Gliklikh Y.E. Global and Stochastic Analysis with Applications to Mathematical Physics

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116 6 Essentials from Stochastic Analysis in Linear Spaces

ω ∈ Ω the curve η(t, ω) is continuous in t. The space of continuous curves

([0, ∞), R

) in this case is called the space of (sample) paths (or trajecto-

ries).

Specify a ﬁnite number t

,...,t

∈ [0, ∞) of time instants and a ﬁnite

collection of Borel sets B

,...,B

⊂ R

.Acylinder set in C

([0, ∞), R

corresponding to the above collections of times and Borel sets, is the set of

curves

,...,t

× (B

,...,B

)={x(·) ∈ C

([0, ∞),R

) | x(t

) ∈ B

i.e., the set of curves that at time t

take a value in the set B

, and arbitrary

values at the other times. The minimal σ-algebra that contains all cylinder

sets for a ﬁnite time interval t ∈ [0,T] ⊂ R coincides with the Borel σ-

algebra on the Banach space C

([0,T], R

) of continuous curves in R

given

on [0,T], with the usual uniform norm. A stochastic process with a.s. contin-

uous sample paths is usually considered as a random variable with values in

([0, ∞), R

) equipped with the σ-algebra generated by cylinder sets, i.e., a

measurable mapping from Ω to C

([0, ∞), R

) with respect to the σ algebra

generated by cylinder sets in C

([0, ∞), R

)andtheσ-algebra F in Ω.

Denote C

([0, ∞), R

)by

Ω and the σ-algebra generated by cylinder sets

F. Then, since a process ξ(·) can be considered as a measurable mapping

from (Ω,F)to(

Ω,

F), it generates a probability measure μ

on (

Ω,

F)inthe

usual way: μ

(A)=P(ξ

−1

(A)) for A ∈

F. This measure is called the measure

generated by ξ(t)on(

Ω,

F)orthedistribution of ξ(t).

For any probability measure μ given on (

Ω,

F), one can construct a

stochastic process η

(t) on the probability space (

Ω,

F,μ) with values in

as follows: η

(t, ω)=ω(t) where the elementary event ω ∈ C

([0, ∞), R

)

is by deﬁnition a continuous curve ω :[0, ∞) → R

. The process ξ

(t)is

called the coordinate process on the probability space (

Ω,

F,μ).

Note that for μ = μ

generated by a stochastic process ξ(t) with continuous

path in R

, the corresponding coordinate process has, by construction, the

same distribution as ξ(t).

Every stochastic process ξ(t)inR

, t ∈ [0,T], given on a probability space

(Ω,F, P), determines three families of σ-subalgebras of the σ-algebra F:

(i) “past” P

, generated by pre-images of Borel sets in R

by all mappings

ξ(s):Ω → R

for 0 <s<t;

(ii) “future” F

, generated by pre-images of Borel sets in R

by all map-

pings ξ(s):Ω → R

for t<s<T;

(iii) “present” (“now”) N

, generated by the mapping ξ(t).

We suppose that all of these families are complete, i.e., contain all sets of

probability zero: P =0.

Let B

be a non-decreasing family of σ-subalgebras of the σ-algebra F.

6.1 Deﬁnitions from probability theory 117

Deﬁnition 6.1. A random process A(t) is said to be non-anticipative with

respect to a ﬁltration B

if A(t) is measurable with respect to B

for every t.

Consider the measure space (

Ω,

F) introduced above. Denote by

the

σ-algebra generated by cylinder sets with bases over [0,t]. Note that for any

probability measure μ on (

Ω,

F) the coordinate process ξ

(t) on the proba-

bility space (

Ω,

F,μ) is non-anticipative with respect to

and moreover,

is its “past”.

6.1.2 Conditional expectation

Consider the Hilbert space L

(Ω,F, P) of square integrable random variables.

Let F

be a σ-subalgebra in F. Consider L

(Ω,F

, P), the Hilbert space of

square integrable random variables that are measurable with respect to F

It is clear that L

(Ω,F

, P) is a closed subspace in L

(Ω,F, P). Denote by

Q : L

(Ω,F, P) → L

(Ω,F

, P) the orthogonal projector. The projector Q

extends to the projector in the corresponding space L

of integrable random

variables.

