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

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236 9 Mean Derivatives on Manifolds

So, we can consider diﬀerential equations and inclusions in terms of forward

(or backward) mean derivatives or current velocities on the one hand and

quadratic mean derivatives on the other hand by analogy with Section 8.4.

In doing this we have to be able to represent a given (2, 0)-tensor ﬁeld α(t, m)

in the form α(t, m)=A(t, m)A

∗

(t, m). Unlike the case of linear spaces (see

Section 8.4), on non-parallelizable manifolds there is a topological obstruc-

tion for obtaining such a presentation for smooth or continuous α(t, m) with

smooth or at least continuous A(t, m):R

→ T

M where n is the dimension

of M (see, e.g., [150]). Nevertheless such a presentation is possible in larger

dimensions under some additional assumptions.

Theorem 9.17 Consider a symmetric (2, 0)-tensor ﬁeld α(t, m) on an n-

dimensional manifold M. There exists a k>nsuch that, if α(t, m) is positive

deﬁnite and continuous (smooth), there exists a continuous (smooth, respec-

tively) ﬁeld of linear operators A(t, m):R

→ T

M such that the relation

α(t, m)=A(t, m)A

∗

(t, m) holds.

Proof. Denote by (α

) the matrix of α(t, m) in the local coordinates of some

chart U

. Introduce an arbitrary Riemannian metric g(·, ·)onM with matrix

) and denote by ¯g the corresponding metric (2, 0)-tensor with matrix

) (see Notation 1.51 and Remark 1.52). Then the (2, 0)-tensor ﬁeld α is

represented in the form α(·, ·)=¯g(b(·), ·)whereb(t, m)(·)isa(1, 1)-tensor

ﬁeld of self-adjoint linear operators acting in the cotangent spaces to M.The

existence and uniqueness of b is derived as follows. In the local coordinates

of U

the expression α(·, ·)=¯g(b(·), ·) takes the form α

= g

where b

are the coeﬃcients of (b)

. Since (g

) is not degenerate, b

are presented in

the form b

= g

. Since the metric tensor is C

∞

-smooth, the ﬁeld b is

continuous (smooth, or Borel measurable) if α is continuous (smooth, Borel

measurable, respectively).

Let a

,...,a

be a ﬁeld of orthonormal frames with respect to the metric

¯g(·, ·) in cotangent spaces at the points of U

. With respect to these frames b

is represented by a matrix (

b) that is symmetric, positive deﬁnite and satisﬁes

the conditions for applying the Gauss decomposition as in the proof of Lemma

8.40. Hence b(t, m) is presented in the form b(t, m)=f(t, m)f

∗

(t, m).

Embed by Nash’s Theorem (Theorem 1.46) the manifold M with metric

g(·, ·) isometrically into a Euclidean space R

(see [186]). k can be chosen

so that it depends only on M (by Nash’s theorem it is determined by the

dimension M), i.e., it is the same for all Riemannian metrics on M .

Denote by P

the orthogonal projector of R

onto its subspace T

M (the

tangent space to M at m ∈ M). Let A(t, m)=f(t, m) ◦ P

: R

→ T

Then one can easily see that α(t, m)=A(t, m)A

∗

(t, m) and by construction

A(t, m) is continuous (smooth, respectively). 

9.5 Mean Derivatives of Itˆo Processes on Manifolds 237

9.5 Mean Derivatives of Itˆo Processes on Manifolds

Let a Riemannian manifold M satisfy the hypothesis of Theorem 7.76 or be

uniformly complete (see Deﬁnition 7.85). Under these assumptions the inte-

gral approach to stochastic diﬀerential equations on manifolds of Section 7.7

is well-posed.

