305 A Obstructions

1150

305 (IX.1 1) ObstructionsA. History The theory of obstructions aims at measuring the extensibility of mappings by means of algebraic tools. Such classical results as the +Brouwer mapping theorem and Hopfs extension and tclassification theorems in homotopy theory might be regarded as the origins of this theory. A systematic study of the theory was initiated by S. Eilenberg [l] in connection with the notions of thomotopy and tcohomology groups, which were introduced at the same time. A. Komatu and P. Olum [L] extended the theory to mappings into spaces not necessarily +n-simple. For mappings of polyhedra into certain special spaces, the +homotopy classification problem, closely related to the theory of obstructions, was solved in the following cases (K denotes an m-dimensional polyhedron): K++S (N. Steenrod [SI), Knt2 4s (J. Adem), Kntk*Y, where ni( Y)=0 for i ( f), any n-cochain dn

1151

305 c Obstructions general, if On( fo, fi) is nonempty, it is a coset of H(K, L; x,(Y)) factored by the subgroup O(f,, fo). Combined with the existence theorem on separation cochains, this cari be utilized to show the following theorem. Assume that O(f ) is nonempty. The set of a11 elements @(f ) that are extensions of an element of @-l (f ) is put in one-to-one correspondence with the quotient group of Hn(I?, L; A,( Y)) modulo On( fo, f) by pairing the obstruction On( f, f ) with each f for a fixed f. Among such elements of @( f ), the set off that are extensible to @+ is in oneto-one correspondence with the quotient group of H(R+1 ,L; ~n(Y))=H(KL; G(Y)) modulo the subgroup On( f, f$+), assuming that fo is extended to fo+ (fkst classification theorem).

of the pair (K, L) with coefficients in n,(Y) is expressible as a separation cochain d= d(fl,f;) wheref/EQ(f) is a suitable mapping such that fil R- =fr 1R- (existence theorem). Therefore if we take an element f- of @-I(f) whose obstruction cocycle c(f-) is zero, the set of a11 obstruction cocycles c(J) of a11 such ~E@(S) that are extensions off- forms a subset of O(f) and coincides with a coset of Z+(K, L; n,(Y)) factored by B+(K, L; rcL,( Y)). Thus a cohomology class ?+r(f-~)EH+~(K, L; rrn( Y)) corresponds to an f- E Qn-l (f) such that c(f-) = 0, and ?+i(f-) = 0 is a necessary and sufftcient condition for f- to be extensible to I? (first extension theorem). For the separation cocycle, d(f,, h-,fJ~ H(K, L;a,(Y)) corresponds to each homotopy h- on I?n-Z such that d-(f,, h-,f,)=O, and 6( f& h-, fi) = 0 is a necessary and sufficient condition for h- to be extensible to a homotopy on R (tirst homotopy theorem). The subset of H+(K, L, K,( Y)) corresponding to on+ (f ) is denoted by On+ (f ) and is called the obstruction to an (n + 1)dimensional extension off . Similarly, the subset On( fo, fi) of H(K, L, n,( Y)) corresponding to o( fo, fi) is called the obstruction to an ndimensional homotopy connecting f. with fi. Clearly, a condition for f' to be extensible to Rn+ is given by 0 E O+l (f ), and a necessary and sufftcient condition for f. 1K = fi 1K (rel L) is given by OeO(fo, fi). A continuous mapping O, Cici= 1, &, dominates 9, and if .?8 is sufficient for Y we can choose a 1?Ameasurable version of dP,/di,,. Conversely, if there exists a a-finite measure i such that we can choose a g-measurable version of dP,/di for all PotzY, then 98 is sufficient. If 9 is dominated by a a-finite 2, .A is sufficient if and only if there exist a d-measurable g,, and an .d-measurable h independent of 0 satisfying dpo -=g&di

a.e. (&,A)

for all PoE9.

This is called Neymans factorization theorem. With a dominated statistical structure, there exists a minimal sufficient c-field, and a o-field containing a sufficient o-held is also sufficient. We say that a o-field g is pairwise sufficient for 9 if it is sufficient for every pair {PO,, P,,} of measures in .Y. A necessary and sufficient condition for 9 to be sufficient for a dominated set B is that 3 be pairwise sufficient for .p. Recently, a more general statistical structure has been studied. Put =,(p)= {A 1A~sd, p(A) < a}. A measure p on d is said to be a localizable measure if there exists ess-sup 9( p) for any subfamily B c &Jp), that is, if there exists a set E E .d such that p(A - E) = 0 holds

for all AEF-, and p(A-S)=O for all Ae,F implies p(E - S) = 0. A a-finite measure is localizable. A measure p is said to have the finite subset property if for any A satisfying 0 < p(A), there exists a B c A satisfying 0 0 and every 0 E 0 there exist a sufticiently small positive number 6 and a sufficiently large number K satisfying n-m Let {c,,} be a maximal order of consistency. This notion was introduced by Takeuchi and Akahira. They studied consistent estimators of location parameters with various orders. Let Z = 0 = RI. Suppose that for every HE 0, Ps has a density function f(x - 0) with respect to the Lebesgue measure. Theorem. Assume that (OCl)f(x)>Oifa a, where 0, is the first coordinate of Q = (O,, , Ok); and (2) H: 0, = a, A: 0, # 11.For example, when the sample is normally distributed with unknown mean p and unknown variance rs2, the Student test (defined in Section G) for a hypothesis H : ,u = p0 against an alternative A : p # fLO is a UMP unbiased test.

400 D Statistical D. Similar

1502 Hypothesis Testing Structure transformation group G on the sample space 3. Denote the set of all invariant 1e:vel tl tests by @[(cc). A test that is uniformly most powerful in CD,(a) is called a uniformly must powerful (in short, UMP) invariant level x test. If there exists a unique UMP unbiased level tl test p*, then a UMP invariant level a test (if it exists) coincides with (p*[9]. When T(x) is tmaximal invariant under G, a necessary and sufficient condition for q(x) to be invariant i:s that q be a function of T(x). For example, suppose that the sample X =(X,, , X,) is taken from N(p, 0) with unknown p and 0. In this situation, Y = (X, S) is a sufficient statistic, where x=&X,/n and S = d-. Let G be the group of transformations (x, s)-(CT, cs) (c > 0) on the range 0) on the parameter space. Both the hypotheses H, :p/02 O) and if there exists a O=Z. fl~fi such that P,.(B)= Pd{(x,la, . . ..x.&)l(x,, . . ..x.)~B)l for any OER, then the real number a is called a scale parameter. The invariance principle states that a test for a testing problem invariant under G should preferably be invariant under G. A test q(x) is called an almost invariant test if cp(gx) = q(x) [Y] for all gE G. Suppose that the testing problem of a hypothesis under consideration is invariant under a