Deﬁnition 6.2. For every ξ ∈ L

(Ω,F, P) the random variable Qξ ∈

(Ω,F

, P) is called the conditional expectation of ξ with respect to F

and is denoted by E(ξ|F

It is important to point out that (up to sets with probability zero) E(ξ|F

)

is the unique random variable in L

(Ω,F, P) such that for every set A ∈F

the equality



ξdP =



E(ξ|F

)dP holds. Recall that the existence of such

a function follows from the Radon-Nikodym theorem. This description of

E(ξ|F

) is equivalent to Deﬁnition 6.2 and is often used as the deﬁnition in

the probabilistic literature (see e.g., [208]). The details of the approach based

on Deﬁnition 6.2 are given, e.g., in [194].

It is not hard to see that the usual mathematical expectation is the condi-

tional expectation with respect to the trivial σ-algebra comprising two sets:

∅ and Ω.

Let A ∈Fand χ

be the indicator of A. The value P(A|F

)=E(χ

)

is called a conditional probability.

Let us describe some properties of conditional expectation. These proper-

ties generally follow from the properties of projectors. A detailed presentation

of this material can be found in [194, 208]

Theorem 6.3

(i) Conditional expectation is a linear operator.

(ii) If η is measurable with respect to F

, E(η|F

)=η.

(iii) If F

⊂F

, E(E(η|F

)|F

)=E(η|F

). In particular, the equality

E(E(η|F

)) = Eη holds.

118 6 Essentials from Stochastic Analysis in Linear Spaces

(iv) If F

⊂F

, E(E(η|F

)|F

)=E(η|F

(v) If η does not depend on F

, E(η|F

)=Eη.

(vi) Let ξ and η be random variables with values in R.Ifξ is measurable

with respect to F

, then for every η the equality E(ξη|F

)=ξE(η|F

)

holds. In the case of vector-valued random variables this property is

valid for inner products.

Let ξ and η be random variables given on a probability space (Ω,F, P)

and taking values in R

.ByE(ξ|η) we denote the conditional expectation

of ξ with respect to the σ-algebra generated by pre-images of Borel sets in

under the mapping η : Ω → R

. One can easily show (see, e.g., [194])

that there exists a unique Borel measurable mapping Y : R

→ R

such that

E(ξ | η)=Y (η). The mapping Y is called the regression of ξ with respect to

η and is usually denoted by the expression Y (x)=E(ξ | η = x).

6.1.3 Markov processes

Let a non-decreasing family B

of σ-subalgebras of a σ-algebra F, t ∈ [0, ∞),

be given. A random process ξ(t) is called a Markovian (or Markov) process

with respect to B

if P-a.s. P(B ∩F |N

)=P(B |N

) · P(F |N

) for every

t ∈ [0, ∞), B ∈B

and F ∈F

(see the Deﬁnition of “past”, “future” and

“present” in Section 6.1.1).

A process ξ(t) is called a simply Markovian (or simple Markov) process if

it is Markovian with respect its own “past” P

The next two conditions are equivalent to each other and to the fact that

the process is Markovian with respect to B

1) For every t ∈ [0, ∞) and every bounded F

-measurable random variable

ϕ with values in R the relation E(ϕ |B

)=E(ϕ |N

) holds.

2) For t ≥ s ≥ 0 and every (measurable) function f(x), for which

sup

x∈R

|f(x)| < ∞, the equality E(f(ξ(t)) |B

)=E(f (ξ(t)) |N

) holds.

A random variable τ(ω) taking values in [0, ∞) is called a random time.

A random time is called a Markov time if for every t ≥ 0 the inclusion

{ω | τ(ω) ≥ t}∈B

holds. If P(τ(ω) < ∞) = 1, the Markov time is called a

stopping time.

6.1.4 Martingales and semi-martingales

A stochastic process η(t) is called a martingale with respect to a non-

decreasing family of σ-algebras B

, t ∈ [0, ∞), if for every t the variable

6.1 Deﬁnitions from probability theory 119

η(t) is measurable with respect to B

(i.e., non-anticipative with respect to

)andforeveryt ≥ s ≥ 0 the equality E(η(t) |B

)=η(s) holds.