Lemma 9.18 Let ξ(t)=R

z(t) where z(t)=



a(s)ds +



A(s)dw(s) is

an ItˆoprocessinT

M.Thenξ(t) satisﬁes the following Itˆo equation in

Belopolskaya-Daletskii form:

dξ(t) = exp

ξ(t)

(Γ

t,0

a(t)dt + Γ

t,0

A(t)dw(t)). (9.20)

From the Deﬁnition 7.83 of parallel translation it easily follows that

Lemma 9.18 is a reformulation of Lemma 7.65. Clearly the equation in local

coordinates (i.e., the Baxendale form) for (9.20) is as follows:

dξ(t)=Γ

t,0

a(t)dt −

trΓ

ξ(t)

(Γ

t,0

A(t),Γ

t,0

A(t))dt + Γ

t,0

A(t)dw(t) (9.21)

(recall that Γ

t,0

is the operator of parallel translation while Γ

(·, ·)isthe

local connector).

Taking into account formulae (9.20)and(9.21), we can apply the results

of the preceding sections to the calculation of the mean derivatives of Itˆo

processes in manifolds. In particular, from Lemma 9.3 we have:

Lemma 9.19 For ξ(t) as in Lemma 9.18, Dξ(t)=E

(Γ

t,0

a(t)).

Also, from Lemma 9.16 we have:

Lemma 9.20 For ξ(t) as in Lemma 9.18, D

ξ(t)=E

(Γ

t,0

(A(t)A

∗

(t))).

For the applications below we have to calculate mean derivatives for an

Itˆo process ξ(t)=R

z(t)wherez(t)isanItˆo process in T

M for t ∈ [0,l]

of the form z(t)=



a(s)ds + σw(t)whereσ>0 is a real constant and a(t)

satisﬁes (8.32). Recall (see Section 7.7.3) that in this case we can deal with

a stochastically complete M (a less restrictive assumption than above).

So, let M be stochastically complete. For the above ξ(t), formula (9.20)

takes the form

dξ(t) = exp

ξ(t)

(Γ

t,0

a(t)dt + Γ

t,0

dw(t)) (9.22)

so that (9.21) takes the form

dξ(t)=Γ

t,0

a(t)dt −

trΓ

ξ(t)

(I,I)dt + Γ

t,0

dw(t). (9.23)

Thus by Lemma 9.20

ξ(t)=σ

I (9.24)

238 9 Mean Derivatives on Manifolds

and by Lemma 9.19

Dξ(t)=E

(Γ

t,0

a(t)) (9.25)

(this also follows from Lemma 9.19)and

∗

ξ(t)=E

(Γ

t,0

a(t)) + D

∗

(Γ

t,0

w(t)). (9.26)

in any chart in M. Consequently

ξ(t)=v

(t, ξ(t)) = E

(Γ

t,0

a(t)) +

∗

(Γ

t,0

w(t)) (9.27)

and

ξ(t)=u

(t, ξ(t)) = −

∗

(Γ

t,0

w(t)) (9.28)

also in any chart.

Lemma 9.21 Formulae (9.11) and (9.12) hold for the above mentioned pro-

cess ξ(t)=R

z(t).

Indeed, the fact that the coeﬃcient at w(t)isσI means that the Rieman-

nian metric is generated by the diﬀusion coeﬃcient as in Section 9.2.

Let a(t) satisfy (8.32) so that the measure μ

corresponding to z(t)=



a(s)ds + w(t)onthespace(C

([0,l],T

M),

F) is absolutely continuous

with respect to the Wiener measure ν with density (8.33). Then formula

(9.26) and consequently formulae (9.27)and(9.28) can be expressed in the

more precise form:

Lemma 9.22

∗

ξ(t)=E

(Γ

t,0

a(t)) + E



t,0



z(t)

− κ(t)



(9.29)

ξ(t)=E

(Γ

t,0

a(t)) +



t,0



z(t)

− κ(t)



(9.30)

ξ(t)=−



t,0



z(t)

− κ(t)



(9.31)

where κ(t) is as deﬁned in Lemma 8.35.

