\8fÛfeR|'S"6w9< 5.3.7 !w Markov Æv?!w>w Markov v?>wZ~UÀÆvw>F*uw^>Àvw!F*uw^!` G?*#F*kmw?.a e ?,kmwp? eim# E wDeR j D?bA a twm ` l x ∈ S, ∑

y∼x |j(x, y)| < ∞w ∑

y∼x j(x, y) = 1a(x). m j <*wKE (j) :=

1

2

∑

(x,y)∈E

j(x, y)2r(x, y).< 5.3.8 (T. Lyons, 1983) #w Markov !qwFqa bA^ k a tVwK*wm

gy Markov o 101Js[ Sn ?|'t S w*^ Bn := Scn, a ∈ S1. in ); a k Bn *"Q<*w^ a wm F* Reff(a, Bn) = E (in),r\ Markov !qwFq supn E (in) < ∞. ` j ?bA^ k a tVwK*wm F* j ?^ a t Bn wm Thompson E (in) ≤ E (j) < ∞. 9 sup E (in) < ∞, %aQ l# (x, y) ∈ E,

in(x, y) : n ≥ 1 E ?er\a ^ kn8v ikn(x, y) : n ≥ 1 .n#F j(x, y) := limn ikn

(x, y), F* j ? a twm w Fatou t E (j) ≤ supn E (in).1 T. Lyons Polya wbAZ69>Q0b=> 5.3.4 Polya Zd +m^Æq d ≤ 2 >q d ≥ 3 3!9E d = 2, 3 wyp69` d = 2, E69AÆvw?>RO l n ≥ 0, ? (x, y) : |x| ∨ |y| = n ÆaEbA an, r.n#ws? 1, an 4(2n + 1) n#: an+1, S c(an, an+1) = 4(2n + 1),^ a0 tw8fReff(a0,∞) =

∑

n≥0

1

4(2n + 1)= +∞,r\Ævw?>w& d = 3, l# (x, y), axs R3 wbA x >3 y wm 3 vxy);>3 (x, y) Hw3 Sxy );H< vxy, Z: (x, y) w# l(O_Bwm 5 Am j(x, y) x Sxy H<wm |6N3uw6+/k3 (x, y) vxy J,sw,h+mwlV:U9q

x 6= 0 3∑

y∼x j(x, y) = 0. r\ j ?twbA%a bA>R A, 8v |j(x, y)| ≤ A · |vxy|−2, dH | · | );3?~a bA>R B8vtwYz n n + 1 *w#wR.Aa Bn2. r\ j <*wKE (j) =

1

2

∑

(x,y)

j(x, y)2 ≤∑

n

A2n−4Bn2 < ∞, j ?K*wSww?!wx l1. & a 6∈ B. ξ ^ a Q tb B rt a wR ξ w$/

gy Markov o 102

2. & x, y 6∈ B, g(x, y) ?^ x Q tb B rt y w__R69c(x)g(x, y) = c(y)g(y, x).

3. bX 0, 1n wA5 kw+m69 Reff(a, b) ?Yzd(a, b) w|'eR

4. CbA# G, r, X ?ww Markov a ∈ S, Z ⊂ S, a 6∈ Z. l x, y ∈ S, x ∼ y, Sxy ); X Xt Z 9r^ x Udt y wRSxy :=

∑

n≥0 1Xn=x,Zn+1=y,n<τ, dH τ ? Z wHH369` a kZ *"QAEbA a t Z wm i, F* E

a(Sxy − Syx) = i(x, y).

5. 69 (1) a ∈ S, 8sw* limn→∞ Ceff(a, Scn) |'t S w^ Sn wKU (2) 8f Reff(a, +∞ w*C a wKU

6. Kirchhoff Ohm 69X+ >Awf\I:7. a ∈ S, # E w^ Π D a ∞ w`` Π v a _ w$7?*z69 (Nash-Williams X%) & Πn ?y.,6w a

∞ w`F*Reff(a,∞) ≥

∑

n

∑

(x,y)∈Πn

c(x, y)

−1

.

8. Tk );.A k Aviw5%P69q k ≥ 2 3 Tk w+m^Æ?!w9. 69_6w7A (() ?#k7A#w+m^Æ?>w

10. l r > 0, M#bAT1*w*# a 8v.A#wf? 1 w Reff(a,∞) = r.

