Chin. Ann. of Math. 21B: 1(2000),109-114.

L I N E A R I S O M E T R I C N O N - A N T I C I P A T I V E T R A N S F O R M A T I O N S O F W I E N E R P R O C E S S

ZORAN IVKOVIC* DRAZEN P A N T I C *

A b s t r a c t

The necessary and sufficient conditions are given so that a non-anticipative transformation in Hilbert space is isometric. In terms of second order Wiener process, these conditions assure that a non-anticipative transformation of Wiener process is a Wiener process, too.

K e y w o r d s Resolution of identity, Separable Hilbert space, Wide sense Wiener

process, Non-anticipative transformation

1991 M R S u b j e c t C lass i f i ca t ion 60G12

C h i n e s e L i b r a r y Class i f i ca t ion O211.6

This paper is an outcome of numerous discusions and exchange of ideas with late Predrag

Peruni~id, whose deep and profound mathematical knowledge and human values gave us an

inspiration and motivation to finalize our joint ideas.

1. We refer to [3,4] for the notion of spectral multiplicity theory in the separable Hilbert

space. Let 7t be a cyclic Hilbert space with the resolution of the identity {7~(t)} of the

maximal spectral type [IP(dt)H 2 = tit-ordinary Lebesgue measure. A non-anticipative linear

transformations is defined by Volterra kernel g(t, u), u <_ t, as

0 t > 0 (0)

t t It means that for ~, ~ E 74, the element P(t){ transforms in fo g( , u)7~(du)~ for each t > 0.

We are going to find the necessary and sufficient conditions so that

~D 1 (t) ~-- g( t , u)7~(du)

defines the resolution of identity {7~1(t)} in 741.7tl C 74, such that again 1[7)l(dt)[[ 2 -- dr, i.e. that the transformation (0) is isometric. Also, we show that there exists the inverse

isometric transformation

P(t) = h(t, ~t)~~ (du)

and determine h(t, u) in terms of g(t, u). In the sequel we shall use the technique of second-order stochastic processes as the curves

in Hilbert space [11.

Manuscript received June 4, 1998.

*Matemati~ki Fakultet, Studentski Trg 16, 11000 Beograd, Yugoslavia.

E -mai l : zivko~mat f.bg.ac.yu

110 CHIN. ANN. OF MATH. Vol.21 Ser.B

2, Let W = {W(t), t > 0} be a standard wide sense Wiener process

( E { W ( t ) } = O, E{W(t)W(s)} = <W(t), W(s)) = I lW(min(s , t)l] 2 -- m in ( s , t ) ) .

Denote be 7/ (W;t) the mean-square linear closure of {W(u) ,u < t},7-/(W) = VT-/(W;t).

Each ~ E ~/(W; t) has the representation

f0 f0 ~---- f(u)W(du), 11~112 = f2(u)du (1)

for some f e L2([0,t]; du). A second order process X = {X(t), t < 0} is a (linear) non-anticipative transformation

of W if X(t) e 7g(W; t) for each t > 0. It follows from (1) that X has the representation

s /0 X ( t ) = g(t,u)W(du), I[X(t)[]2 = g2(t,u)du (2)

for some Volterra kernel g(t, u) E L2([0, t]; du). L~vy [5] gave some examples of processes X which are Wiener processes too. Hida [2]

treated these examples in the framework of the theory of canonical representation of Gauss-

Jan processes.

In this paper we give the necessary and sufficient conditions on the Volterra kernel g(t, u) such that Wt = {Ha(t) . t >_ 0} defined by

/: Wl(t) = 9(t,u)W(du) (3)

is a Wieneer process, i.e. that the linear non-anticipative transformation defined by g(t, u) is isometric. In this case, g(t, u) is called a Wiener kernel.

E x a m p l e 1. Consider a family of second order processes {Xa, a > 0} defined by

, a -~ )W(du). (4)

Then Xa = {Xa(t) , t > 0} is a Wiener process for every a > 0.

For a = 1 we have the well-known L~vy's example

Xl(t)= ( - (5) Treating linear combinations of Wiener kernels from (4), one can prove that for every n _> 1

and ~ 1 , " " , a,~ > 0 we can determine the constants ck(O < k < n) so that the process

k=l

is a Wiener process. For n = 2 and a r we get

x(t) = fo 1 , a - / 3 (ga(t,u) - go(t,u)) + 1

where ga(t, u) is the Wiener kernel defined by (4).