Proof. In order to derive (9.29) note that the probability of the event ξ(t) ∈

A,whereA ⊂ M is a Borel set, is equal to μ

(ξ(t)

−1

A), where ξ(t)

−1

A is the

set of curves from C

([0,l],T

M) such that the values of their developments

at t belong to A. Denote by Ξ(t)thesetinT

M consisting of the values

at t of all curves from ξ(t)

−1

A. Generally speaking, N

and N

do not

coincide, in particular Ξ(t) may not belong to N

(but it certainly belongs

to P

by construction). But we can calculate μ

(ξ(t)

−1

A)byintegrating

the probability density of z(t)overΞ(t). This means that the value ξ(t)is

distributed as a map from

Ω into T

M which is measurable with respect

9.6 Equations and Inclusions with Mean Derivatives 239

to N

and has the same probability distribution as z(t). Taking into account

the relation between the probability distributions and mean derivatives (see

Lemma 8.17) as well as the construction of the Itˆo development, we obtain

(9.29). Formulae (9.30)and(9.31) follow immediately from (9.29)andfrom

(9.27)and(9.28), respectively. 

Formulae (9.25)and(9.26), and consequently (9.29)and(9.30), can also

be derived from the following general statement. Let M be a Riemannian

manifold and z(t)beanItˆo process in a certain T

M such that ξ(t)=R

z(t)

exists. Let y(t) be another process given on the same probability space.

Theorem 9.23

(i) D

ξ(t) exists if and only if D

z(t) exists and D

ξ(t) is parallel to

z(t) along ξ(·).

(ii) D

∗

ξ(t) exists if and only if D

∗

z(t) exists and D

∗

ξ(t) is parallel to

∗

z(t) along ξ(·).

This statement for ξ(t)=R

z(t) is an analog of the characteristic prop-

erty of the curve Sv(t)fromTheorem3.43. It follows directly from the con-

struction of R

z(t) (see Section 7.6.1). The principal point here is that the

conditional expectation with respect to the same σ-algebra N

is used in the

mean derivatives before and after the parallel translation. Note that Dξ(t)

and Dz(t) (as well as D

∗

ξ(t)andD

∗

z(t), respectively) are not, generally

speaking, parallel to each other along ξ(·)sinceN

and N

may not coin-

cide.

9.6 Equations and Inclusions with Mean Derivatives

In this section we introduce diﬀerential equations and inclusions with mean

derivatives on manifolds and prove some simple existence of solution theo-

rems.

Let M be a Riemannian manifold. In this section we use the Levi-Civit´a

connection H of the Riemannian metric.

Take t ∈ [0,T]. Consider a vector ﬁeld a(t, m) and a symmetric positive

semi-deﬁnite (2, 0)-tensor ﬁeld α(t, m)onM.

By a ﬁrst order diﬀerential equation with mean derivatives we mean a

system of the form



Dξ(t)=a(t, ξ(t)),

ξ(t)=α(t, ξ(t)).

(9.32)

Deﬁnition 9.24. We say that (9.32)hasaweak solution on [0,T] with initial

condition ξ(0) = m

if there exists a probability space (Ω,F, P) and a process

ξ(t) given on (Ω, F, P), taking values in M, such that P-a.s. and for almost

all t in [0,T]system(9.32) is satisﬁed.

240 9 Mean Derivatives on Manifolds

We shall mainly look for weak solutions of (9.32) among solutions of some

equations of type (7.18). Taking into account Theorem 9.5 and Lemma 9.16

it is clear that the ﬁrst equation of (9.32) determines the drift of a solution

of some equation of type (13.5) while the second equation determines the

diﬀusion coeﬃcient. However, we do not assume apriorithat a solution of

(9.32) is also a solution of an equation of the form (7.18). This is why we do

not consider the notion of strong solutions.

For each (t, m) denote by A

a,α

(t, m) the generator determined by the

formulae HA

a,α

(t, m)=a(t, m)andQA

a,α

(t, m)=α(t, m) (see formulae

(2.45)and(2.44)).

Theorem 9.25 Assume that in (9.32) a(t, m) and α(t, m) are C

-smooth,

α(t, m) is positive deﬁnite and A

a,α

(t, m) satisﬁes the conditions of Theorem

7.43. Then for any initial condition ξ(0) = m

∈ M equation (9.32) has a

weak solution that exists for all t ∈ [0,T].