Directly from the deﬁnition of martingale and property (iii) of conditional

expectation (see Section 6.1.2) it follows that the mathematical expectation

of a martingale is constant.

A process η(t) is called a local martingale if there exists a non-decreasing

sequence of Markov times (stopping times) τ

such that lim τ

= ∞ and for

every τ

(ω) the process η(t∧τ

) is a martingale, where t∧τ

=min(t, τ

(ω)).

A stochastic process η(t) is called a semi-martingale if η(t)=A(t)+M(t)

where M(t) is a local martingale and A(t) is a process whose sample paths a.s.

have bounded variation in t (i.e., the Stieltjes integral of integrable functions

is well-deﬁned with those paths as integrators).

It is clear that a martingale is a local martingale and that a local martin-

gale is a semi-martingale. Note that there is a construction of the integration

of random functions with respect to semi-martingales (see [176]), a particular

case of which is the stochastic integral with respect to the Wiener process

that is described below in Section 6.2.

Under a smooth change of coordinates in R

martingales and local mar-

tingales do not transform into analogous processes, but semi-martingales are

transformed into semi-martingales. A decomposition into the sum of a local

martingale and a process of bounded variation is possible but in this decom-

position the summands are not the results of transformation of corresponding

summands in the previous coordinate system. From this property it follows

that the notion of a semi-martingale with values on a manifold is well-deﬁned.

A stochastic process ξ(t) is called a backward martingale with respect to a

non-increasing family of σ-algebras F

if for every t the variable ξ(t)ismeasur-

able with respect to F

and for every s ≥ t ≥ 0 the equality E(η(t)|F

)=η(s)

holds.

6.1.5 Weak convergence of probability measures

A detailed presentation of this material can be found in, e.g., [25, 194, 208].

Everywhere in this Section the symbol X denotes a separable complete metric

space and B is its Borel σ-algebra (i.e., the minimal σ-algebra generated by

open sets). Recall that a measure μ on (X, B) is called a probability measure

if μ(X)=1.

Deﬁnition 6.4. A sequence of probability measures μ

on (X, B) weakly con-

verges to a probability measure μ



f dμ

→



f dμ

for every continuous

bounded function f : X → R.

120 6 Essentials from Stochastic Analysis in Linear Spaces

Deﬁnition 6.5. A family of probability measures {μ

} on (X, B) is called

weakly relatively compact if every sequence of measures from {μ

} has a

weakly convergent subsequence.

The measure to which the subsequence converges in Deﬁnition 6.5 may

notbelongtotheset{μ

}. For short we shall often call weakly relatively

compact set of measures weakly compact, omitting the word “relatively”.

Theorem 6.6 (Prokhorov’s theorem) A family of probability measures M

on (X, B) is weakly relatively compact if and only if for every ε>0 there

exists a compact K

⊂ X such that μ

) > 1 −ε for every μ

∈M.

6.2 A Survey on Stochastic Integrals and Equations

In this section we brieﬂy describe the basic facts from the theory of stochastic

integrals and stochastic diﬀerential equations necessary for understanding

their geometric properties below. We consider only integrals with respect to

a Wiener process since they play the main role in the forthcoming sections.

A complete and detailed exposition of this material can be found in many

monographs and textbooks (see, e.g., [76, 83, 84, 162, 165, 175, 176, 216]).

We should particularly highlight the excellent introductory paper [50]which

illuminates those aspects of the theory that are especially important for our

approach.

6.2.1 White noise and Wiener processes

We begin this section with a physically motivated observation that leads to

an intuitive introduction to Wiener processes (for details, see [160]).

Consider an ordinary diﬀerential equation ˙x(t)=F (t, x(t)) in R

and

suppose that its right-hand side is subjected to an additive random inﬂuence

that satisﬁes the following physically natural assumptions:

(a) the mechanism that produces the randomness is the same at all times;

(b) the randomness occurs at any time t independently of the other times;

the dispersion equals 1.

We can interpret (a) as saying that the process has independent values at

diﬀerent times and (b) as saying that the distributions of the values at all

times are the same. Assumption (c) is given for simplicity; one can consider

more general processes satisfying (a) and (b).

The above-mentioned process is denoted by ˙w(t) and is called white noise.