G Levy $r6B"w^ÆaG?z3*w^ÆaG^R?^ÆIaw>M9+m5awe9C9+mr_$ ?e\p?J3*w^ÆaG?.b[waw ^ÆI_w qew.s12 v6B"yLwJ3*w^ÆaG Poisson aG Brown & (Ω, F , P) ?8j* X = (Xt : t ≥ 0) D?bA d- ^ÆaG` l t ≥ 0, Xt ?bA d- ^Æ&q d = 1 3+D5=^ÆaG§6.1 St?D%h+mw^ÆaG?_'aGa?^Æ&kw|Z&

ξn : n ≥ 1?s$/^ÆIX0 = 0, Xn = ξ1 + · · ·+ ξn. F* X = (Xn :

n ≥ 0) (d(1) k n > m ≥ 0, ' Xn − Xm X1, · · · , Xm;(2) k n > m ≥ 0, ' Xn − Xm Xn−m − X0 s$/dHwbACFD'CACFD_'Cnhu XD?_'Ir\J3*^ÆaG1"6_wm 6.1.1 d- ^ÆaG X = (Xt : t ≥ 0) D?_'aG`a(d(1) l t > s ≥ 0, Xt − Xs Fs := σ(Xu : u ≤ s);

(2) l t > s ≥ 0, Xt − Xs Xt−s − X0 s$/./bC_'aG1b?& X0 = 0, R?aGb?^0 Q_'aG^Dj*gaGa?hLwby^ÆaGdH M_N8w Poisson aG Brown < 6.1.1 & X = (Xt : t ≥ 0) d- _'aG k t ≥ 0, mµt Xt w$/F* Rd w89e µt : t ≥ 0 (d l t, s ≥ 0,

µt+s = µt ∗ µs. (6.1.1)

103

gp Levy le 104

Proof. r Xt+s = (Xt+s − Xs) + Xs, _'C#A^Æ&w Xt+s − Xs w$/R? µt, S µt+s = µt ∗ µs.< 6.1.2 & µt : t ≥ 0 ? Rd w89e(d l t, s ≥ 0, µt+s =

µt ∗ µs, F*a 8j* (Ω, F , P) idw d- _'aG X =

(Xt : t ≥ 0) 8v l t ≥ 0, Xt w$/? µt.

Proof. 69%?E8 Kolmogorovw,CH# lw8$/ µ, x ∈ Rd, B ∈ B(Rd), m p(x, B) = µ(B − x), F* p ?bAjT x, p U B ?bA89T B, p U x ?e9eRr\0 < t1 < t2 < · · · < tn, mFj* (Rd)n w89 µt1,··· ,tn

µt1,··· ,tn

(A) (6.1.2)

:=

∫

(Rd)n

1A(x1, · · · , xn)µt1(dx1)µt2−t1(dx2 − x1) · · ·µtn−tn−1(dxn − xn−1)

=

∫

(Rd)n

1A(x1, x1 + x2, · · · , x1 + · · · + xn)µt1(dx1)µt2−t1(dx2) · · ·µtn−tn−1(dxn),dH A ? (Rd)n w k Borel µt wC.HZ689eµt1,··· ,tn

: n ≥ 1, 0 < t1 < · · · < tn? Rd w,w*$/er\ Kolmogorov ,a 8j*(Ω, F , P) d^ÆaG X = (Xt : t ≥ 0), 8v l n ≥ 1, 0 < t1 < · · · < tn,

(Rd)n w Borel A P((Xt1 , · · · , Xtn

) ∈ A) (6.1.3)

=

∫

(Rd)n

1A(x1, · · · , xn)µt1(dx1)µt2−t1(dx2 − x1) · · ·µtn−tn−1(dxn − xn−1)

(6.1.4)F* (Xt1 , Xt2 − Xt1 , · · · , Xtn− Xtn−1) wn$/

P((Xt1 , Xt2 − Xt1 , · · · , Xtn− Xtn−1) ∈ A)

=

∫

(Rd)n

1A(x1, · · · , xn)µt1(dx1)µt2−t1(dx2) · · ·µtn−tn−1(dxn),^\:eijz|Q X ?_'aGw Xt w$/? µt.^6AteiaQ_'aGxsbA(d (6.1.1) w89e

gp Levy le 105

§6.2 Levy-Khinchin ybC Rd 0[bA89e µt, µt ); µt wh3eRF*Cih3eRwCFvµt+s(x) = µt(x) · µs(x), x ∈ Rd. (6.2.1)0>[1 t 7→ µt(x) >ReRwFCqa ?>ReR+mwRL$86eiv8q µt U t JCn/3a ?bA>ReR 6.2.1 bA_'aG X = (Xt) D Levy aG`q t ↓ 0 3

Xt 8Fr Xt 8Fx% $/Fr\_'aGX? Levy aGqwFqdww9e µt (dq t ↓ 0 3 µt Fwm9 δ0. 6.2.2 Rd bA89e µt : t ≥ 0D?[`a(d (6.1.1)wq t ↓ 0 3 µt F δ0, l Rd JeR f µt(f) = f(0).' & µt ?bA[F*q t ↓ 0 3 µt → 1. Sa eR φ(x)8vµt(x) = exp(−tφ(x)), x ∈ Rd, (6.2.2)rh3eR?JwS φ ? Rd w0R=JeRr |µt| ≤ 1, φ w51?#3weR φD[ µt wwaGw Levy>Rh3eRbC Levy>Rb[ µt. F*bA_wjR? Levy>R4*[w);A:#aA^> 6.2.1 bA^?L~w Rd Ws G

