7/25/2019 Non-Unique Solution for SDE

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Non-uniqueness o[

a solution

o[

Ito s

stochastic

equation

325

hence

k

[]

[o3

k

Ok

7/25/2019 Non-Unique Solution for SDE

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326

I.V.

Girsanov

, 1)

for

a

Markov diffusion

process.

Here

a s,

m)

and

re s,

are

real

functions,

s(o)

is a

Wiener

pro-

cess.

In

the

proof,

Ito assumes

that

a and m

satisfy

a

Lipschitz

condition

in

m

and

that

2)

These

results

were

sharpened

by

a

series

of

authors. In

a

recent

work

[2]

A. V. Skorokhod

showed

the

existence

of a

solution

of

equation

1)

for continuous

coefficients

a

and m

satisfying

condi-

tion

2).

Assuming in addition

that

a and m

satisfy

a

H61der

condition

in

x

with

exponent

>

1/2,

Skorokhod showed the

uniqueness

of

the

solution

obtained.

It was

shown

by

the

author

of

the

present

note

[3]

that

the

uniqueness

of

the solution of

1)

holds

also

for

weaker

assumptions,

in

particular,

for

any

>

0

if

only

a

>

a

0

>

0.

This result

is

easily

carried over

to

n-dimensional diffusion

processes.

An

example

will

be

cited

below of

an

equation

for

which the

uniqueness theorem

does

not

hold,

and

all its

sufficiently

nice solutions

are described.

2.

Fundamental

concepts

and

statement

of the

problem

The stochastic

equation

1+

I,,, o.,)

is

considered.

First

we make

more precise

what

we

mean

by

the

solution

of

the

equation.

Follow-

ing

[],

by

a

Markov process

we

mean

a set X

{at a

), (w),

,,

Ps,

a(d)}

consisting

of

a

func-

tion

at w)

on

Q

with values

in

the

measurable

phase space

E ,

),

its domain

of definition

0

v p,n

>=pilog.

i=I i=i

The

exact

vlue

of

v(p,

n)

is

equl

to

the

minimum

of

i=lmii

for

mi

which

are

integral

and sub-

ject

to

Che

limitatiou

()

+

+

--2

see,

for

example,

[2],

Theorems

7,

8).

The

purpose

of the

present

note

is to

obtain

nil

exact

expression

for

v

p,

n)

in

the case

when

Pl

Pn-1

+

Pn

Lemma

1.

There

exists an

optimal

i n

the

sense

o[

Hu[[man)

method

o[

coding

[o r

which

m