31

32 CHAPTER 7

THEOREM 7.1 Let „¦ be the probability space of the di¬erences of the

two-sided Wiener process. There is a unique isometric operator from

L2 (‚, σ 2 dt) into L2 („¦), denoted by

∞

f’ f (t) dw(t),

’∞

such that for all ’∞ < a ¤ b < ∞,

∞

χ[a,b] (t) dw(t) = w(b) ’ w(a).

’∞

∞

The set of f (t) dw(t) is Gaussian.

’∞

If E is any set, χE is its characteristic function,

1, t ∈ E

χE (t) =

0, t ∈ E.

∞

b

We shall write, in the future, f (t) dw(t) for χ[a,b] (t)f (t) dw(t).

’∞

a

Proof. Let f be a step function

n

f= ci χ[ai ,bi ] .

i=1

Then we de¬ne

n

∞

ci [w(bi ) ’ w(ai )].

f (t) dw(t) = (7.1)

’∞ i=1

If g also is a step function,

m

g= dj χ[ej ,fj ] ,

j=1

THE WIENER INTEGRAL 33

then

∞ ∞

E f (t) dw(t) g(s) dw(s)

’∞ ’∞

n m

ci [w(bi ) ’ w(ai )] dj [w(fj ) ’ w(ej )]

=E

i=1 j=1

n m

ci dj σ 2 |[w(bi ) ’ w(ai )] © [w(fj ) ’ w(ej )]|

=

i=1 j=1

∞

2

=σ f (t)g(t) dt.

’∞

Since the step functions are dense in L2 (‚, σ 2 dt), the mapping extends

by continuity to an isometry. Uniqueness is clear, and so is the fact that

the random variables are Gaussian. QED.

The Wiener integral can be generalized. Let T , µ be an arbitrary

measure space, and let S0 denote the family of measurable sets of ¬nite

measure. Let w be the Gaussian stochastic process indexed by S0 with

mean 0 and covariance r(E, F ) = µ(E © F ). This is easily seen to be of

positive type (see below). Let „¦ be the probability space of the w-process.

THEOREM 7.2 There is a unique isometric mapping

f’ f (t) dw(t)

from L2 (T, µ) into L2 („¦) such that, for E ∈ S0 ,

χE (t) dw(t) = w(E).

The f (t) dw(t) are Gaussian.

The proof is as before.

If H is a Hilbert space, the function r on H — H that is the inner

product, r(f, g) = (f, g), is of positive type, since

¯ 2

≥ 0.

ζi (fi , fj )ζj = ζj fj

j

34 CHAPTER 7

Consequently, the Wiener integral can be generalized further, as a purely

Hilbert space theoretic construct.

THEOREM 7.3 Let H be a Hilbert space. Then there is a Gaussian

stochastic process, unique up to equivalence, with mean 0 and covariance

given by the inner product.

Proof. This follows from Theorem 6.1. QED.

The special feature of the Wiener integral on the real line that makes

it useful is its relation to di¬erentiation.

THEOREM 7.4 Let f be of bounded variation on the real line with com-

pact support, and let w be a Wiener process. Then

∞ ∞

f (t) dw(t) = ’ df (t) w(t). (7.2)

’∞ ’∞

In particular, if f is absolutely continuous on [a, b], then

b b

f (t) dw(t) = ’ f (t)w(t) dt + f (b)w(b) ’ f (a)w(a).

a a

The left hand side of (7.2) is de¬ned since f must be in L2 . The right

hand side is de¬ned a.e. (with probability one, that is) since almost ev-

ery sample function of the Wiener process is continuous. The equality in

(7.2) means equality a.e., of course.

Proof. If f is a step function, (7.2) is the de¬nition (7.1) of the Wiener

integral. In the general case we can let fn be a sequence of step functions

such that fn ’ f in L2 and dfn ’ df in the weak-— topology of measures,

so that we have convergence to the two sides of (7.2). QED.

References

See Doob™s book [15, §6, p. 426] for a discussion of the Wiener integral.

The purely Hilbert space approach to the Wiener integral, together with

applications, has been developed by Irving Segal and others. See the

following and its bibliography:

THE WIENER INTEGRAL 35

[16]. Irving Segal, Algebraic integration theory, Bulletin American Math.

Soc. 71 (1965), 419“489.

For discussions of Wiener™s work see the special commemorative issue:

[17]. Bulletin American Math. Soc. 72 (1966), No. 1 Part 2.