∂u

∂t=

1

2∆uw ?

pt(x) =

(

1√2πt

)d

exp

(

−|x|22t

)

, t > 0, x ∈ Rd.a? ; t w5d$/w4eRm89µt(dx) = pt(x)dx.

gp Levy le 106F* µt : t ≥ 0 ?[=5EZ6 l t, s > 0, pt+s = pt ∗ ps.0R?Uw ;$* t, s w5d$/^Æ&wk? ; t + s w^Æ&05d$/w*C|QvM_N8µt(x) = pt(x) = e−t |x|2

2 ,r\w Levy >R? φ(x) = |x|22 .A^w:`^?Nw Poisson& λ > 0, t > 0, µt ? λtw Poisson$/ Poisson$/w*Cv l t, s > 0 µt∗µs = µt+s, µt ?[D Poissondh3eR µt(x) = exp(−tλ(1− eix)), r\d Levy >R φ(x) = λ(1 − eix).> 6.2.2 (0n Poisson ) & J ? Rd wbA89 λ > 0 ?bA>R t ≥ 0,

µt := e−λt∞∑

n=0

(λt)n

n!J∗n,dH J∗n ? n- L[ J∗0 := ǫ0, jZ6 µt ?8w t, s ≥ 0, µt+s =

µt ∗ µs, f ∈ Cb(Rd), kEF

limt↓0

µt(f) = limt↓0

e−λt(f(0) +∞∑

n=1

(λt)n

n!J∗n(f)) = f(0).r\ µt ?bA[D?0n Poisson λ DaGwt JD?$/ µt wh3eR

µt = e−λteλtJ = e−λ(1−J)t,dH J ? J w Fourier &r\0n Poisson w Levy >R? λ(1 − J).5 lbA[w Levy >RR? Brown 0n Poisson A1$gEw"61R d = 1 3uCQaw);jZ6A[w[?[r\ Levy >Rwk^? Levy >R1te$$/a5?[~bA:[ Rd w$/ µ D?e$w` l n ≥ 1, a $/ µn 8v µ = µ∗nn . ww$/eRDe$$/eRwwh3eRD?e$h3eR%bAh3eR φ e$qwFq l n a h3eR φn 8v φ = φn

n. %`πt ?[XC% l πt ?e$$/#CQbAtaeiPxw.x: 1 − cos 2α ≤ 4(1 − cosα) |QC/k

gp Levy le 107< 6.2.1 & f ?bAh3eRw51 u, % ξ ∈ R,

1 − u(2ξ) ≤ 4(1 − u(ξ))."6w?Lw< 6.2.1 & fn ?h3eR%eR fnn bAJ* ( f) qwFq n(1 − fn) J* ( φ). 03u f = e−φ.J#& n(1− fn) bAJ* φ, % fn −→ 1 w **bCF* l a > 0, L$ew n, |1− fn| < r < 1 [−a, a]Er\ Taylor(` |ξ| ≤ a,

n log fn(ξ) = n log(1 − (1 − fn(ξ)))

= −n(1 − fn(ξ)) − n

2(1 − fn(ξ))2 − · · · ,#b1w* −φ(ξ), dwbCC aw kC|Q n log fn F −φ, fn

n → e−φ.9& fnn bAJ* f , 1#6 f .Krh3eR |fn|2nw*? |f |2, S.!& fn, f ?5w#3wr f Jw fn

n l**bCFSa a > 0, f [−a, a] R?5F*q n L$e3 *[−a, a] fn

n 5w bA5Yzr\ −n log fn, s3 n(1− fn), [−a, a]bCt 6.2.1 |Q n(1 − fn) [−2a, 2a] bC^OTaylor (`|Q −n log fn [−2a, 2a] bCr\ f [−2a, 2a] R?5L00AaG69 f R R?5' R w l** I fn

n bCFJw f . r f R?5Sn log fn I bCF log f . 0 [ fn F 1 w l**bC6w Taylor (`|Q

−n log fn(ξ) = n(1 − fn(ξ))(1 + ∆n(ξ)),dH ∆n(ξ) = 12 (1 − fn(ξ)) + 1

3 (1 − fn(ξ))2 + · · · −→ 0, r\ n(1 − fn(ξ)) →− log f .0AGw| H#` f ?e$h3eR% f = fn

n ,r\ f = limn en(fn−1), f ?0n Poisson h3eRw*~` f ?A fnn wJ*% l t ≥ 0, 0n Poisson h3eR exp(tn(fn − 1))

gp Levy le 108J* f t, f t ?h3eR|Q f ?e$ww` f t ww$/ πt,F* πt ?[oK 6.2.1 "6w> E(1) h3eR f e$qwFqa h3eR fn 8v fn

n F f .