F. Minimax

Tests and Most Stringent

Tests

Minimax tests and most stringent tests are sometimes used as alternatives to 1JMP tests. Suppose that B = {P, 1@EQ} is a tdominated family and 23 is generated by a countable number of sets. A level a test (p* is called a minimax level a test if for any level a test cp, inf E,(cp*) > inf E,(q). BEWl flEWA Such a test exists for any ac(0, 1). If a group G of measurable transformations on F leaves a testing problem invariant, then an intimate relation exists between the minimax property and invariance. Concerning this relation, we have the following theorem: For each ae(0, 1) there is an almost invariant level a test that is minimax if there exists a a-field % of subsets of G and a sequence {v} of probability measures on (G,cLI) such that (i) BE% implies {(x,g)IgxEB}E% x!!l; (ii) AELI, gc:G implies AgEzZ; and (iii) lim,,,lv,(Ag)-v,,(A)I=O for any AE% and gEG. Fundamental in the invariant testing problem is the Hunt-Stein lemma: Under the condition just stated, for any cp there exists an almost invariant test $ such ~ that

The following six types of transforlmation groups satisfy the condition of the theorem: (1) the group of translations on R, (2:l the group of similarity transformations on R, (3) the

1503

400 G Statistical

Hypothesis

Testing

group of transformations q=(a, h):(x,, ,x,)E R+(ax,+h....,ax,+h)~R(O g, with 00 > 0, we can use S = {xIx2(x)>c(z)), where x2=s2/ot. (4) To test the hypothesis g2 > 0: against the alternative cr2 l/2. All tests (l)-(5) are UMP unbiased, and (3)-(5) are UMP invariant under the translations (xi, . . . ,x,)+(x, +a, . . . . ~,+a) (-Q p2) is the alternative, S= {(x, y)]) T(x,y)l 3 C(M)} (S = {(x, y) I T(x, y) > c(a)}) can be used as a critical region, where T(x, y) = (x-y)JhTiq(JJ~). Both tests are UMP unbiased and are invariant under (xi, ,x,, y,, . . , y,)+(uxi + b ,..., ax,+b,ay,+b ,..., ay,+b)(-aO (i= 1, . . ..k) and C~zlpi(B)= 1; (2)pi(Q) is twice continuously differentiable with respect to the coordinates of 8; and (3) the rank of the matrix (api/aOj) is k. Then the system of equations above has a unique solution 0 = &(x1, , x,), and g,, converges in probability to 0, when @ = &. The asymptotic distribution of x2(x)= Cl=,((xinp,(&))Z/npi(&)) is the chi-square distribution with n-s - 1 degrees of freedom

Likelihood ratio tests and chi-square tests of goodness of fit are consistent tests under conditions stated in their respective descriptions. In general, there are many consistent tests for a problem. Therefore it is necessary to consider another criterion that has to be satisfied by the best test among consistent tests. Pitmans asymptotic relative efftciency is such a criterion. Other notions of efficiency have also been introduced. A completely specified form of distribution is rather exceptional in applications. More often we encounter cases where distribution of the sample belongs in a large domain. Various tests independent of the functional form of distribution have been proposed, and the asymptotic theory plays an important role in those cases (- 371 Robust and Nonparametric Methods). The following concept of asymptotic efficiency is due to R. R. Bahadur [l 11: Let {T,} be a sequence of real-valued statistics defined on %-(). { 7) is said to be a standard sequence (for testing H) if the following three conditions are satisfied. (I) There exists a continuous probability distribution function F such that for each 8EwH, lim,,, Pg){ T, < t} = F(t) for every teR. Table 2. ContingencyB, B,x,2 XI2

Table...... .

B,XIS x2.v

TotalXl. x2.

A,A2

x11 x21

k,

xi,

xi2

XL,

x;.

Total

x.r

x.~

1..

x.,

n

1507

400 Ref. Statistical

Hypothesis

Testing

(II) There exists a constant a, O 0. Suppose that {T,} is a standard sequence. Then T, has the asymptotic distribution F if H is satisfied, but otherwise T,+ co in probability. Consequently, large values of T, are significant when T, is regarded as a test statistic for H. Accordingly, for any given XE.%(), 1 - F( T,(x)) is called the critical level in terms of T,, and is regarded as a random variable defined on Z() [ 11. It is convenient to describe the behavior of this random variable as n+ m in terms of K,, where K,(x) = - 2 log[ lF(T,(x))]. Then for each OEW~, K, is asymptotically distributed as a chi-square variable xi with 2 degrees of freedom and for OE K,/n-tab2(8) in probability as n-ma. n-w,, The asymptotic slope of the test based on {T,} (or simply the slope of { T,}) is defined to be c(0)=ab2(0). Note that the statistic K, is equivalent to T, in the following technical sense: (i) {K,} is a standard sequence; (ii) for each OE~, the slope of {K,!) equals that of {T}; and (iii) for any given n and x, the critical level in terms of K,!* equals the critical level in terms of T,. Since the critical level of K,* is found by substituting K,* into the function representing the upper tail of a fixed distribution independent of F, {KA} is a normalized version of {T,}. Suppose that {T,(l)} and {T,*)} are two standard sequences defined on Xc), and let F()(x), a,, and hi(Q) be the functions and constants prescribed by conditions (I)-(III) for i= 1,2. Consider an arbitrary but fixed /3 in 0 -wH, and suppose that x is distributed according to P,. The asymptotic efficiency of {T,)} relative to {T,*)} is defined to be ~12(0)=c,(Q)/c,(O), where c,(O)=a,b~(O) is the slope of {T,)}, i = 1, 2. The asymptotic efficiency is called Bahadur efficiency. Several comparisons of standard sequences are given in [ 111. The relationship between Bahadur efficiency and Pitman efficiency for hypothesis-testing problems has also been studied. Under suitable conditions the two efficiencies coincide.

then X,, etc. At each stage a decision is made on the basis of the previously obtained data whether the observation should be stopped and a judgment made on the acceptability of the hypothesis. Such a test is called a sequential test. Let X, , X2,. . be independent and identically distributed by&(x). For testing a simple hypothesis H : 0 = 0 against a simple alternative A: 0 = 1, we have the sequential probability ratio test: Let G,(x,, x2, . . . , xn) = l-I:=, fi (xi)/l-& fO(xi), and preassign two constants a, < ul. After the observations of X 1, . . . , X, are performed, the next random variable X,,, is observed if a, < G,,(x,, , x,) 0, F(u, 0) = F(0, 0) = 0. If the interaction is translationally invariant (i.e., y,@(l) = @(I + a) for all UE Z and I) and if we restrict our attention to translationally invariant states (i.e., cp(y,(A))= q(A) for all A~11 and UEZ), then the following conditions are also equivalent to the above. 6. Variational principle: [k(cp)- s(q) < be($) -s($) for all translationally invariant II, (the minimality of the mean free energy), where e(cp)=limN(A)~cp(U(A))=limN(A)~~(H(A)) (the mean energy), s(v) = lim N(A)-lS,,(v) (the mean entropy), the infimum value Be(cp) - s(v) is -P(/?@) with P([j@)=limN(A)-- logT,(e-Pu(A)) (the pressure), and the limits exist if A 7 Z is taken in the following van Hove sense: For any given cube C of lattice points, the minimal number ni (C) of translations of C that cover A and the maximal number n,(C) of mutually disjoint translations of C in A satisfy II; (C)/n, (C)d 1 as A 7 Z. 7. Tangent to the pressure function: P(Q) is a continuous convex function on the Banach space of translationally invariant Q, with I/@(/ < cry. A continuous linear functional c( on this Banach space is a tangent to P at (I, if I(@ + Y) B P(Q) + X(Y) for all Y. For a translationally invariant state $, we define cc,(Y) = $(C,,O N(I)- Y(I)). The condition is that - rrp is a tangent to P at p@. (Conversely, any tangent CI to P at [j@ arises in this manner.) The set I 2 while 11@11 w and c(, defined if c(> 1. There is < more than one KMS state (with spontaneous magnetization) for 2 >, r > I and large /U > 0, and hence a phase transition exists (F. J.