Proof. By Theorem 9.17 we can construct a ﬁeld of linear operators A(t, m)

that is C

-smooth and such that α(t, m)=A(t, m)A

∗

(t, m). Consider equa-

tion (7.18) with these a(t, m)andA(t, m). Since its coeﬃcients are C

-smooth

(i.e., locally Lipschitz continuous), for any initial condition ξ(0) = m

∈ M ,

by Theorem 7.36 the equation has a strongly unique local strong solution

that exists, by Theorem 7.43, on the entire interval t ∈ [0,T]. From Theorem

9.5 and Lemma 9.16 it follows that this solution satisﬁes (9.32). 

Now consider inclusions in mean derivatives on M .Leta(t, m)beaset-

valued vector ﬁeld on M , i.e., for every point m ∈ M aseta(t, m) ⊂ T

is speciﬁed. Let also α(t, m) be a set-valued symmetric positive semi-deﬁnite

(2, 0)-tensor ﬁeld on M (this means that for all t and m any tensor from the

set α(t, m) is symmetric and positive semi-deﬁnite). Consider the problem



Dξ(t) ∈ a(t, ξ(t)),

ξ(t) ∈ α(t, ξ(t)).

(9.33)

Deﬁnition 9.26. We say that (9.33)hasaweak solution on [0,T] with initial

condition ξ(0) = m

if there exists a probability space (Ω,F, P) and a process

ξ(t) given on (Ω, F, P), taking values in M, such that P-a.s. and for almost

all t in [0,T] the inclusions (9.33) are satisﬁed.

Denote by A

a,α

(t, m) the set-valued second order vector ﬁeld with images

a,α

(t, m)={A

a,α

(t, m) | a ∈ a(t, m),α ∈ α(t, m)} determined by formulae

a,α

(t, m)=a(t, m)andQA

a,α

(t, m)=α(t, m) as above (see formulae

(2.45)and(2.44)).

Theorem 9.27 Let α(t, m) and a(t, m) be an upper semicontinuous set-

valued symmetric positive semi-deﬁnite (2, 0)-tensor ﬁeld and a vector ﬁeld

on M, respectively, with closed convex images. Let in addition for every com-

pact K ⊂ M the sets a([0,T], K) and α([0,T], K) be compact and assume

9.6 Equations and Inclusions with Mean Derivatives 241

that at every (t, m) the generator A

a,α

from a neighborhood V of the graph

of A

a,α

(t, m) satisﬁes the conditions of Theorem 7.43 with the same proper

function ϕ. Then for any initial condition ξ(0) = m

there exists a weak

solution of (9.33), well-deﬁned on the entire interval [0,T].

Proof. Introduce a sequence of positive numbers ε

→ 0. By Theorem 4.11

for every ε

there exists a single-valued continuous ε

-approximation a

(t, m)

of a(t, m) and a single valued continuous ε

-approximation ˜α

(t, m)(·, ·)of

α(t, m) such that the sequences of those approximations point-wise converge

to a Borel measurable selection a(t, m)ofa(t, m)andα(t, m)ofα(t, m),

respectively, as q →∞.

Note that the tensor ﬁelds ˜α

(t, m)(·, ·) are symmetric and positive semi-

deﬁnite. Introduce another sequence

(t, m)(·, ·)=˜α

(t, m)(·, ·)+ε

g(m)(·, ·)

where g(m)(·, ·)isthe(2, 0)-metric tensor (Riemannian metric). It is evident

that the tensors α

(t, m)(·, ·) are continuous, positive deﬁnite and symmetric

and that the sequence α

(t, m)(·, ·) point-wise converges to α(t, m)(·, ·)as

q →∞. Since continuous ﬁelds can be approximated by smooth ones, without

loss of generality we may suppose that all ﬁelds a

(t, m)andα

(t, m)(·, ·)are

smooth. For simplicity denote by A

(t, m) the generator A

,α

(t, m).

From Theorem 9.17 it follows that there exist a suﬃciently large integer

K and a sequence of smooth ﬁelds of linear operators A

(t, m):R

→

M such that α

(t, m)=A

(t, m)A

∗

(t, m) for all q, t and m.Notethat

K depends only on the dimension of M since R

is a space in which M

with arbitrary Riemannian metric can be isometrically embedded by Nash’s

Theorem (Theorem 1.46).