(2) h3eR?e$wqwFqa?0n Poisson h3eRw*(3) e$h3eRwJ*?e$w(4) le$$/ µ, a bw[ πt 8v π1 = µ.?SP Levy>Rw); SPe$h3eRw);?x%w"6?Q:w Levy-Khinchin I:< 6.2.2 Rd 0=eR φ ?bA[w Levy >RqwFq φ e);

φ(x) = i(a, x) +1

2(Sx, x) +

∫

Rd\0

(

1 − ei(x,y) +i(x, y)

1 + |y|2)

J(dy), (6.2.3)dH a ∈ Rd, S ? Rd D#3+C\^ J ? Rd\0 9w(d∫ |x|2

1 + |x|2 J(dx) < ∞,D[w Levy 9~ S, J φ bJHO+m1.!& d = 1. #& φ O);% φ ?Jww l n > 1, J x : |x| > 1n w*E?*90[ φ e:

φ(x) = limn

[

∫

|y|> 1n

(1 − eixy)J(dy) + ix ·∫

|y|> 1n

y

1 + y2J(dy)

]

+ iax +1

2Sx2.# shJ?bA0n Poisson Levy >RkbAbC_d Levy >RwkS? Levy >R6w| (3), φ ^? Levy >R"669 φ b S, J . m

ν(dy) :=y2

1 + y2J(dy)w ν(0) := S, F* ν ?bA*9w

φ(x) = iax +

∫

R

(

1 − eixy +ixy

1 + y2

)1 + y2

y2ν(dy),

gp Levy le 109dH6w$eR y = 0 V?eJwm\Vd*= x2

2 8eRJ:Vz` φ ;e%;sR? (1 + x2)ν(dx) w Fourier &bz ν. bz1m φ w;;$θ(x) :=

∫ 1

0

[φ(x) − 1

2(φ(x + h) + φ(x − h))]dh

= −∫ 1

0

∫

R

eixy(1 − coshy)1 + y2

y2ν(dy)dh

= −∫

R

eixy 1 + y2

y2ν(dy)

∫ 1

0

(1 − coshy)dh

= −∫

R

eixy 1 + y2

y2(1 − sin y

y)ν(dy), k(y) := 1+y2

y2 (1 − sin yy ), %a c > 0 8v c ≤ k(y) ≤ 1

c . r\ k · ν ?*9wd Fourier&? θ, |Qa θ bS ν θ ^ φ b' & φ? Levy>R% φ?Jww 6.2.1,a h3eR fn,w$/ µn, 8v n(1 − fn) → φ. %φ(x) = lim

nn

∫

R

(1 − eixy)µn(dy)

= limn

n

∫

R

(

1 − eixy +ixy

1 + y2

)

µn(dy) − ix · n∫

R

y

1 + y2µn(dy).

an := −n

∫

R

y

1 + y2µn(dy), νn(dy) :=

ny2

1 + y2µn(dy), φn := n(1 − fn).1;;$

θn(x) :=

∫ 1

0

[φn(x) − 1

2(φn(x + h) + φn(x − h))]dh = −

∫

R

eixyk(y)νn(dy).r φn J*S θn ^J* θ. Fourier&wJCk ·νnF|Q νn FbA*9 ν, F*∫

R

(

1 − eixy +ixy

1 + y2

)1 + y2

y2νn(dy) −→

∫

R

(

1 − eixy +ixy

1 + y2

)1 + y2

y2ν(dy),s3|Q an F5R a.Hw Levy 9_(dwn/x%A (r\_) ǫ > 0,

∫

|x|<1

|x|2J(dx) < ∞, J(x : |x| > ǫ) < ∞.

gp Levy le 110r\ Levy >Rw)b:wA:r>^:Eφ(x) = i(a, x) +

1

2(Sx, x) +

∫

Rd\0

(

1 − ei(x,y) + i(x, y)1|y|<1)

J(dy), (6.2.4)r.HZ6∫∣

∣

∣

∣

i(x, y)

[

1

1 + |y|2 − 1|y|<1

]∣

∣

∣

∣

J(dy) < ∞. 0A)b:H a w=eK?.swqeiGd`wx%)b:^U9 Levy aGw_=s d Levy >R§6.3 Poisson %+mzU Poisson ww_'aG? Poisson aG 6.3.1 ^ÆaG N = (Nt) D?7R λ w Poisson aG`

(1) N0 = 0 a.s.;

(2) N ?_'aGw Nt ?7R λt w Poisson $/(3) N ?Jwp?bw_'aG,JCp?1ei69mH0[w Poisson aG?a w& Tn : n ≥ 1 ?#3^ÆId$/eR F4eR f . Sn ?d1$kaG