Dyson, Comm. Muth. Phys., 12 (1969); J. FrGhlich and T. Spencer, Comm. Math. Phys., 83 (1982)). If a l-dimensional interaction has a finite range (i.e., @(I) = 0 if the diameter of I exceeds some number rO) or if it is classical 1)11@(1)11< 00 for and Cls,, N(l)-(diaml-t d=2, then q(A) for cp~K, and AE~I(A) for a finite A is real analytic in a and any other analytic parameter in the potential (Araki, Comm. Math. Phys., 14 (1969); [22]; M. Cassandro and E. Olivieri, Comm. Math. Phys.,80 (1981)).

For a 2-dimensional king model with the nearest-neighbor ferromagnetic interaction [23], K, consists of only one point for 0 /& has exactly two extremal points corresponding to positive and negative magnetizations (M. Aizenman, Comm. Math. Phys., 73 (1980); Y. Higuchi, Colloquia Math. Sot. J&OS Bolyui, 27 (1979)). In this case, all KMS states are translationally invariant, while there exist (infinitely many) translationally noninvariant KMS states for sufficiently large b if v = 3 (Dobrushin, Theory Prob. Appl., 17 (1972); H. van Beijeren, Comm. Math. Phys.,40 (1975)).

The accumulation points of b-KMS states as [j- +cr, (or -m) provide examples of ground (or ceiling) states defined by any one of the following mutually equivalent conditions I +, 2 + (or 1 - ,2 -) (0. Bratteli, A. Kishimoto, and D. W. Robinson, Comm. Math. Phys., 64 (1978)): 1 + (1 -). Positivity (negativity) of energy: For any A E uA %(A), -@(A*&,(A)) is real and positive (negative). 2, (2-). Local minimality (maximality) of energy: For any finite subset A of Z and for any state $ with the same restriction to %(A) as the state cp under consideration, cp(H(A)) < 44HbV) (v(H(N)>+(H(A))). For translationally invariant potentials and states, the following condition is also equivalent to the above: 3 + (3 ). Global minimality (maximality) of energy: e(v)e($)) for all translationally invariant states $. The totality of KMS, ground, and ceiling states can be characterized by the following formulation of the impossibility of perpetual motion: Let P, = PF E 91 be a normdifferentiable function of the time t E R with a compact support, representing (external) timedependent perturbations. Then there exists a unique perturbed time evolution X: as a oneparameter family of *-automorphisms of \LI satisfying (d/&)x:(A) = x/(&,(A) + i[Pt, A]) for all A ELI in the domain of 6,. A state cp changes with time t as q,(A)= cp(s,(A)) under the perturbed dynamics $, and the total

402 H Statistical

1518 Mechanics as forming an electron gas, in which electron scattering by lattice vibrations or by impurities is more important than electron-electron scattering. Following the example of gas theories H. A. Lorentz set forth a simple theory of irreversible processes of metallic electrons. His theory was, however, not quite correct, since metallic electrons are highly quantummechanical and classical theories cannot be applied to them. Quantum-mechanical theories of metal electrons were developed by A. Sommerfeld and F. Bloch.

energy given to the system (mechanical work performed by the external forces) is given by LP(~)=~Zn. cp,(dPJdt)dr. For KMS states at any /I, as well as ground and ceiling states, L(cp)>O for any P,. If cp is a factor state, the converse holds, i.e., L(q) > 0 for all P, implies that cp is either a KMS, ground, or ceiling state. The condition L(q) > 0 for all P, is equivalent to - icp( U*&,( U)) > 0 for all unitary U in the domain of 6, and in the identity component of the group of all unitaries of Il. A state cp satisfying this condition is called passive, and a state cp whose n-fold product with itself as a state on ~Jl@ IS passive relative to at for all n is called completely passive. The last property holds if and only if cp is a KMS, ground, or ceiling state (W. Pusz and S. L. Woronowicz, Comm. Math. Phys., 16(1970)).

1. Master

Equations

The totality of KMS, ground, and ceiling states can be characterized by a certain stability under perturbations (P, considered above) under some additional condition on z, (R. Haag, D. Kastler, and E. B. TrychPohlmeyer, Comm. Math. Phys., 38 (1974); 0. Bratteli, A. Kishimoto, and D. W. Robinson, Comm. Math. Phys., 61 (1978)). When a lattice spin system is interpreted as a lattice gas, an operator Nn~YtU: (such as ($1 + 1)/2) is interpreted as the particle number at the lattice site n and N(A) = &,, N, is the particle number in A. It defines a representation of a unit circle T by automorphisms Z, of % defined as r,,(A) = lim eiN()BAe-iN(h)B (A /*Z), called gauge transformations (of the first kind). The grand canonical ensemble can be formulated as a /I-KMS state with respect to t(,r@, (instead of x,), where the real constant p is called the chemical potential. It can be interpreted as an equilibrium state when the gauge-invariant elements {A E% 1T,(A) = A} instead of 91 are taken to be the algebra of observables or as a state stable under those perturbations that do not change the particle number.