Now introduce ˆa

(t, m) which, in a chart on M , has coordinates ˆa

−

where Γ

are the Christoﬀel symbols of H and (α

)isthema-

trix of α

.NotethatA

(t, m) is the ﬁeld of generators for the Itˆo equation

(ˆa

). By construction, for q large enough (m, A

(t, m)) belongs to V.Then

one can easily derive from the hypothesis that the Itˆo equations (ˆa

)sat-

isfy the conditions of Lemma 7.57 and so the equations (ˆa

)havestrong

and strongly unique solutions ξ

, well-deﬁned on the entire interval [0,T], and

the set of corresponding measures {μ

} on (

Ω,

F) is weakly compact. Hence

we can choose a subsequence that weakly converges to some measure μ.For

convenience we suppose that the sequence μ

itself weakly converges to μ.De-

note by ξ(t) the coordinate process on the probability space (

Ω,F,μ). Recall

that this means that for every x(·) ∈ C

([0,T],M)wehaveξ(t, x(·)) = x(t).

We show that ξ(t) is the solution that we are looking for.

Notice that the “present” σ-algebra N

of ξ(t)isaσ-subalgebra of P

and

so for conditional expectations the equality

E(E(·|P

) |N

)=E(·|N

) (9.34)

holds.

242 9 Mean Derivatives on Manifolds

Consider the chart U, the global chart in R

and the symbols



in-

troduced in the proof of Lemma 7.57. Deﬁne ¯a

(t, m, x),

(t, m, x)and

¯α

(t, m, x)(·, ·) by formula (7.33)where˘a,

A and (˜α

) are replaced by a(t, m),

A(t, m)and(α

), respectively. Deﬁne ¯a(t, m)and¯α(t, m)(·, ·)bythesame

formulae via the replacements a(t, m)andα(t, m), respectively.

As in the proof of Lemma 7.57, one can easily show that the equation

(t)=¯a



(t)



−





(t)



(t)





(t)





dt (9.35)



(t)



dw(t)

(analogous to (7.34)) is well-deﬁned on R

and that in U it transforms into

the system



dξ

(t)=a

(t, ξ

(t))dt −

tr(Γ

))dt + A

(t, ξ

(t))dw(t),

(t)=0

(9.36)

where trΓ (A

)=Γ

, and so with probability 1 the solution of (9.35)

with initial condition

(0) = m

∈ M lies in M and coincides with the

corresponding solution of (ˆa

) for all q. Thus the measures on the path

space in R

corresponding to the solutions are located on C

([0,T],M)and

coincide there with μ

for all q.

By construction the sequence ¯a

(t, x(·)) = ¯a

(t, x(t)) point-wise converges

to ¯a(t, x(·)) = ¯a(t, x(t)). Hence it converges almost surely with respect to all

λ ×μ

where λ is the normalized Lebesgue measure on [0,T].

The fact that





(t + Δt) ∧ θ

x(·)



− x



t ∧θ

ξ(·)



−

(t+Δt)∧θ

x(·)



t∧θ

x(·)



¯a (s, ξ(·)) −





Γ (¯α)









is proved by a simple modiﬁcation of the proof of Lemma 8.47. Introduce com-

pact W

as in Lemma 7.57.Takingintoaccount(9.34)thefactthata(t, m)

and α(t, m) are Borel measurable and that the above arguments are valid

for every p and ∪

= M, and so every m belongs to W

for large enough

p, by transition to (9.36) one can easily derive from the last expression that

the regression (8.5) takes the form Y

(t, m)=a(t, m) −

tr(Γ

(α(t, m)).

Since in the normal chart of every m ∈ M with respect to H all Christof-

fel symbols Γ

(m) are equal to zero, we obtain from Deﬁnition 8.2 that

ξ(t)=a(t, ξ(t)).