S0 = 0, Sn =n∑

k=1

Tk.mJ3*4R=^ÆaG N = (Nt : t ≥ 0) " ω ∈ Ω,

Nt(ω) :=∞∑

n=0

n1t∈[Sn(ω),Sn+1(ω)), t ≥ 0. (6.3.1)F* l t ≥ 0, Nt ?4R=^Æ& l ω ∈ Ω, t 7→ Nt(ω) ? [0,∞) 4R=w|'eR^ÆaG N = (Nt) DE;aG Sn aE n AE; N eiaEE;wa [Sn, Sn+1) w=? n. ` Tn?A-^bEbA-_Ew3* SnR?E n A-w3* Nt R? t 3f6^bgLE-wRC0[wME;aGr>T=4*n/"E;aG?'aGI

gp Levy le 111< 6.3.1 `E;aG N = (Nt)?_'aG% Tn ?>R$/wau` Tn ?7R λ > 0 w>R$/% N ?_'aGw Nt?7R λ w Poisson $/Proof. &E;aG N ?_'wF* t, s > 0,

P(Nt+s = 0) = P(Nt+s − Ns = 0, Ns = 0)

= P(Nt+s − Ns = 0)P(Ns = 0)

= P(Nt = 0)P(Ns = 0).r Nt = 0 qwFq T1 = S1 > t, r\P(T1 > t + s) = P(T1 > t)P(T1 > s),v T1 ?>R$/wau T1 ?>R$/% Sk ? Gamma $/w l t > 0, #34R

k, P(Nt = k) = P(Sk ≤ t < Sk+1) = P(Sk ≤ t < Sk + Tk+1)

=

∫ t

0

λk

(k − 1)!xk−1e−λxdx

∫ ∞

t−x

λe−λydy

=λk

(k − 1)!

∫ t

0

xk−1e−λxe−λ(t−x)dx

=(λt)k

k!e−λt."6169 P(Nt+s − Ns = k, Ns = n) = P(Nt = k)P(Ns = n), bw=nC/kjrE;aG?Jw6wt Poisson aG5?*3*?>R$/wE;aG PoissonEbw?0n PoissonaG_z0n

Poissonww_'aG?0n PoissonaG0n PoissonaG? Poisson aGuM#waw:VV/?0[wbAF A`9 xlbA7R λw>R$/w3*v-ATw$/ j*HK$bAoavxlb3*\00[wdD0n Poisson aG Kolmogorov a 8j* (Ω, F , P), 0A8j* (1) bA7R λ w>R$/ws$/^ÆI Tn : n ≥ 1; (2) = Rd w$

gp Levy le 112/ J ws$/^ÆI ξn : n ≥ 1; (3) I Tn I ξn &N = (Nt : t ≥ 0) ? Tn sw7R λ w Poisson aGa ξn. vm

Xt =

Nt∑

n=1

ξn, t ≥ 0. (6.3.2)

Proposition 6.3.1 X = (Xt : t ≥ 0) ? d- _'aGwd[?0n Poisson µt =

∞∑

n=0

eλt tnJ∗n

n!,dH J∗n ?[w n- 5 J∗0 = δ0.

V Brown `RL8POtw~F* Brown^wOta?*L! R.BrownH#iQuSPwbK'4z% ak)6wv7hewL!9b A. Einstein SPWs'4w3uCQdUd4eRhJm4wRL!kE Y9 N.Wiener 69 duwJC69 a?z%wbAnqwRL=q Brown ^qdOta^?8 HhLvhwbA^ÆaG§7.1 Brown ! 7.1.1 d- ^ÆaG B = (Bt : t ≥ 0) D d- (X Brown `

(1) B0 = 0 a.s.;

(2) B ?_'aGw Bt w[?(3) B ?Jwa 8 N , 8v l ω 6∈ N , t 7→ Bt(ω) ?Jwt 6.1.2 a bA(d (1) (2) w d- ^ÆaG?jwU.?69s3(d (3) w^ÆaGwa C0? Wiener wK(< 7.1.1 (N. Wiener) R

d a (X Brown Proof. 1& d = 1, =w69?yXw H#1w Kolmogorov ,CM#bA8j* (Ω, F , P) dw^ÆaGaG X = (Xt) 8vaw*$/? (??) w#CQ jZ6 (Xt)t≥0 (d BrownmHSJC~wdan/hLw?"6wUx:

E|Bt − Bs|2n = (2n − 1)!!|t − s|n. (7.1.1)r\1ED7 Xt 8vaJ& D = j2n : j ∈ Z

+, n ∈ N #3$k% D ? R+ weRO^m

H =

∞⋃

N=1

∞⋂

l=1

∞⋃

n=l

N2n

⋃

j=1

(

∣

∣

∣X j

2nX j−1

2n

∣

∣

∣ ≥ 1

2n/8

)

.