Note that the spaces 7-/(W0;t) and 7-/(W1;t) coincide for each t > 0 if and only if

/: Wl(t) = h(u)Wo(du), (6)

where h2(u) = 1, a.s. We say that W0 and W1 are equivalent and in fact we deal with classes

of equivalences Wiener processes (or Wiener kernels).

No.1 Z. IVKOVIr & D. PANTIr TRANSFORMATIONS OF WIENER PROCESS 111

T h e o r e m 1. Let g(t, u) be an analytic function on {0, u < t}. Then g(t, u) is a Wiener kernel if and only if

(i) g2(t,t) = 1;

(ii) g(t, u) is a homogeneous function, i.e. g(t, u) = g(u/t); (iii) random variable ~ = f t ug'(u/t)dW(u) belongs to 7-t(W; t) e n(W1; t), i.e.

fo (u) ug I -~ g du = O for aU s < t.

Proof . (==~) Differentiating in s and t the relation fo g(t, u)g(s, u)du = s one gets

g~(t,u)g(s,u)du = O, g(t ,s)g(s,s) + g(t,u)g~(s,u)du = 1

respectively. By continuity, letting s $ t we get

/0 /: tg~(t ,u)g(t ,u)du = O, g2(t,t) + g(t ,u)g~(t,u)du = 1,

or g2(t, t) = 1. In the sequel we shall suppose that g(t, t) = 1, due to equivalence classes of

Wiener kernels.

Consider the kernel g(xt, zu), x > 0 and the processes

w~(t) = g(~t, xu)W(du). (7)

It is easy to verify that

/: (w~(8), w~(t)} = g(~t ,~u)g(~8,~u)du = 8, s <_ t.

It means that W~(t) is a Wiener process too. Denote

/0" f ( z , y , s , t ) = {W~(t),W~(s)} = g(:et, xu)g(ys, yu)du, s < t, z , y > 0. (8)

We are going to show that f i x, y, s, t) satisfies the following linear partial differential equa-

tion

x f ~ ( x , y , s , t ) + y f s f ( x , y , s , t ) = O . (9)

Indeed

~0 8 f~ (x, y, s, t) = (tg~ (xt, xu) + ugl2(xt, xu))g(ys, yu)du, (10)

/: f~(x, y, 8,t) = xg~(xt, ~u)g(ys, yu)du, (11)

f ' ( x , y , 8 , t ) = g(~t,~8) + g(, , t ,~u)yg~(ys, yu)du. (12)

We have from (10) and (11)

t f~ ( z , y , s , t ) ] ' ( x , y , s , t ) fo 8 ' = - ug 2 (xt, xu)g(ys, yu)du, X

and by partial integration

I" y g(zt, zu)ug~(ys, yu)du = t f ~ ( z , y , s , t ) - x f ~ ( x , y , s , t ) + s f ~ ( x , y , s , t ) - f ( x , y , s , t ) . (13)

so, adding (12) and (la) we have

u f~ ( z , y , s , t ) = t f~(x ,y , s , t ) - z f ~ ( z , y , s , t ) - f ( z , y , s , t ) ,

112 CHIN. ANN. OF MATH. Vol.21 Ser.B

which is equivalent to (9).

The general solution of Equation (9) is f ( x, y, s , t ) = r where r .) is an

arbi t rary function. If s "t t, by continuity we have

(W~(t), W~(t)) - k~(tx, ~) , x

where ~(. , .) is an arbitrary function. From symmetry in x and y it follows that

x

- - - - ( ) Y Y = ~ ty, �9 (ty, or �9 x y

Put t ing tx = ~ and y / x = ~, we see that the pervious relation becomes

(14) oo

a c~ " ~ Condition (14) rn~n=O

As kl/((~,fl) is an analytic function, we have ~(a , /9) --

yields

E a ~ m N n + l a ~ r n N m - n

m~n~0 m~r~--~0

This equality could be satisfied only if, for n + 1 ~ m - n, i.e. m ~ 2n + 1, coefficients am,~

are equal to zero. Hence, the function ~ ( ~ , ~ ) must be of form k~(a,~) = ~ b~a2'~+l~ n. n=0

Recalling the definition of the function ~ we have

(wx(t) , war)) = b.t2"+lxny" (15) n=0

oQ from ~( t x , y / x ) = ~ bnt2n+lxn+ly n.