The Boltzmann equation gives the velocity distribution function of a single particle in the system. This line of approach can be extended in two directions. The first is the so-called master equation. For example, consider a gaseous system consisting of N particles, and ask for the probability distribution of all the momenta, namely, the distribution function fN(pI, ,pN; t), where p,, . . . ,pN are the momenta of the N particles. The equations of motion are deterministic with respect to the complete set of dynamical variables (x 1, p ,,..3 xN, pN). The equation for ,f(p,, , pN, t) may not be deterministic, but it may be stochastic because we are concerned only with the variwith all information about ablesp,,...,p,, the space coordinates x1, , xN disregarded. This situation is essentially the sa.me in both classical and quantum statistical mechanics. If the duration of the observation process is limited to a finite length of time and the precision of the observation to a certain degree of crudeness, the time evolution of the momentum distribution function fN can be regarded as a +Markov process. In general. an equation describing a Markov process of a certain distribution function is called a master equation. Typically it takes the following form for a suitable choice of variables x:

H. The Boltzmann

Equation

Statistical mechanics of irreversible processes originated from the kinetic theory of gases. Long ago, Maxwell and Boltzmann tried to calculate viscosity and other physical quantities characterizing gaseous flow in nonequilibrium. The +Boltzmann equation is generally a nonlinear +integrodifferential equation. On the basis of this equation mathematical theories were developed by D. Enskog, S. Chapman, and D. Hilbert [2]. Free electrons in a metal can be regarded

where W(x, x) is the transition probability from x to x. By expanding the first integrand into a power series in x-x, with x fixed and by retaining the first few terms, we obtain the Fokker-Planck equation:(alat)f(x, t) = - (dlo?x)(a, (x)f(x, + (~2/~x2H%(xMx.~ a,(x) = s W(x, x + r)rdr. t))

=dx(t)W(x, x)f(x, - t)), s W(x,x).f(x,(25)

t))/Z

(26)(27)

1519

402 Ref. Statistical of Particle Distribution

Mechanics

J. The Hierarchy Functions

and the random force cannot be independent, but are related by a theorem asserting that cr my= s0 O. Let &d,r be the totality of functions a(t, w)= (olj(t, w)): [0, co) x Wd+Rd @ R (:= the totality of d x r real matrices) such that each component olj(x,w)(i=1,2 ,..., d;j=1,2 ,..., r)is@[O,x?)) x B( Wd)-measurable and Br:( Wd)-measurable for each fixed t 2 0. In general, cxj(t, is called w) nonanticipative if it satisfies the second property above. An important case of c(E&,~ is when it is given as cc(t, w) = o(t, w(t)) by a Bore1 function a: [0, co) x Rd-+Rd @ R. In this case, c1is called independent of the past history or of Markovian type. For a given c(E G?,~ and ,%E dd, I, we consider the following stochastic differential equation: (1) dX(t)= ij=l

$(t,X)dB(t)+/?(t,X)dt, i=l,2 ,..., d,

also denoted

simply

as

dX(t)=x(t,X)dB(t)+&,X)dt. Here X(t)=(X(t), , Xd(t)) is a d-dimensional continuous process. B(t) = (B(t), , B*(t)) is a r-dimensional Brownian motion wlith B(0) = 0. A precise formulation of equation (1) is as follows. X=(X(t)) is called a solution of equation (1) if it satisfies the following conditions: (i) X is a d-dimensional, continuous, and {@I)adapted process defined on a probability space (QF-, P) with an increasing family {&}, i.e., X : R + Wd which is R/ac,( Wd)-measurable for every t>O; (ii) c$(t,X)e6Cp, @(t,X-)EL?p, i=l,..., d,j=l,..., r(-SectionBforthe definition of spd); (iii) there exists an r-

D,F(X)odX.

This chain rule for stochastic differentials takes the same form as in the ordinary calculus. For this reason symmetric multiplication plays an

1537

406 D Stochastic

Differential

Equations

dimensional {e}-Brownian motion B(O)=0 such that the equality X(t)-X(O)= i ccj(s,X)dBj(s)

j=l

f s X)ds, +rB(s,0

B(t) with

s0

i=!,2

,..., d,

holds with probability 1. Thus a solution X is always accompanied by a Brownian motion B. To emphasize this, we often call X a solution with the Brownian motion B or call the pair (X, B) itself a solution of (1). In the above definition, a solution is given with reference to an increasing family {&}. The essential point is that c-fields a(B(u) -B(u); ~>u>t) and 0(X(s), i?(s);OQs< t) are If X satisfies the conindependent for every ditions of solutions stated above, then the specified independence is obvious, and conversely, if this independence is satisfied, then by setting z = nE,e 0(X(s), B(s); O 0 a.s. In this de!? nition also, the solutions can be restricted to those having nonrandom initial values. We say that equation (1) has a unique strong solution if there exists a function F(x, w): Rd x H+Wd(M/;;={w~WIIw(0)=O}) such that tbfollowing are true: (i) For any solution (X, r of(l), X = F(X(O), B) holds as.; (ii) for any Rd-valued random variable X(0) and an rdimensional Brownian motion B =(B(t)) with B(0) = 0 which are mutually independent, X = F(X(O), B) is a solution of (1) with the Brownian motion B and the initial value X(0). If this is the case, F(x, w) itself is a solution of (1) with the initial value x, and with respect to the canonical Brownian motion B(t, w) = w(t) on the r-dimensional Wiener space ( Wg, 9, P), 9 is the completion of a( W,l) with respect to the r-dimensional Wiener measure P. If equation (I) has a unique strong solution, then it is clear that pathwise uniqueness holds. Conversely, if pathwise uniqueness holds for (1) and if a solution exists for any given initial law, then equation (1) has a unique strong solution,

CC251.The existence of solutions was discussed by A. V. Skorokhod [20]. If the coefficients c1and /j are bounded and continuous on [O, co) x Wd, a solution of (1) exists for any given initial law. This is shown as follows [6]. We first construct approximate solutions by Cauchys polygonal method and then show that their probability laws are ttight. A limit process in the sense of probability law can be shown to be a solution. The assumption of boundedness above can be weakened, e.g., to the following condition: For every T> 0, a constant KT > 0 exists such that (4)

ll4t,w)ll + II/m 4

GKTU + IIW~ fE[O, 7-1, WE Wd

Here ll~ll,=maxO~,,, Iw(s)l. In the case of the

406 E Stochastic

1538 Differential Equations increasing family {.c} such that 9=Vvr,,,& we set up an so-,-measurable, d-dimensional random variable X(0) with a given law and a d-dimensional {,9j}-Brownian motion B =(&t)) such that &O)=O. Set X(t)=X(O) +@t) and M(t)=exp[fo/3(s,X)d&s)~foI[~(s,X)12ds]. Then M(t) is an {.e}martingale, and the probability P on (Q F) is determined by P(A)= E(M,; A), A E 9$ By Girsanovs theorem, i?(t) = X(t) - X(0) j fi(s, X)ds is a d-dimensional {&I-Brownian motion on (Q 9, p), and hence (X.. B) is a solution of (6). Any solution is given in this way and hence the uniqueness in the sense of law holds. But the pathwise uniqueness does not hold in general; an example was given by Tsirelson [ 1,6] as follows. Let it,,) be a sequence such that 0 < < t, < t,-, < t, = 1 and lim + t, = 0. Set (0,

Markovian equation (2) it is sufficient to assume that g(t, x) and h(t, x) are continuous: (5) lla(t,x)ll + Ilb(t,x)ll dK,(l +lxl), t~[0, T], xeR.

