Some remarks on boundary value problems for linear stochastic differential equations

R ^ENDICONTI

del

S ^EMINARIO M ^ATEMATICO

della

U NIVERSITÀ DI P ^ADOVA

A ^NTONI L. D ^AWIDOWICZ K RYSTYNA T WARDOWSKA

Some remarks on boundary value problems for linear stochastic differential equations

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Boundary

for Linear Stochastic Differential Equations.

ANTONI L. DAWIDOWICZ - KRYSTYNA TWARDOWSKA

(*)

SUMMARY - In this paper ^two

negative

^results

dealing

^with

boundary

^condi-

tions for stochastic differential

equations

^{are examined.}

1. Introduction.

For _years there have been considered some theorems in which the

uniqueness

of solutions of some

boundary

^value

problems implies

the existence of solutions of these

problems.

^{We mean such theorems}

for

algebraic equations, integral equations (the

Fredholm’s alterna-

tive),

deterministic differential

equations ( [1] ).

The authors have al-

ready

^{made an}

attempt

^to obtain similar results for random diffe- rential

equations ([2])

in which the stochastic

integral

^{did not occur.}

The

consequent

continuation of the researches

originated

ⁱⁿ the paper mentioned above is an

attempt

^at

obtaining

^the

analogous

^results

for the stochastic differential

equations.

In the one paper

([3])

known to the authors on

boundary

^value

problems

with the It6 stochastic

integral,

^this

integral

appears in the solution formula

only,

whereas the

equation

^{is a random one without}

stochastic

integral

^{and the}

boundary

condition is a deterministic one.

(*)

^Indirizzo

degli

AA.: A. L. DAWIDOWICZ : Institute of Mathematics,

Jagiellonian University,

^ul.

Reymonta

4, ^30-059 Krak6w, Poland; ^{K. TwAR-}

DOWSKA :

Department

^of Informatics,

Jagiellonian University,

^ul.

Kpernika

27, 31-501 Krak6w, ^Poland.

6

In the

present

paper ^we have

attempted

to obtain some results

concerning

the relation between the existence and the

uniqueness

^of

the solutions for linear stochastic differential

equations

^{with the}

stochastic

boundary

conditions.

Unfortunately,

y we have received

negative

results which may,

however,

^be

interesting

from the metho-

dological point

^{of view. We have found that the reason of the situa-}

tion is the fact that the diffusion _process can be

uniquely

^determined

by

^{means of the}

boundary

condition which is measurable with

respect

to

c-algebra

much poorer than this

ar-algebra

to which refers the dif- fusion process.

In this paper the authors have used the

topological properties

^of

the Hilbert spaces

([4]).

2. Notations.

Let

(Q, Y, P)

^{be a}

^complete probability

space, where is a

right

continuous filtration.

We consider the linear stochastic differential

equation

with a

general

stochastic

boundary

^condition

where _xt: is a d-dimensional stochastic process and 1vt = ... , is and

n-dimensional, {Ft}-adapted

Wiener pro-

cess in S~.

i ⁼

1; ... ; n.

Furthermore we shall denote

E21.Q Y)

⁼

Y; Rd),

0([0, T])

⁼

0([0, T],

By ~~ t~ t&#x3E;o

^{we define the}

family

^{of the}

or-algebras generated by

^the

process Wt.

We take the

mapping A : L2(Q, ^0([0, T])) ^{~ L2(S~,}

^{such that}

3. Existence and

uniqueness ^theorems.

We shall prove the

following

THEOREM 1. Let the functions

A(t), a(t), bi(t)

^{for i =}

17

... , n

be

progressively

measurable. Let

t

^for

every t

^E

[0, T].

Then the set of the random variables y ^E

L2(Q, :Fp)

^{such that}

problem (1), (2)

^{has a} solution is dense in

T,2 (S?

PROOF. Let us consider the

mapping

which to every z ^E Rd

applies

the solution of

equation (1)

with condition

The

composition

^{AT: .Rd -7}

L2(Q, YT)

^{is a} linear and continuous

mapping. Therefore,

^the

image

^{of this}

mapping

^{is a nonwhere dense}

set in the space

L2(Q, ^Y,).

^{But if x} is the solution of

equation (1),

then it is because the

family

^is

generated by

^the

process

Therefore,

xo must be such that

Xo E Rd.

^{This means that}

for every solution x of

equation (1)

^{we have}

But im AT is a finite dimensional

subspace

^of

L2(S~,

^{and hence}

the theorem is

proved.

Now we may state

THEOREM 2. If

A(t), a(t), Bi(t), bi(t)

^are stochastic processes pro-

gressively

measurable with

respect

^to then there exists such a

that

problem (1), (2)

^{has no solution.}

PROOF. Let ¹ Then the

process xt

^satisfies

equation

8

which results from the

properties

^{of the} conditional

expectation.

From the factorization

condition

for the Hilbert spaces there exists such a

1: L2(Q, ^{T; C([0,} Z’])~ ^{- L2(Q, ,~T)}

^that

where

This

equation (5)

^{with the} condition , fulfills the as-

sumptions

of Theorem

1,

^which

completes

^the

proof.

REMARK. Theorem 2 remains true if

only .A(t),

^are

{F’t}-progressively

measurable and

a(t),

^are

{Ft}-progressively

measurable.

4. Conclusions.

The above theorems

imply that,

ⁱⁿ

general,

^it is useless to con-

sider the

boundary

^value

problems

for the stochastic differential

equations.

An

exception might

^be

boundary

conditions like

as well as some

exceptional

situations

when,

^for

example,

^{we can}

construct an

isomorphism

^between the spaces

L2(Q, Yo)

^and

L2(S~, yr).

Such an

isomorphism

exists when

in

the space

Yo)

^{is determined}

the

canonical {F0}-measurable ^Wiener

^process.

REFERENCES

[1]

S. N. CHOW - A. LASOTA, On

boundary

^value

problems for ordinary dif- ferential equations,

^{J. Diff.}

Eqs.,

¹⁴

(1973),

pp. 326-337.

[2] ^{A. L.} DAWIDOWICZ - ^K. TWARDOWSKA, On random

boundary

^value

prob-

lems

for ordinary differential equations,

Rend. Sem. Mat. Univ. Padova, 76

(1986),

pp. 163-169.

[3]

^A. RUSSEK, Gaussian n-Markovian processes and stoachastic

boundary

value

problems,

Z. Wahrscheinlichkeitstheorie und Verw.

Gebiete,

⁵³

(1980),

pp. 117-122.

[4]

^{R. E. EDWARDS,} Functional

Analysis, Theory

^and

Applications, ^Holt,

Rinehart and Winston, ^1965.

[5]

^L. ARNOLD, Stochastic

Differential Equations,

^J.

Wiley

^and

Sons,

^New York, ^1974.

Manoscritto

pervenuto

in redazione il 25

giugno

^1986.