It turns out that this process takes values in generalized functions and so it

6.2 A Survey on Stochastic Integrals and Equations 121

is rather diﬃcult to deal with. We shall avoid the use of generalized func-

tions in the usual way: we introduce the process w(t)=



˙w(s)ds.From

the properties (a)–(c) of ˙w(t) we intuitively derive that w(t)musthavea.s.

continuous sample paths and independent increments such that for a given

diﬀerence t − s = δ all increments w(t

) − w(s

) with t

− s

= δ have the

same distribution. Finally, for all such increments, E(w(t) − w(s)) = 0 and

E(w(t) − w(s))

= t −s.

The diﬀerential equation we started with, with the above random inﬂuence,

has the form ˙x(t)=F (t, x(t)) + ˙w(t). Avoiding the use of generalized func-

tions, we transform it into the integral equation x(t)=x



F (s, x(s))ds +

w(t). For a diﬀerential equation without random inﬂuence this transformation

yields the equivalent integral equation. In the presence of random inﬂuence

the transformation makes the equation easier to work with since it is given

in terms of processes with continuous sample paths.

w(t) has all the properties of a certain process, called a Wiener process,

that we want to formally introduce in this Section. The precise deﬁnition

requires the following abstract scheme.

Let (Ω,F, P) be a probability space and B

, t ∈ [0, ∞), be a nondecreasing

family of σ-subalgebras of the σ-algebra F. In what follows, we assume that

the σ-algebras B

are complete, i.e., they contain all sets from F of measure

zero.

Here we consider only stochastic processes (random variables) on (Ω,F, P)

with values in a Euclidean space with an inner product (·, ·). Specifying a

basis, we shall describe its vectors by columns of coordinates, i.e., we identify

this space with R

Deﬁnition 6.7. A stochastic process w(t) is called a Wiener process (relative

to the family B

)if:

1) the sample paths of w(t) are almost surely (a.s.) continuous in t;

2) w(t) is a square integrable martingale with respect to B

;

3) w(0) = 0 and E((w(t) − w(s))

)=t −s for t ≥ s.

In this case it is said that the Wiener process w(t)isadapted to B

From Deﬁnition 6.7 we deduce that the Wiener process has the (intuitive)

properties that we listed the beginning of this Section:

Theorem 6.8 (Levi, see, e.g., [175]) If w(t) is a Wiener process, then it has

stationary independent Gaussian increments. Furthermore, w(t) satisﬁes the

following conditions: E(w(t) − w(s)) = 0 and E((w(t) −w(s))

)=t − s for

t ≥ s.

In other words, for t ≥ s, the increment w(t) −w(s) is independent of B

and has the same probability distribution as w(t − s).

The distribution density ρ

(t, x) of a Wiener process in R

is described

by the formula (see, e.g., [162])

122 6 Essentials from Stochastic Analysis in Linear Spaces

(t, x)=

(2πt)

−

. (6.1)

One can easily see that P

⊂B

where P

is the “past” of w(t)(see

Section 6.1.1).

Consider the space

Ω = C

([0, ∞), R

) of continuous curves in R

given on

the semi-inﬁnite interval [0, ∞). Introduce in

Ω the σ-algebra

F generated by

cylinder sets (see Section 6.1.1). As for the other processes with continuous

sample paths, every Wiener process can be regarded as a mapping of the

measure space (Ω,F) into the measure space (

Ω,

F). Thus, P gives rise to a

measure ν on (

Ω,

F) according to the construction given in Section 6.1.1.The

measure ν, called the Wiener measure, depends only on the inner product on

, not on a speciﬁc Wiener process w(t). The Wiener measure enables one to

introduce the probability distributions of w(t)in

Ω = C

([0, ∞), R

), i.e., the

conditional probability distributions of the random variables w(t

),...,w(t

)

in R

for all collections t

,...,t

Consider (

Ω,

F,ν) as a probability space and deﬁne the coordinate process

¯w(t)on(

Ω,

F,ν) (see Section 6.1.1)via ˜w(t, ω)=ω(t) (here the elementary

event ω ∈ C

([0, ∞), R

) is by deﬁnition a continuous curve ω :[0, ∞) → R

Consider the σ-algebra

generated by the cylinder sets with base over [0,t],

i.e.,

= P

¯w

. Clearly, ¯w(t) is a Wiener process relative to the family

The process ¯w(t) is called a Brownian motion process or a standard Wiener

process.