If these conditions are violated, a solution X(c) does not exist globally in general but exists up to a certain time e, called the explosion time, such that lim,,, Ix(t)] = m if e < x?. To extend the notion of solutions in such cases, we have to replace the path space Wd by the space tid that consists of all continuous functions w: [0, m)+Rd (= RdU {A) = the one-point compactification) satisfying w(t) = A for every t>e(w)(=inf{t]w(t)=A}). Now, we list some results on the uniqueness of solutions. First consider the equations of the Markovian type (2), and assume that the coefficients are continuous and satisfy the condition (5). (i) If (T, h are Lipschitz continuous, i.e., for every N > 0 there exists a constant K, such that llo(t,x)--a(t,y)ll + llb(t,x)-

tat, and t=O,wk+2)

h(t,y)l/ dK,lx-yl,

tE[O, Tl, x, yEB,:=

w(4+1)B(t>w)= t,+,-ti+* > O bI tE Ch+, cl >i=O,1,2 ,...,

[ZE Rdl ]z] < N}, then the pathwise uniqueness of solutions holds for equation (2). Thus the unique strong solution of (2) exists, and this is constructed directly by Picards successive approximation (ItB [7,8]). (ii) If d= 1, a is Holder continuous with exponent l/2 and h is Lipschitz continuous, i.e., for every N > 0, K, exists such that

where 0(x)=x - [xl, x E R, is the decimal part of x. Time changes (- Section B) are also used to solve some stochastic differential equations

C61.E. Stochastic Differential Diffusion Processes Equations and

then the pathwise uniqueness of solutions holds for equation (2) (T. Yamada and Watanabe [25]). (iii) If the matrix a(t, x)= ~~(t,x)a(t,x)* (i.e., uj(t,~)=J$=~ &(l,x)&(t,x)) is strictly positive definite, then the uniqueness in the sense of law of the solution for (2) holds (D. W. Stroock and S. R. S. Varadhan [21]). (iv) An example of stochastic differential equations for which the uniqueness in the sense of law holds but the pathwise uniqueness does not hold was given by H. Tanaka as follows: d=r=l, b(t,x)=O and o(t,x)=Zlxao) -I Ix0, (I.-,4)(C,2(Rd)) is a dense subset of C,(Rd) (= the totality of continuous functions f on Rd such that I$,,,, f(x) = 0) then the ttransition semigroup of the diffusion is a tFeller semigroup on C,(Rd), and its infinitesimal generator A is the closure of (A, Ci(Rd)). Hence u(t, x) = E,[f(w(t))], ,~EC~(R~), is the unique solution of the evolution equation duldt = Au, u 11=0 =f: Generally, if the coefficients 0 and b are sufficiently smooth, we can show, by using the stochastic differential equation (3), that u(t,x) is also smooth for a smooth f and satisfies the heat equation au/i% = Au. Taking the expectations in (7), we have the re-

example, consider a reflecting +Brownian motion on the half-line [0, co). This is a diffusion process X=(X,) on [O, GO)obtained by setting X,=(x,1 from a l-dimensional Brownian motion xt. The corresponding differential operator is A = id21dx2, and the boundary condition is Lu = du/dx lxzO = 0, that is, the transition expectation u(t, x) = E,[,f(X,)] is determined by du/& = Au, Lu = 0, and u 11=0 =,f: In constructing such diffusion processes with boundary conditions, stochastic differential equations can be used effectively. In the case of reflecting Brownian motion, it was formulated by Skorokhod in the form (9) dX(t)=dB(t)+dq(t).

lation ~,Cf(wWl =.fW +Sb4CAfbW)l&which implies that the transition probability P(t, x, dy) of the diffusion satisfies the equation ap/cit = A*p in (t, y) in a weak sense, where A* is the adjoint operator of A. If a/&-A* is +hypoelliptic, we can conclude that P(t, x, dy) possesses a smooth density p(t, x, y) by appealing to the theory of partial differential equations. Recently, P. Malliavin showed that a probabilistic method based on the stochastic differential equations can also be applied to this problem effectively, [6, 16, 173. If c(t, x) is continuous and u(t, x) is s&iciently smooth in (t, x) on [0, co) x Rd, then the following fact, more general than (7), holds: (8)

u(t, w(t))exp[

I:c(,s, w(s))ds]-r(O,x)

-Jiev[ J~~(u,-(u~~du]x (au/& +(A +c)u)(s, w(s))ds is a local martingale (i.e., E&Z) with respect to {me, PI}. By applying the toptional sampling theorem to (8) for a class of {&}-stopping times, we can obtain the probabilistic representation in terms of the diffusion of solutions for initial or boundary value problems related to the operator A [3,4]. F. Stochastic Differential Boundary Conditions Equations with

Here B(t) is a 1-dimensional Brownian motion (B(O)=O), X(t) is a continuous process such that X(f)>,O, and cp(t) has the following property with probability 1: q(O) = 0, t-r q(t) is continuous and nondecreasing and increases only on such t that X(t) = 0, i.e., &Iloj(X(s))dq(s)=cp(t). Given a Brownian motion B(t) and a nonnegative random variable X(0) which are mutually independent, X(t) satisfying (9) and with the initial value X(0) is unique and given by X(t) = X(0) + B(t), t a, (P. L&y, Skorokhod; - C6,181). In the case of multidimensional processes, possible boundary conditions were determinea by A. D. Venttsel [24]. Stochastic differential equations describing these diffusions were formulated by N. Ikeda [S] in the 2-dimensional case and by Watanabe [23] in the general case as follows. Let D be the upper half-space R~,={x=(x~,...,x~)(x~>O),~D={~(X~=O), and d = {x 1xd > 0). The general case can be reduced, at least locally, to this case. Suppose that the following system of functions is given: a(x):D-tRd x R, b(x):D-rRd, z(x):aD +Rdm x R, B(x):aD-tRd-, and p(x):ciD+ [0, cc), which are all bounded and continuous. Consider the following stochastic differential equation: dX(t) = i cr;(X(t))lb(X(t))dB(t) j=l +b(X(t))Zb(X(t))dt

IdXd(t)=

As we saw in the previous section, diffusion processes generated by differential operators can be constructed by stochastic differential equations. A diffusion process on a domain with boundary is generated by a differential operator that describes the behavior of the process inside the domain, and a boundary condition that describes the behavior of the process on the boundary of the domain. For

+BiWWPW>i=l,2 ij=l

,..., d-l.