113

gt Brown i 114T54R N , Al =

∞⋃

n=l

N2n

⋃

j=1

(

∣

∣

∣X j

2nX j−1

2n

∣

∣

∣≥ 1

2n/8

)

.1269 ⋂∞l=1 Al w8r\kde+ P(H) = 0.wUx: (7.1.1) H n = 2 w=n

P

N2n

⋃

j=1

(

∣

∣

∣X j

2n− X j−1

2n

∣

∣

∣ ≥ 1

2n/8

)

≤N2n

∑

j=1

P

(

∣

∣

∣X j

2n− X j−1

2n

∣

∣

∣ ≥ 1

2n/8

)

= N2nP

(

∣

∣

∣X 12n

∣

∣

∣ ≥ 1

2n/8

)

≤ N2n(

2n/8)4

E

∣

∣

∣X 12n

∣

∣

∣

4

=(

2n/8)4

N2n3

(

1

2n

)2

=3N

2n/2,S

P(Al) ≤∞∑

n=l

P

N2n

⋃

j=1

(

∣

∣

∣X j

2nX j−1

2n

∣

∣

∣ ≥ 1

2n/8

)

≤ 3N

∞∑

n=l

1

2n/2

=3N

√2√

2 − 1

1(√

2)l

.r\P

(∞⋂

l=1

Al

)

= liml→∞

P Al

≤ 3N√

2√2 − 1

liml→∞

1(√

2)l

= 0 .

gt Brown i 115\|Q P(H) = 0, P(Hc) = 1. b 6 De Morgan Hc =

∞⋂

N=1

∞⋃

l=1

∞⋂

n=l

N2n

⋂

j=1

ω :∣

∣

∣X j

2n(ω) X j−1

2n(ω)∣

∣

∣<

1

2n/8

r\ ω ∈ Hc, % l N , a l 8v l n > l j = 1, · · · , N2n ∣

∣

∣X j

2n(ω) − X j−1

2n(ω)∣

∣

∣ <1

2n/8.1r\eikbA$j69 l ω ∈ Hc, Xt(ω) l*bCJS l t ≥ 0, q s W[ D ~ t 3 Xs(ω) w*a D

[0,∞) HOr\ l t ∈ [0,∞) 1eimBt(ω) = lim

s∈D→tXs(ω) if ω ∈ Hc ω ∈ H , m Bt(ω) = 0. m (Bt)t≥0 Jw l t ≥ 0, Xt wVVF Bt. -"w=yR?69 (Bt)t≥0 ? X wD5r\a? Brown kj1EbA$>`i69 (Xt(ω) : t ∈ D ∩ [0, n]) l n bCJEV 7.1.1 & α > 0 w f ? D weR(d l N , a l 8v l n > l j = 1, · · · , N2n

∣

∣

∣

∣

f(j

2n) − f(

j − 1

2n)

∣

∣

∣

∣

≤(

1

2n

)α

.F* l n > 0, a >R Cn 8v l s, t ∈ D ∩ [0, n],

|f(s) − f(t)| ≤ Cn|s − t|α, f ? α- ;T1 Holder Jw§7.2 ^=

Brown ^bAq3`9? Brown d Markov C q3VEr\ Brown Xt MarkovC0?bA#>LwCFH#?Paul Levy i#wA:"iw#? Markov CFEQxwbA8P12w0Au\ Brown weGw$/

gt Brown i 116 Gw5Hh* tH1EP^ÆaGweGw$/ Brown B = (Bt)t≥0 uUeG sups∈[0,t] Bs w$/ei#w usQ& B = (Bt)t≥0 8j* (Ω, F , Ft, P) b(X Brown & b > 0 b > a, +&Tb = inft > 0 : Bt = b .F* Tb ?q3w Brown ^tb b `9v5R?(X Brown " b, r\

P

sups∈[0,t]

Bs ≥ b, Bt ≤ a

= P

sups∈[0,t]

Bs ≥ b, Bt ≥ 2b − a

= P Bt ≥ 2b − adHbAx:R? Brown Tb 9v (^ D b `9 BTb= b) WvRX Brown _iaU y = b ?DwRs(X Brown UDAx:R?0DCw| ^R?_Æw# 6w Gei:E

P Tb ≤ t, Bt ≤ a = P Tb ≤ t, Bt ≥ 2b − a (7.2.1)

= P Bt ≥ 2b − a ,U#wR?69t Brown wt Markov C< 7.2.1 l (F 0t+)- q3 T , (Bt+T − BT : t ≥ 0) ?bA F 0

T+ w(X Brown /U l Borel A, P(Bt+T − BT ∈ A|FT ) = P(Bt ∈ A) a.s. on T < ∞. (7.2.2)