~-~0 oo

If x = y, then (Wx(t), Wx(t)) = t, so t = ~ b~t2n+lx 2n, from which it is obvious that

b0 - 1 and bn = 0, n > 1. Then (15) becomes (W~(t) ,Wy(t)) = t, and ]lW~(t)-Wu(t)l l = O.

So, g(xt, xu) does not depend on x, i.e. g(t, u) is a homogeneous function of the form g(u/t) .

The condition (iii) follows from fo g(u / t )g(u /s )du = s, s <_ t, via differentation in t:

os u ~ u u

(r From (iii) we have fo g(u / t )g(u /s )du = r and, as t $ s, fo g2(u/s) du = r

Differentation in s yields r = 1, because of conditions (i) and (iii). Now, r = s -{- C,

where C is an arbi t rary constant. But, since S fo g2(z)dz = ~(s) , we have C = 0 and

/I~(s) = s, which means that g(u/ t) is Wiener kernel. The proof is complete.

Now we shall consider the existence of the inverse transformation of (3). As 7/(W1; t) is

a proper subspace of 7-/(W; t), such a transformation is anticipative.

T h e o r e m 2. Let g be an analytic Wiener kernel. Then 7-l(W~) -- 7-/(W), and the inverse

isometric transformation of (3) is

o

No.1 Z. IVKOVIr & D. PANTIr TRANSFORMATIONS OF WIENER PROCESS 113

or T~(t) = J~o' (j~oU a v 1)T~l(du ) oo ~(~)~+ +/. (/o '~

P r o o f . Since [ tAs - "L~ -

~8(W1(8)'W(')) joftAs 0 U

it is evident that .pw, (W(t)) (the projection of W(t) on 7-/(W1)) is equal to the right-hand

side in (16). So we must prove that

7 )W' (W(t)) = W(t). (17)

First of all, as

and O g ( ~ ) E L2([O,t];du), it follows that ~ E L2([t, oo),du), so

,~,(~,,,,__ (/o,, ~ u Xvg(7.) w(d,~)) ~v + w(d,.,),

Putt ing x = ~ , we can easily see that oo 0 /.v~,(~)~ v F o o .=.o (~)~=_ o ( ,~

Therefore

/7~ = _ f 0 ~ O u

From the relations (19) and (20) follows

W,(dt) = _ ( ~~176 0 t

Hence

~ + = ~ , + + / o ' ( / 7 ~ o

/o= (/7 (u)) = Wl(t) + -~vg du Wl(dv)

-~ ~ot ( ~ooU o ~_~u g(v)dv-}- 1)Wl(du) + foo ( ~o t ~___~g(V)dv)Wl(du) = v w , (w(t))

(18)

(19)

(2o)

114 CHIN. ANN. OF MATH. Vol.21 Ser.B

and the statement is proved. R e m a r k . Note that if we consider Wiener process {W(t), 0 < t < T} on the finite

interval, the inverse transformation does exist.

Denote

~ ( s , t) = IIW(t) - p wl (t))ll ~, pw1 is the projection operator on ~-~(W1;8 ). For s >_ t we have a2(s , t ) = t r where

where r could be determined from the equation

/0 ) 1 sg'(s)ds 2 r = ~

E x a m p l e 3. Consider L~vy's kernel (5)

In that case

x > 0 .

o g = - 2 + 3 ~ , O < u < t .

If we project W(t) on 7/(W1; s), then for s >_ t,

~W*(w( t ) ) = - ~ l(t) 3t2 Wl(du), 3t 2

a2(s,t) = 4s 2.

REFERENCES

[ 1 ] Cram~r, H., Stochastic processes as curves in Hilbert space, Theory Probability Appl., 9:2(1964), 169- 179.

[ 2] Hida, T., Canonical representations of Gaussian processes and their applications, Mere. College Sei. Univ. Kyoto., A33:1(1960), 109-155.

[3] Stone, M. H., Linear transformations in Hilbert space and their applications to analysis, Am. Math. Soc. Colloquium Publications, 25, New York, 1932.

[4] Plessner, A. I. & Rokhlin, V. A., Spectral theory of linear operators, Uspekhi Mat. Nauk., 1(1964), 71-191 (in Russian).

[ 5] L~vy, P., Sur une class~ de courbes de l~space de Hilbert et sur une ~quation integral non-lin~aire, Ann. Sci. Eeole Normale Superieure, 73(1956), 121-156.