~f(X(t))l~(X(t))dBj(t)

I

+bd(X(t))l~(X(t))dt+d~(t),

406 G Stochastic

1540Differential Equations

By a solution of this equation, we mean a system of continuous semimartingales 3E= (X(t), i?(t), M(t), q(t)) over a probability space (Q9, P) with an increasing family {&} satisfying the following conditions: (i) X(t)=(X(t), ,Xd(t)) is D-valued, i.e., Xd(t)>O; (ii) with probability 1, ip(O)=O, t-+q(t) is nondecreasing, and SolaD(X(~))d~(s)=cp(t); (iii) B(t) and M(t) are r-dimensional and s-dimensional systems of elements in &PC, respectively, such that (B, Bj),=ht, (B,M*),=O, and (Mm, M), = Smnq(t), i, j = 1, , r, m, n = 1, , s; and finally (iv) the stochastic differentials of these semimartingales satisfy (10). The processes B(t), M(t), and q(t) are subsidiary, and the process X(r) itself is often called a solution. We say that the uniqueness of solution holds if the law of X=(X(t)) is uniquely determined from the law of X(0). As before, the existence and the uniqueness of solutions imply that solutions define a diffusion process on D, and these are guaranteed if, for example, min,,?,add(x) > 0 and c, b, 7, fi are Lipschitz continuous, [6,23]. Here, we set a(x) = XL=1 c$(x)a,!(x) and &(x) = & T:(x)z{(x). It is a diffusion process generated by the differential operator A =;. i aij(x)DiDj+ I,, I ii=l

the following on M: (11)

stochastic

differential

equation

dX,=A,(X,)odwk(t)+A,(X,)dt.

(Here, the usual convention for the omission of the summation sign is used.) A precise meaning of equation (11) is as follows: We say that X =(X,) satisfies equation (11) if X is an (g}adapted continuous process on M admitting explosions such that, for any C-function f on M with compact support (we set f(A) = 0), f(X,) is a continuous semimartingale satisfying

(12) df(X,)=(A,f)(X,)odwk(t)+(A,f)(X,)dt,where o is It6s circle operation defined in Section C. This is equivalent to saying that X,=(X,, , Xf), in each local coordinate, is a d-dimensional semimartingale such that (13) dX;=r$(X,)odwk(t)+bi(X,)dt = cr;(X,)dwk(t) + ;k$lDjq$j+bi

1

(X,)dt,

b(x)Di, condition, /3(x)Diu(x)i=l

and by the Venttsel Lu(x)=;.~~ ., 1

boundary

aj(x)D,o,u(~)+~~ +Ddu(x)-p(x)(Au)(x)=O

on 8D.

G. Stochastic Manifolds

Differential

Equations

on

Let M be a connected a-compact C-manifold of dimension d, and let W= C( [0, a)+M) be the space of all continuous paths in M. If M is not compact, let fi = MU {A} be the onepoint compactilication of M and PM be the space of all continuous paths in &! with A as a ttrap. These path spaces are endowed with the a-fields g( W,) and &?( GM), respectively, which are generated by Bore1 cylinder sets. By a continuous process on M we mean a ( W,, .?8(W,))-valued random variable, and by a continuous process on M admitting explosions we mean a ( wM, a( eM))-valued random variable. In this section the probability space is taken to be the r-dimensional Wiener space (WJ, 9, P) with the increasing family {E}, where z is generated by 9&( W,l) and P-null sets. Then w =(w(t)), WE WG, is an rdimensional {Ft}-Brownian motion. Suppose that we are given a system of C vector fields A,, A,, . . . . A, on M. We consider

where AL(x) = a,!(x)Di, k = 1, 2,. , r, and A,(x) = bi(x)Di. By solving the equation in each local coordinate and then putting these solutions together, we can obtain for each XE M a unique solution X, of (11) such that X0 = x. We can also embed the manifold M in a higher-dimensional Euclidean space and solve the stochastic differential equation there. We denote the solution by X(t, x, w). The law P, on fiM of [t-X(t, x, w)] defines a diffusion process on M which is generated by the differential operator A =~~;=, AZ + .4,. Next, if we consider the mapping x--t X(t, x, w); then, except for w belonging to a set of P-measure 0, the following is valid: For all (t, w) such that X(t, x0, W)E M, the mapping x-+X(t, x, w) is a diffeomorphism between a neighborhood of x0 and a neighborhood of X(t, x0, w). This is based on the following fact for stochastic differential equations on Rd. If in equation (3) the coefficients 0; and b are C-functions with bounded derivatives of all orders a, /al> 1, then, denoting by X(&x, w) the solution such that X(0)=x, we have that x+X(t, x, w) is, with probability 1, a diffeomorphism of Rd for all t [ 131. Example 1: Stochastic moving frame [6,15]. Let M be a Riemannian manifold of dimension d, O(M) be the orthonormal frame bundle over M, and L,, L,, . . . , L, be the basic vector fields on O(M), that is, 1 (Lif)(x,e)=lim-Cf(x,,e,)-.f(x,e)I1, t-0 t i= 1, . ..d.

1541

406 Ref. Stochastic

Differential

Equations

where e = (e, , . , ed) is an orthonormal basis in T,(M), x, = Exp(tei), i.e., the geodesic such that x0 =x and i = ei, and e, is the parallel translate of e along x,. Let b be a vector field on M and L,, be its horizontal lift on O(M), i.e., L, is a vector field on O(M) determined by the following two properties: (i) L, is horizontal and (ii) &(L,)=b, where n:O(M)+M is the projection. Consider the following stochastic differential equation on O(M): fir(t)= L,(r(t))odw(t)+ L,(r(t))dt.

Solutions determine a family of (local) diffeomorphisms r+r(t, r, w)=(X(t, r, w), e(t, r, w)) on O(M). The law of [t-X(t,r, w)] depends only on x = n(r), and it defines a diffusion process on M that is generated by the differential operator +A,,, + b (AM is the LaplaceBeltrami operator). Using this stochastic moving frame r(t, r, w), we can realize a stochastic parallel translation of tensor fields along the paths of Brownian motion on M (a diffusion generated by )A,) that was first introduced by It8 [lo], and by using it we can treat heat equations for tensor fields by means of a probabilistic method. Example 2: Brownian motion on Lie groups. Let G be a Lie group. A stochastic process {g(t)} on G is called a right-invariant Brownian motion if it satisfies the following conditions: (i) With probability 1, y(0) = e (the identity), and t-g(t) is continuous; (ii) for every t > s, g(t)g(s)- and o(g(u); u s, g(t)g(s)- and g(t - s) are equally distributed. Let A,, A,, . , A, be a system of rightinvariant vector fields on G, and consider the stochastic differential equation

(14) dgt=Ai(gt)odwi(t)+A,(g,)dt.Then a solution of (14) with go = e exists uniquely and globally; we denote this solution by g(t, w). It is a right-invariant Brownian motion G, and conversely, every right-invariant Brownian motion can be obtained in this way. The system of diffeomorphisms g+g(t, g, w) defined by the solutions of (14) is given by Pv(L 9, 4 = SOk w)g. Generally, if M is a compact manifold, the system of diffeomorphisms g,:x+X(t, x, w) defined by equation (11) can be considered as a right-invariant Brownian motion on the infinite-dimensional Lie group consisting of all diffeomorphisms of M [2].