Remark 6.9. Often in the probability literature one ﬁnds that the Wiener

process is described as unique. This phrase means only that the standard

Wiener process is unique since the Wiener measure ν (as well as the coordi-

nate process) is unique on (

Ω,

F). But there are plenty of concrete realiza-

tions of Wiener processes on various probability spaces. In particular diﬀerent

Wiener processes can be independent.

To facilitate further references, we summarize here some results on Wiener

processes.

Theorem 6.10 Any Wiener process w(t) has the following properties:

1) A sample path of w(t, ω) is a.s. (i.e., with probability 1) non-diﬀerenti-

able for all t and has unbounded variation on any arbitrarily small in-

terval.

2) The coordinates w

(t) of w(t) are one-dimensional Wiener processes

that are mutually independent and the orthogonal projection of w(t) to

any k-dimensional subspace of R

is a k-dimensional Wiener process.

3) Let a be an orthogonal operator in R

.Thena◦w(t) is a Wiener process.

In particular, if ¯w(t) is a standard Wiener process, then so is a ◦ ¯w(t),

i.e., the Wiener measure is invariant under the action of orthogonal

operators on R

6.2 A Survey on Stochastic Integrals and Equations 123

Note that assertion 1) of Theorem 6.10 clariﬁes the fact that white noise,

the “derivative” of a Wiener process, takes values only in generalized func-

tions.

6.2.2 Stochastic integrals

Our goal in this section is to deﬁne the stochastic integral with respect to a

Wiener process. For the sake of simplicity, we restrict our attention to the

construction based on a Riemann integral. An approach involving a Lebesgue

integral can be found in [76, 83, 84, 162, 175].

Specify a positive constant l<∞.LetA :[0,l] × Ω → L(R

, R

)bea

random operator function, i.e., A(t) is a random linear operator from R

for every t ∈ [0,l]. In what follows the main role will be played by the

particular case k = n, i.e., where A(t) is a random linear operator in R

Consider a Wiener process w(t) with respect to B

with values in R

.To

deﬁne the ItˆointegralofA(t), pick a partition q =(0=t

<... <t

= l)

of the interval [0,l] and consider the integral sum

q−1



i=0

A(t

)



w(t

i+1

) −w(t

)



. (6.2)

Note that we have selected the argument in A(·)astheleftendof[t

i+1

]in

the i-th summand. The limit (if it exists) of such sums as diam q → 0 (usually

in the space L

((Ω,F, P), R

) but possibly with respect to some other type

of convergence of random variables) is called the Itˆointegralof A(t)andis

denoted by



A(t)dw(t). Since the trajectories of w(t) have a.s. unbounded

variation, the Itˆo integral cannot be deﬁned as the Stieltjes integral along

every trajectory.

It turns out that under certain boundedness hypotheses, the Itˆo integral

does exist as the L

-limit of the integral sums when A(t) is non-anticipative

with respect to B

. In particular, it exists (as a Lebesgue type integral) if the

entries A

(t)ofA(t) satisfy the equality

⎧

⎨

⎩





)

(t, ω)dt<∞

⎫

⎬

⎭

=1. (6.3)

The Itˆo integral with varying upper limit is the process deﬁned by



A(τ)dw(τ)=



A(τ)dw(τ),

124 6 Essentials from Stochastic Analysis in Linear Spaces

where χ

is the characteristic function of [0,t]. Note that



A(τ)dw(τ)is

linear in A and dw. Some other important properties of the integral are

given in the next theorem.

Theorem 6.11 The process



A(τ)dw(τ) has the following properties:

(1) it is non-anticipative with respect to B

;

(2) it is a martingale relative to B

;

(3) its sample paths are a.s. continuous in t.

Let us outline the proof of Theorem 6.11(3) for future reference. Consider

processes with continuous trajectories of the form



(t)=

q−1



i=1

i+1

)A(t

)



w(t

i+1

) −w(t

)



+ A(t

)



w(t) − w(t

)



where k =max{i |χ

)=1}. Under certain hypotheses, the sequence



(t)

contains a subsequence that converges a.s. uniformly to



A(τ)dw(τ). This

yields assertion (3).