9.7 Stochastic Diﬀerential Inclusions in Terms of Inﬁnitesimal Generators 243

The fact that





(t + Δt) ∧ θ

x(·)



−



t ∧θ

x(·)



⊗



(t + Δt) ∧ θ

x(·)



−



t ∧θ

x(·)



−

(t+Δt)∧θ

x(·)



t∧θ

x(·)

¯α



ξ(·)







is proved by a slight modiﬁcation of the proof of Lemma 8.48. From the last

relation one easily deduces that D

ξ(t)=α(t, ξ(t)). 

9.7 Stochastic Diﬀerential Inclusions in Terms of

Inﬁnitesimal Generators

In this Section we deal with a slight generalization of the notion of an in-

ﬁnitesimal generator which is well-deﬁned for non-Markovian stochastic pro-

cesses. For a process ξ(t) with values in a manifold M (in particular, in R

)

we introduce the generator as a ﬁeld of second order semi-elliptic diﬀerential

operators acting on the (suﬃciently smooth) function f according to the rule

A(t, m)f

= lim

Δt→+0



f(ξ((t + Δt)) ∧ τ

) −f (ξ(t ∧ τ

))

Δt



ξ(t)=m



(9.37)

where τ

is the Markov time that ξ ﬁrst hits the boundary of a given chart

containing m. The diﬀerence between (9.37) and Deﬁnition 6.34 is that here

we use the regression (see Section 6.1.2) instead of the unconditional expec-

tation. Note that if ξ(t) is Markovian, both (9.37) and Deﬁnition 6.34 deﬁne

thesameobject.

Clearly the generator deﬁned by (9.37) is a second order tangent vector. As

in Lemma 9.4 and Lemma 9.16 one can easily prove that if A is the generator

of ξ(t) for a given connection H, the formulae

ξ(t)=(HA)(t, ξ(t)) (9.38)

and

ξ(t)=2(QA)(t, ξ(t)) (9.39)

hold where Q and H are deﬁned by formulae (2.44)and(2.45), respectively.

Suppose that the ﬁeld of second order tangent vectors A(t, m) is set-valued,

i.e., in every second order tangent space τ

M to the manifold M asetA(t, m)

244 9 Mean Derivatives on Manifolds

depending on t ∈ [0, ∞) is given. We want to ﬁnd a stochastic process ξ(·)

such that for every t its generator A(t, m) a.s. satisﬁes the inclusion

A(t, ξ(t)) ∈ A(t, ξ(t)). (9.40)

Problems of this sort naturally arise if the process is described in terms of its

generator.

If the set-valued ﬁeld A(t, m) has a continuous selector with some regular-

ity properties, one can ﬁnd the process having that selector as generator. This

process is a solution of (9.40).Forexample,ifA(t, m) is lower semi-continuous

and has convex closed values, by Michael’s theorem it has a continuous selec-

tor. If the selector has a positive deﬁnite (2, 0)-tensor component, the above

argument applies, proving the existence of solutions of (9.40).

If the selectors do not exist, the proof of solvability for (9.40) becomes much

more complicated. We consider a rather general case of this sort, important

for applications, where A(t, m) is upper semi-continuous and not necessarily

positive deﬁnite.

Theorem 9.28 Let A(t, m), t ∈ [0,T], be an upper semi-continuous set-

valued second order vector ﬁeld on a manifold M with closed convex values

such that:

(i) for every t ∈ [0,T], m ∈ M ,andforeachA∈A(t, m), the (2, 0)-tensor

A is symmetric and positive semi-deﬁnite;

(ii) for every compact K ∈ M the set A([0,T], K) is compact in τM;

(iii) there exist a proper function ψ : M → R, a constant C>0 and a

neighborhood V of the graph of A in [0,T] ×τ (M) such that for every

(t, m, A) ∈Vthe inequality |Aψ| <C holds.

Then for every m

∈ M there exists a probability space and a stochastic

process ξ(t) with initial condition ξ(0) = m

, well-deﬁned for all t ∈ [0,T],

given on the probability space and taking values in M , such that, for its in-

ﬁnitesimal generator, inclusion (9.40) is a.s. satisﬁed.