References

[l] B. S. Tsirelson (Cirelson), stochastic differential equation

An example having no

of

strong solution, Theory Prob. Appl., 20 (1975), 416&418. [2] K. D. Elworthy, Stochastic dynamical systems and their flows, Stochastic Analysis, A. Friedman and M. Pinsky (eds.), Academic Press, 1978, 79-95. [S] A. Friedman, Stochastic differential equations and applications I, II, Academic Press, 1975. [4] I. I. Gikhman and A. V. Skorokhod, Stochastic differential equations, Springer, 1972. [S] N. Ikeda: On the construction of twodimensional diffusion processes satisfying Wentells boundary conditions and its application to boundary value problems, Mem. Coll. Sci. Univ. Kyoto, (A, Math.) 33 (1961), 367427. [6] N. Ikeda and S. Watanabe, Stochastic differential equations and diffusion processes, Kodansha and North-Holland, 1981. [7] K. It& Differential equations determining Markov processes (in Japanese), Zenkoku Shija Sfigaku Danwakai, 1077 (1942), 1352% 1400. [S] K. It& On stochastic differential equations, Mem. Amer. Math. Sot., 4 (1951). [9] K. It& Stochastic differentials, Appl. Math. and Optimization, 1 (1975), 347-381. [ 101 K. It& The Brownian motion and tensor fields on Riemannian manifold, Proc. Intern. Congr. Math., Stockholm 1962,536-539. [ 1 l] K. It8 and S. Watanabe, Introduction to stochastic differential equations, Proc. Intern. Symp. SDE, Kyoto, 1976, K. It8 (ed.), Kinokuniya, 1978, i&xxx. [ 123 T. Jeulin, Semi-martingales et grossissement dune filtration, Lecture notes in math. 833, Springer, 1980. [ 131 H. Kunita, On the decomposition of solutions of stochastic differential equations, Proc. LMS Symp. Stochastic Integrals, Durham, 1980. [ 141 H. Kunita and S. Watanabe, On square integrable martingales, Nagoya Math. J., 30 (1967), 209-245. [15] P. Malliavin, GCometrie difftrentielle stochastique, Les Presse de luniversiti: de MontrCal, 1978. [16] P. Malliavin, Stochastic calculus of variation and hypoelliptic operators, Proc. Intern. Symp. SDE, Kyoto, 1976, K. It8 (ed.), Kinokuniya, 1978, 195-263. [ 171 P. Malliavin, Ck-hypoellipticity with degeneracy, Stochastic Analysis, A. Friedman and M. Pinsky (eds.), Academic Press, 1978, 199%214,327-340. [ 181 H. P. McKean, Stochastic integrals, Academic Press, 1969. [ 191 P. A. Meyer, Un tours sur les inttgrales stochastiques, Lecture notes in math. 511, Springer, 1976,245-400.

407 A Stochastic

1542 Processes butions. Now, consider two stochastic processes .%= {X,},,, and 02 = { II;JfE7.. ?q is called a modification of .?? if they are defined over a common probability space (Q8, P) and P(X, = x) = 1 (t E T). Regardless of whe.ther .ot and O?/are defined over a common probability space or over different probability spaces, X and Y are said to be equivalent or each is said to be a version of the other if their iinitedimensional distributions are the same. According to Kolmogorovs extension theorem, every stochastic process has a version over the space W=R. The function X(w) oft obtained by fixing w in a stochastic process {X,},,, is called the sample function (sample process or path) corresponding to o. In applying various operations to stochastic processes and studying detailed properties of stochastic processes, such as continuity of sample functions, the notions of measurability and separability play important roles. We assume that T is an interval in the real line, and (if needed) that the probability measure P is tcomplete. Denote by 3 the class of all +Borel subsets of T. A stochastic process is said to be measurable if the function ix&T X,(w) of (t, w) is 3 x B-measurable. Continuity in probability defined in the next paragraph gives a sufficient condition for a stochastic process to have a measurable modification. A stochastic process {X,},,, is said to be separable if there exists a countable subset S of T such that

[20] A. V. Skorokhod, Studies in the theory of random processes, Addison-Wesley, 1965. [21] D. W. Stroock and S. R. S. Varadhan, Diffusion processes with continuous coeflicients I, II, Comm. Pure Appl. Math., 22 (1969), 345 - 400,479%530. [22] D. W. Stroock and S. R. S. Varadhan, Multidimensional diffusion processes, Springer, 1979. [23] S. Watanabe, On stochastic differential equations for multi-dimensional diffusion processes with boundary conditions, I, II, J. Math. Kyoto Univ., 11 (1971), 169-180, 545551. [24] A. D. Venttsel (Wentzell), On boundary conditions for multidimensional diffusion processes, Theory Prob. Appl., 4 (1959), 164- 177. [25] T. Yamada and S. Watanabe, On the uniqueness of solutions of stochastic differential equations, J. Math. Kyoto Univ., 13 (1973), 4977512.

407 (XVll.4) Stochastic ProcessesA. Definitions The theory of stochastic processes was originally involved with forming mathematical models of phenomena whose development in time obeys probabilistic laws. Given a basic tprobability space (Q, d, P) and a set T of real numbers, a family {Xt}ttT of real-valued trandom variables defined on (0, !B, P) is called a stochastic process (or simply process) over (Q 93, P), where t is usually called the time parameter of the process. For each finite t-set {t,, , t,}, the +joint distribution of (Xrl, , X,) is called a finite-dimensional distribution of the process {X,),,T. Stochastic processes are classified into large groups such as tadditive processes (or processes with independent increments), +Markov processes, +Markov chains, tdiffusion processes, +Gaussian processes, +stationary processes, imartingales, and +branching processes, according to the properties of their finite-dimensional distributions. This classification is possible because of the following fact, a consequence of Kolmogorovs textension theorem (- 341 Probability Measures I): Given a system .p of finitedimensional distributions satisfying certain tconsistency conditions, we can construct a suitable probability measure on the space W= R of real-valued functions on T so that

< lim sup X,(w) s-tt,ses

forany

tcT

=l.

It was proved by J. L. Doob that every stochastic process has a separable modification

C61.Various types of continuity are considered for stochastic processes. {Xt}ttT is said to be continuous in probability at SE T if P( 1X,-X,1 >E)+O (t+s, tE T) for each E>O; it is said to be continuous in the mean (of order 1) at SE T if E(lX,+X,l)-0 (t+s, TV T). Continuity in the mean of order p (>l) is defined similarly. Continuity in the mean of any order implies continuity in probability. Suppose thalt {X,},,T is separable. Then

are measurable events. If P(Q) > 0, then SE T is called a fixed point of discontinuity. The con~ dition P(u seT O,s)= 0 means that almost all

the stochastic process ix,),,,,

obtained by

sample functions are continuous. Regularityproperties of sample functions of processes, such as continuity or right continuity, have

setting X,(w) = the value of WE W at t, has 9 as its system of finite-dimensional distri-

1543

407 B Stochastic

Processes

been studied by many people. The following theorem is due to A. N. Kolmogorov: Let T= [O, 11. If

for constants y > 0, E> 0, and c > 0, then jXt}tr7. has a modification {s,),,, for which almost all sample functions are continuous, and

for any 6(0 < fi 0. Constants (30) are stopping times. If 0 and z are stopping times, then min(g, z) and max(cr, T) are also stopping times. The limit of an increasing sequence of stopping times is a stopping time, while the limit of a decreasing sequence of stopping times is a stopping time with respect to {!&+}, where !B,+ = n,>,%J3,. Let 23, be the class of A E% such that A n {t 0, the shift point function 0,p is defined by DB,P = {s ~(0, co); s + teD,} and (O,p)(s)=p(s+t). Let l7, be the totality of point functions on S and g(Z7s) be the smallest a-field on us with respect to which all p+N,((o, t] x U, t>O, UE~J are measurable. A point process on S is a (Z7,, a(Z7s))-valued random variable. Then there exists a point process p on S such that (i) for any t >O, p and 0,p have the same probability law, and (ii) N, is a Poisson random measure associated with ((0, a) x S, @O, co) x B(Z7,), dt x m(ds)). The point process p is called the stationary Poisson point process with the characteristic measure m.