Proof. & T (n) ? T wzT (n) =

∑

k≥1

k

2n1k−1

2n ≤T< k2n .F* l H ∈ F 0

T+, H ∩ T < t ∈ Ft, r\ l n ≥ 1,

P(Bt+T (n) −BT (n) ∈ A; H, T < ∞)

gt Brown i 117

=∑

k≥1

P(Bt+k/2n − Bk/2n ∈ A; H ∩ T (n) = k/2n)

=∑

k≥1

P(Bt+k/2n − Bk/2n ∈ A)P(H ∩ T (n) = k/2n)

=∑

k≥1

P(Bt ∈ A)P(H ∩ T (n) = k/2n)

= P(Bt ∈ A)P(T < ∞) .Ax:E?r (Bt+s −Bs : t ≥ 0) ? F 0s w(X Brown

n ~ Brown JC|Q (7.2.2).169# (7.2.1)

P(Tb ≤ t, Bt ≤ a) = P(Tb ≤ t, B(t−Tb)+Tb− BTb

≤ a − b)

= P(Tb ≤ t)P(B(t−Tb)+Tb− BTb

≤ a − b)

= P(Tb ≤ t)P(B(t−Tb)+Tb− BTb

≥ −(a − b))

= P(Tb ≤ t, Bt ≥ 2b − a).Ax:u_(X Brown wDC# (7.2.1) evP

sups∈[0,t]

Bs ≥ b, Bt ≤ a

=1√2πt

∫ +∞

2b−a

e−x2

2t dt ,aCQ Brown deGwn$/8u41 a b s< 7.2.2 & B = (Bt)t≥0 ?b(X Brown t > 0. F*^Æ&(Mt = sups∈[0,t] Bs, Bt) wn4eR

P Mt ∈ db, Bt ∈ da =2(2b − a)√

2πt3exp

− (2b − a)2

2t

dadb,dH (b, a) R2 w (b, a) : a ≤ b, b > 0 Hh*z l b > 0,

P

sups∈[0,t]

Bs ≥ b

= P Tb ≤ t

gt Brown i 118

=2√2πt3

∫ ∫

a≤c,c≥b(2c − a) exp

− (2c − a)2

2t

dadc

=2√2πt3

∫ +∞

b

∫ c

−∞(2c − a) exp

− (2c − a)2

2t

dadc

=2√2πt3

∫ +∞

b

∫ +∞

c

x exp

−x2

2t

dxdc

=2√2πt

∫ +∞

b

exp

(

−x2

2t

)

dxaCQeG sups∈[0,t] Bs w$/q3 Tb) w$/EV 7.2.1 69E(

e−sTb)

= e−√

2sb.

§7.3 '~$& B = (Bit)t≥0 (i = 1, · · · , d)? R

d (X Brownd_ (Ft)t≥0.

Proposition 7.3.1 1) l p > 0, .A Bt ? p- ;eww t > s

E(|Bt − Bs|p) = cp,d|t − s|p/2 . (7.3.1)

2) (Bt)t≥0 J_ er3) l i, j, Mt = Bi

tBjt − δijt ^?Jr

Proof. b1$ir69aq t > s 3r Bt −Bs Fs. 1r\E(Bt − Bs|Fs) = E(Bt − Bs) = 0.F*

E(Bt|Fs) = E(Bs|Fs) = Bs^R?U (Bt)t≥0 ?Jr%1Eb Brown 69 3). 0KyA"E(B2

t − B2s |Fs) = E((Bt − Bs)

2 |Fs)

+ E(2Bs (Bt − Bs) |Fs)

= E((Bt − Bs)2) + 2BsE((Bt − Bs) |Fs)

gt Brown i 119

= E (Bt − Bs)2

= t − sSE(B2

t − t|F 0s ) = E(B2

s − s|Fs)

= B2s − s0gL69 B2

t − t ?bAr< 7.3.2 & B = (Bt)t≥0 ? R J^ÆaG8v B0 = 0. F* (Bt)t≥0 ?b(X Brown qwFq l ξ ∈ R t > s

E

exp(√

−1〈ξ, Bt − Bs〉)

|Fs

= exp

(

− (t − s)|ξ|22

)

. (7.3.2)

Proof. 1V:t (7.3.2) [ Bt − Bs Fs w? ; t − s w5d$/au` (Bt)t≥0 ?b(X Brown F* Bt −Bs Fs, wBt − Bs ? ; t − s w5d$/S

E

exp(√

−1〈ξ, Bt − Bs〉)

|Fs

= E

exp(√

−1〈ξ, Bt − Bs〉)

=1

√

2π(t − s)

∫

R

e√−1〈ξ,x〉− |x|2

2(t−s) dx

= exp

(

− (t − s)|ξ|22

)