Proof. In this proof we are working in the Banach manifold C

([0,T],M)

equipped with the σ-algebra F generated by cylinder sets. By P

we denote

the σ-subalgebra of F generated by cylinder sets with bases over [0,t] ⊂ [0,T].

Specify an arbitrary complete Riemannian metric g(·, ·)onM with chart

components g

. This metric turns M into a metric space with respect to the

corresponding Riemannian distance. Denote by H the Levi-Civit´a connection

of g(·, ·). Recall that g

(·, ·) is an inner product in T

M, m ∈ M. This inner

product determines the inner products g

(·, ·)inthespaceof(2, 0)-tensors at

m ∈ M by the rule g

((α

), (β

)) = g

. Deﬁne the inner products

in τ

M by the formula

g(A

, A

)=g(HA

, HA

)+g(QA

, QA

9.7 Stochastic Diﬀerential Inclusions in Terms of Inﬁnitesimal Generators 245

Thus in all T

M, τ

M and the space of (2, 0)-tensors over M the corre-

sponding Euclidean norms smoothly depending on m are given.

Let ε

→ 0 be a positive sequence. By Theorem 4.11 for every ε

there

exists a single-valued ε-approximation

(t, m)ofA(t, m) such that the se-

quence A

(t, m) point-wise converges to a Borel measurable selector A(t, m)

of A(t, m)asq →∞.

Consider the sequences of vector ﬁelds a

(t, m)=HA

(t, m)andof

(2, 0)-tensor ﬁelds ˜α

(t, m)(·, ·)=2QA

(t, m), respectively. It is evident that

(t, m) point-wise converges to a(t, m)=HA(t, m), ˜α

(t, m) point-wise con-

verges to α(t, m)=2QA(t, m)asq →∞and that both a(t, m)andα(t, m)

are Borel measurable.

Note that the tensor ﬁelds ˜α

(t, m)(·, ·) are symmetric and positive semi-

deﬁnite. Introduce another sequence

(t, m)(·, ·)=˜α

(t, m)(·, ·)+ε

g(m)(·, ·).

It is clear that the tensors α

(t, m)(·, ·) are continuous, positive deﬁnite

and symmetric and that the sequence α

(t, m)(·, ·) point-wise converges to

α(t, m)(·, ·)asq →∞. Since continuous ﬁelds can be approximated by smooth

ﬁelds, without loss of generality we may suppose that all ﬁelds a

(t, m)and

(t, m)(·, ·) are smooth. Denote by A

(t, m) the smooth second order tan-

gent vector ﬁeld corresponding to the pair (a

(t, m),α

(t, m)). By construc-

tion A

(t, m)isa2ε-approximation of A(t, m) and the sequence A

(t, m)

point-wise converges to A(t, m)asq →∞.

Note that the properties of a

(t, m)andα

(t, m)arethesameasinthe

proofofTheorem9.27. Hence by imitating that proof we can show that

there exists a stochastic process ξ(t), deﬁned for all t ∈ [0,T], such that

ξ(t)=a(t, ξ(t)) ∈ (HA)(t, ξ(t)) and D

ξ(t)=α(t, ξ(t)) ∈ 2(QA)(t, ξ(t)).

By construction this is the solution of (9.40) that we are looking for. 

The inclusion (9.40) most often arises in applications in a linear space.

For the case in hand, we prove an existence theorem of another sort, whose

hypothesis is formulated in terms of estimates of Itˆotype.

Let A(t, m) be a set-valued second order vector ﬁeld in R

.Thenwecan

consider the set-valued vector ﬁeld a(t, m), the vector part of A(t, m), and its

tensor part, the set-valued symmetric (2, 0)-tensor ﬁeld α(t, m) taking values

in positive semi-deﬁnite tensors.

Theorem 9.29 Let the set-valued vector ﬁeld a(t, x) be upper semi-continu-

ous, have closed convex values and satisfy the estimate

a(t, x) <K(1 + x) (9.41)

for some K>0.

Let the set-valued (2, 0)-tensor ﬁeld α(t, x) be upper semi-continuous, take

closed convex values in symmetric positive semi-deﬁnite tensors and be such

that for each α(t, x) ∈ α(t, x) the estimate