Note the following discrepancy between the ordinary Riemann and stochas-

tic Itˆo integrals. The former calculated for, say, a bounded continuous func-

tion f(t) with respect to some power α>1ofdt always equals zero since the

integral sum



q−1

i=1

f(t

∗

)(t

i+1

− t

)

, t

∗

∈ [t

i+1

], tends to zero as diam q

tends to zero. However, this is not the case for the latter. One can deﬁne

the multiple stochastic integral



a(τ)dw

(τ) ··· dw

(τ) of a given stochas-

tic process a(·) to be the limit of the integral sums



q−1

i=1

a(t

)



i+1

) −

)



···



i+1

) −w

)



which may not be equal to zero.

Later on, we shall use the following result on the existence and properties

of multiple integrals.

Theorem 6.12 Let α(t) be a random real-valued function and w(t) be a

Wiener process with values in R

, i.e., w(t)=(w

(t),...,w

(t)), where the

coordinates w

(t), i =1,...,n are mutually independent one-dimensional

Wiener processes. Then:

(i)



α(τ)dw

(τ)dw

(τ)=0 i = j;

(ii)



α(τ)(dw

(τ))



α(τ)dτ;

(iii)



α(τ)dτdw

(τ)=0;

(iv)



α(τ)(dw

(τ))

=0;

(v) all integrals of higher order in dτ and dw

(τ) exist and are equal to

zero.

6.2 A Survey on Stochastic Integrals and Equations 125

Here we just outline the proof. Assertion (i) follows immediately from the

hypothesis that w

(t)andw

(t) are independent. To prove (ii), it suﬃces to

observe that for any Wiener process w(t)wehaveE





w(t)−w(s)





= |t−s|.

Assertions (iii)–(v) result from the fact that the multiple Riemann integral

with respect to (dt)

, k>1, is equal to zero.

Remark 6.13. It is sometimes useful to use white noise (the “derivative” of

a Wiener process, see Section 6.2.1) to give a better physical interpretation

of solutions of certain equations but, as mentioned in Section 6.2.1,itisdif-

ﬁcult to use in practice since it takes values in generalized functions. The

use of the Itˆo integral allows one to avoid dealing with white noise by us-

ing integral (rather than diﬀerential) equations and taking into account that



A(t)˙w(τ)dτ =



A(t)dw(τ). Then, if an ordinary diﬀerential equation

˙x(t)=a(t, x(t)) is subjected to a random perturbation and takes the form

˙x(t)=a(t, x(t)) + A(t, x(t)) ˙w(t)whereA(t, x) is a linear operator, a mathe-

matically exact description of the perturbed equation is given by transition

to the integral equation x(t)=x



a(τ,x(τ ))dτ +



A(τ,x(τ ))dw(τ).

The latter equation is called the stochastic diﬀerential equation in Itˆoform

or Itˆo stochastic diﬀerential equation. Such equations are considered in detail

below.

It should be pointed out that the value of a stochastic integral depends

on the point in [t

i+1

] that is substituted as the argument value of the

integrand in the summands of the integral sum. Recall that in integral sums

of Itˆo integrals we choose the left end of [t

i+1

Alternative choices yield some other versions of stochastic integrals. In

particular, we introduce the so-called backward (or anticipative) integral



A(τ)d

∗

w(τ)[152] as the limit of the following integral sums

q−1



i=0

A(t

i+1

)



w(t

i+1

) −w(t

)



, (6.4)

(i.e., choosing the right end of [t

i+1

]) if, of course, the limit exists. This

integral diﬀers, in general, from the Itˆointegral.Forexample,thebackward

integral with varying upper limit is not a martingale relative to B

Considering the integral sums

q−1



i=0

A(t

i+1

)+A(t

)

(w(t

i+1

) −w(t

)) (6.5)

we arrive at the Stratonovich integral



A(τ) ◦ dw(τ )(or



A(τ)d

w(τ))

deﬁned as the limit of these sums (if it exists) (see [50, 216].) It is easy to

see that the Stratonovich integral is equal to half of the sum of the Itˆoand

backward integrals, provided that all three integrals exist. The Stratonovich