References [l] A. N. Kolmogorov, Grundbegriffe der Wahrscheinlichkeitsrechnung, Erg. Math., 1933; English translation, Foundations of the theory of probability, Chelsea, 1950. [2] N. Wiener, Differential-space, J. Math. Phys., 2 (1923) 131-174. [3] P. Levy, Processus stochastiques et mouvement brownien, Gauthier-Villars, 1948. [4] P. Levy, Theorie de laddition des variables aleatoires, Gauthier-Villars, 1937. [S] M. S. Bartlet, An introduction to stochastic processes, Cambridge Univ. Press, 1955. [6] J. L. Doob, Stochastic processes, Wiley, 1953. [7] J. L. Doob, Stochastic processes depending on a continuous parameter, Trans. Amer. Math. Sot., 42 (1937), 1077140. [S] K. Ito, Stochastic processes, Aarhus Univ. lecture notes 16, 1969. [9] P. A. Meyer, Une presentation de la theorie des ensembles sousliniens; application aux processus stochastiques, Sem. Thtorie du Potentiel, 196221963, no. 2, Inst. H. Poincare, Univ. Paris, 1964. [lo] C. Dellacherie and P. A. Meyer, Probabilites et potentiel, Hermann, 1975, chs. I-IV. [ 1 l] C. Dellacherie, Capacites et processus stochastiques, Springer, 1972. [12] P. Levy, Le mouvement brownien fonction dun ou de plusieurs parametres, Rend.

Sem. Math. Univ. Padova, (5) 22 (1963), 24101. [ 131 H. P. McKean, Jr., Brownian motion with a several-dimensional time, Theory Prob. Appl., 8 (1963), 335-354. [ 143 R. L. Dobrushin, The description of the random field by its conditional distributions and its regularity conditions, Theory Prob. Appl., 13-14 (1968), 1977224. (Original in Russian, 1968). [ 151 I. M. Gelfand, Generalized random processes (in Russian), Dokl. Akad. Nauk SSSR (N.S.), 100 (1955) 8533856. 1161 I. M. Gelfand and I. Ya. Vilenkin, Generalized functions IV, Academic Press, 1964. (Original in Russian, 1961.) [17] K. Ito, Stationary random distributions, Mem. Coll. Sci. Univ. Kyoto, (A) 28 (1954), 209-223. [ 181 K. Ito, Isotropic random current, Proc. 3rd Berkeley Symp. Math. Stat. Prob. II, Univ. of California Press, 1956, 125- 132. [19] A. M. Yaglom (Jaglom), Some classes of random fields in n-dimensional space related to stationary random processes, Theory Prob. Appl., 2 (1957), 273-320. (Original in Russian, 1957.) [20] K. Urbanik, Stochastic processes whose sample functions are distributions, Theory Prob. Appl., 1 (1956), 132-134. (Original in Russian, 1956.) [21] K. Urbanik, Generalized stochastic processes, Studia Math., 16 (1958), 2688334.

408 (X1X.7) Stochastic ProgrammingA. General Remarks

Stochastic programming is a method of finding optimal solutions in mathematical programming in its narrow sense (- 264 Mathematical Programming), when some or all coefftcients are stochastic variables with known probability distributions. There are essentially two different types of models in stochastic programming situations: One is chance-constrained programming (CCP), and the other is a two-stage stochastic programming (TSSP). The difference between them depends mainly on the informational structure of the sequence of observations and decisions. For simplicity, let us here consider stochastic linear programming, which is the best-known model at present. Let A,, A be m x n-dimensional matrices and x, c E R and b, b. E R. Suppose further that components of A, b, c are random variables, while those of A,, b, are constants. Consider

408 B Stochastic

1546 Programming being allowed to compensate for it after the specification of those values. Second stage: One obtains an optima1 compensation ye R for the given x and the realized values of the random variables. Assuming that q E R is a random vector in addition to A, b, c, we can formulate TSSP as follows. First stage: min,E,{(c(w)x + Q(x, w) 1XEX)}; second stage: Q(x,~)=min~{q(o)yI Wy=A(w)xb(w),y>O}, where X=X,flK, K :={xlQ(x,o) < +co with probability 1) and q(tu)y is a loss function for the deviation A(w)x -- b(o). The m x n matrix W is called a compensation matrix. Several theorems have been proved: (i) K is a closed convex set; (ii) Q(x) = E,Q(x, co) is a convex function on K if the random variables in A(w), b(w), q(w) are square integrable; (iii) if P has a density function, then Q(x) has a continuous gradient on K; (iv) when P has a finite discrete probability distribution, a. TSSP problem is reduced to a linear programming problem having a dual decomposition structure.

the following formally defined linear programming problem: min,{ cx 1Ax < b, x E X0}, X0 = {x~A,xO}. Let (n,%,P) be a probability space (- 342 Probability Theory) such that {A(w), h(w), ~((0)) is a measurable transformation on 0 into R xnimtn.

B. Chance-Constrained

Programming

(CCP)

This method is based on the assumption that a decision x has to be made in advance of the realization of the random variables. Suppose that A,(w) is the ith row of A(w), and b,(w) is the ith component of b(w). We call P( {w 1A,(w)x < h,(w)}) > cli a chance constraint, where rxi is a prescribed fractional value determined by the decision maker according to his attitude toward the constraint A,(w)x < hi(o): if he attaches importance to it, he will take C(~ great as possible. Defining feasible sets as X,(cci) and X by X,(cc,)={xIP({wIA,(w)x< b,(w)}) > ai}, X=X,, n {n:, X,(cc,)}, we can formulate CCP as follows: min,{ F(x) 1x~X}, where F: X + R, is the certainty equivalent of the stochastic objective function cx. We have four models of CCP corresponding to different types of F(x): (i) E-model: F(x)=?x, C= E,c(w)x. (ii) I/-model: F(x) = Var(c(w)x) = xV,x, where V, is a variance-covariance matrix of c(w). (iii) P, -model: F(x) =A P( {w 1c(w)x