.oK 7.3.1 & (Bt) ? Rd (X Brown ` ξ ∈ R

d, F*Mt ≡ exp

(√−1〈ξ, Bt〉 +

|ξ|22

t

)?r^R 7.3.1 Tkt (7.3.2) w#? ξ w?eRSx: l0= ξ ^Eh*z −√−1ξ il ξ, 1v

E exp (〈ξ, Bt − Bs〉) |Fs = exp

(

(t − s)|ξ|22

)

.

gt Brown i 120r\ l3 ξ

exp

(

〈ξ, Bt〉 −|ξ|22

t

)?bAJr0A> ^ei|Zt Rd w3= ξ = (ξ(t)), \vtwx:D Cameron-Martin I:§7.4 x~s1atwF[b Brown Bt Mt ≡ B2

t − t ?rF*B2

t = Mt + AtdH At = t. r\J"r B2t ?bAr bA'waGwk12at0A$?? Ito ^Æ$ vi1wU.< 7.4.1 &

D = 0 = t0 < t1 < · · · < tn = t?* [0, t] w*|$w&VD =

n∑

l=1

|Btl− Btl−1

|2D B $| D w&;a?#3^Æ&F*EVD = tw VD w ;

E

(VD − EVD)2

= 2

n∑

l=1

(tl − tl−1)2 .

Proof. =5EVD =

n∑

l=1

E|Btl− Btl−1

|2

=n∑

l=1

(tl − tl−1)

= t .

gt Brown i 122< 7.4.1 & B = (Bt)t≥0 ?b(X Brown F* l t > 0,

limm(D)→0

∑

l

|Btl− Btl−1

|2 = t in L2(Ω, P)dH D ?* [0, t] w*$|wm(D) = max

l|tl − tl−1| .r\

limm(D)→0

∑

l

|Btl− Btl−1

|2 = t in probability.

Proof. r6wt1E

∣

∣

∣

∣

∣

∑

l

|Btl− Btl−1

|2 − t

∣

∣

∣

∣

∣

2

= E |VD − E (VD)|2

= 2

n∑

l=1

(tl − tl−1)2

≤ 2m(D)

n∑

l=1

(tl − tl−1)

= 2tm(D),r\lim

m(D)→0E

∣

∣

∣

∣

∣

∑

l

|Btl− Btl−1

|2 − t

∣

∣

∣

∣

∣

2

= 0 .6wF? 8FpHw$|vEfw~6wFei&E?wVVwProposition 7.4.2 & (Bt)t≥0 ?b(X Brown F* l t > 0, q n~3

2n

∑

j=1

∣

∣

∣B j

2n t − B j−12n t

∣

∣

∣

2

→ t a.s. (7.4.1)

Proof. & Dn ? [0, t] w$$|Dn = 0 =

0

2nt <

1

2nt < · · · <

2n

2nt = t .

gt Brown i 123w Vn ); VDn. F*t Lemma 7.4.1, EVn = t and

E |Vn − EVn|2 = 2

2n

∑

l=1

(

l

2nt − l − 1

2nt

)2

= 2n+1

(

1

2nt

)2

=1

2n−1t2 .r\ Markov .x:

P

|Vn − EVn| ≥1

n

≤ n2E |Vn − EVn|2

=n2

2n−1t2S

∞∑

n=1

P

|Vn − EVn| ≥1

n

= t2∞∑

n=1

n2

2n−1< +∞ .0[ Borel-Cantelli tv Vn → t wVVF=5wVVFv> m$|EEvzU l

n &Dn = 0 = t0,n < t1,n < · · · < tkn,n = t? [0, t] w*$|` Dn+1 ⊃ Dn w

limn→∞

m(Dn) = limn→∞

max |tni,n − tni−1,n| = 0,F*q n ~3n∑

i=1

∣

∣

∣Btki,n

− Btki−1,n

∣

∣

∣

2

→ t a.s. (7.4.2)\> "6w;j #3rF::|Q697-jProposition 7.4.3 Mn ); (7.4.2) wi#%3*UwI

· · · · · · , Mn, · · · , M2, M1?bA#3r

|5sz[1] Billingsley, P., Probability and Measure, John Wiley & Sons, 1986

[2] Billingsley, P., Convergence of Probability measures, John Wiley & Sons, 1968

[3] Bremaud, Markov Chains

[4] Doyle, P.G., Snell, J.L., Random Walks and Electric Networks, Mathe-

matical Association of America, Washington, DC, 1984

[5] Feller, W., Probability Theory and its Application, Vol. I(1959: Third edition),

Vol. II(1970), Wiley & Son

[6] Kallengberg, O., Foundations of Modern Probability, dNR% Springer,

2001

[7] Lyons, R., Probability on Trees and Networks, Probability web. 2003

[8] Shreve, S., Stochastic Calculus and Finance, Probability web, 2003

[9] y)<m^ 8HI, 2ogNR%b 2005

[10] y)<D2 T?;,>-, 2ogNR%b 2005

124

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