We are assuming a knowledge of the Wiener process. For an exposition

of the simplest facts, see Appendix A of:

[18]. Edward Nelson, Feynman integrals and the Schr¨dinger equation,

o

Journal of Mathematical Physics 5 (1964), 332“343.

For an account of deeper facts, see:

[19]. Kiyosi Itˆ and Henry P. McKean, Jr., “Di¬usion Processes and their

o

Sample Paths”, Die Grundlehren der Mathematischen Wissenschaften in

Einzeldarstellungen vol. 125, Academic Press, Inc., New York, 1965.

Chapter 8

A class of stochastic

di¬erential equations

By a Wiener process on ‚ we mean a Markov process w whose in-

¬nitesimal generator C is of the form

‚2ij

C= c , (8.1)

‚xi ‚xj

i,j=1

where cij is a constant real matrix of positive type. Thus the w(t) ’ w(s)

are Gaussian, and independent for disjoint intervals, with mean 0 and

covariance matrix 2cij |t ’ s|.

THEOREM 8.1 Let b : ‚ ’ ‚ satisfy a global Lipschitz condition; that

is, for some constant κ,

|b(x0 ) ’ b(x1 )| ¤ κ|x0 ’ x1 |

for all x0 and x1 in ‚ . Let w be a Wiener process on ‚ with in¬nitesimal

generator C given by (8.1). For each x0 in ‚ there is a unique stochastic

process x(t), 0 ¤ t < ∞, such that for all t

t

b x(s) ds + w(t) ’ w(0).

x(t) = x0 + (8.2)

0

The x process has continuous sample paths with probability one.

If we de¬ne P t f (x0 ) for 0 ¤ t < ∞, x0 ∈ ‚ , f ∈ C(‚ ) by

P t f (x0 ) = Ef x(t) , (8.3)

37

38 CHAPTER 8

where E denotes the expectation on the probability space of the w process,

then P t is a Markovian semigroup on C(‚ ). Let A be the in¬nitesimal

generator of P t . Then Ccom (‚ ) ⊆ D(A) and

2

Af = b · f + Cf (8.4)

2

for all f in Ccom (‚ ).

Proof. With probability one, the sample paths of the w process are

continuous, so we need only prove existence and uniqueness for (8.2) with

w a ¬xed continuous function of t. This is a classical result, even when w is

not di¬erentiable, and can be proved by the Picard method, as follows.

Let » > κ, t ≥ 0, and let X be the Banach space of all continuous

functions ξ from [0, t] to ‚ with the norm

ξ = sup e’»s |ξ(s)|.

0¤s¤t

De¬ne the non-linear mapping T : X ’ X by

s

b ξ(r) dr + w(s) ’ w(0).

T ξ(s) = ξ(0) +

0

Then we have

s

’»s

Tξ ’ T· ¤ |ξ(0) ’ ·(0)| + sup e [b ξ(r) ’ b ·(r) ] dr

0¤s¤t 0

s

’»s

¤ |ξ(0) ’ ·(0)| + sup e |ξ(r) ’ ·(r)| dr

κ

0¤s¤t 0

s

’»s

e»r ξ ’ · dr

¤ |ξ(0) ’ ·(0)| + sup e κ

0¤s¤t 0

= |ξ(0) ’ ·(0)| + ± ξ ’ · , (8.5)

where ± = κ/» < 1. For x0 in ‚ , let Xx0 = {ξ ∈ X : ξ(0) = x0 }. Then

Xx0 is a complete metric space and by (8.5), T is a proper contraction

on it. Therefore T has a unique ¬xed point x in Xx0 . Since t is arbitrary,

there is a unique continuous function x from [0, ∞) to ‚ satisfying (8.2).

Any solution of (8.2) is continuous, so there is a unique solution of (8.2).

Next we shall show that P t : C(‚ ) ’ C(‚ ). By (8.5) and induction

on n,

T n ξ ’ T n · ¤ [1 + ± + . . . + ±n’1 ] |ξ(0) ’ ·(0)| + ±n ξ ’ · . (8.6)

A CLASS OF STOCHASTIC DIFFERENTIAL EQUATIONS 39

If x0 is in ‚ , we shall also let x0 denote the constant map x0 (s) = x0 ,

and we shall let x be the ¬xed point of T with x(0) = x0 , so that

x = lim T n x0 ,

n’∞

and similarly for y0 in ‚ . By (8.6), x ’ y ¤ β|x0 ’ y0 |, where β =

1/(1 ’ ±). Therefore, |x(t) ’ y(t)| ¤ e»t β|x0 ’ y0 |. Now let f be any

Lipschitz function on ‚ with Lipschitz constant K. Then