WEAK SOLUTIONS OF STOCHASTIC DIFFERENTIAL INCLUSIONS AND THEIR COMPACTNESS

Differential Inclusions, Control and Optimization 29 (2009 ) 91–106

WEAK SOLUTIONS OF STOCHASTIC DIFFERENTIAL INCLUSIONS AND THEIR COMPACTNESS

Mariusz Michta

Faculty of Mathematics, Computer Science and Econometrics University of Zielona G´ ora

Prof. Z. Szafrana 4a, 65–516 Zielona G´ ora, Poland

Dedicated to Prof. M. Kisielewicz on the occassion of his 70th birthday.

Abstract

In this paper, we consider weak solutions to stochastic inclusions driven by a semimartingale and a martingale problem formulated for such inclusions. Using this we analyze compactness of the set of so- lutions. The paper extends some earlier results known for stochastic differential inclusions driven by a diffusion process.

Keywords: semimartingale, stochastic differential inclusions, weak solutions, martingale problem, weak convergence of probability mea- sures.

2000 Mathematics Subject Classification: 93E03, 93C30.

1. Introduction

The major contributions in the field of stochastic inclusions have been con- nected with stochastic control problems (see e.g., [1, 2, 3, 10, 11, 12, 9, 20, 21]

and references therein) and with the existence and properties of their strong

solutions. In [13, 14, 15, 16] and [18] the existence and compactness property

of weak solutions to Brownian motion driven stochastic differential inclusions

were studied. In this work we present a martingale problem approach as a

useful tool in the study of weak solutions of an inclusion driven by a con-

tinuous semimartingale, in which the multivalued integrand also depends

on the driving process. We also consider the case of a stochastic inclusion

driven by Levy’s process. It extends the cases studied earlier in [13, 16, 18]

and [17]. We recall at first main definitions and known facts needed in the paper. Let (Ω, F, {F

}

, P ) be a complete filtered probability space satisfying the usual hypothesis, i.e., {F

}

is an increasing and right continuous family of sub σ-fields of F. By Comp() we denote the space of nonempty and compact subsets of the underlying space, equipped with the Hausdorff distance δ. Let G = (G(t))

be a set-valued stochas- tic process with values in Comp(IR

⊗ IR

), i.e., a family of F-measurable set-valued mappings G(t) : Ω → Comp(IR

⊗ IR

), each t ∈ [0, T ]. For the notions of measurability, continuity, lower and upper continuity (l.s.c. and u.s.c) of set-valued mappings we refer to [6]. Similarly, G is F

-adapted, if G(t) is F

-measurable for each t ∈ [0, T ]. We call G predictable, if it is measurable with respect to predictable σ-field P(F

) in [0, T ] × Ω. For a stochastic process R we introduce the following notation: R

= sup

|R

| and R

= sup

|R

|. For a stopping time η, by R

we denote the stopped process, i.e., R

= R

. Let S

[0, T ], (p ≥ 1) denote the space of all F

- adapted and c´adl´ag processes (R

)

, such that ||R||

:= ||R

||

< ∞, with L

= L

(Ω, R

). A semimartingale R = A + N is said to be a H

[0, T ]- semimartingale (1 ≤ p ≤ ∞), if it has a finite H

[0, T ] − norm, defined by: ||R||

= inf

j

(N, A), where j

(N, A) = || [N, N ]

1 2

T

+ ^R

|dA

|

||

, ([N, N ]

) is a quadratic variation process of local martingale part N, and |A

| = ^R

|dA

| represents the total variation on [0, t] of the measure induced by the paths of the finite variation process A. Given a predictable set-valued process G = (G

)

and a d dimensional semimartingale R adapted to the filtration (F

)

, R

= 0, let us denote

S

(G) := {g ∈ P(F

) : g(t) ∈ G(t) for each t ∈ [0, T ] a.e.

and g is R integrable}.

For conditions of integrability with respect to semimartingales see e.g. [22].

Recall a set-valued stochastic process G = (G

)

is R-integrably bounded if there exists a predictable and R-integrable process m such that the Hausdorff distance δ(G

, {0}) ≤ m

a.s., each t ∈ [0, T ].

2. Weak solutions

Let (Ω, F, (F

)

, P ) be a given filtered probability space. For any ran-

dom element R : Ω → Θ with values in a measurable space Θ, we de-

note by P

the measure on Θ being the distribution of R (under P ). Let

(A

, C

, ν

) denote the local characteristics of a semimartingale R, with respect to the fixed truncation function h : IR

→ IR

(see e.g. [8] for de- tails). For H : [0, T ] × Ω → IR

⊗ IR

being any predictable and bounded (or locally bounded) mapping we will denote a stochastic integral ^R HdR as H ·R. Let h

:IR

→ IR

be a fixed truncation function. For y ∈ IR

, let Hy denote an m dimensional process with (Hy)

= ^P

H

y

, for i ≤ m.

As in [8] let:

A

=



 

 

A

+ [h

(y, Hy) − h

(y)] · ν

if i ≤ d P

j≤d

H

◦ A

+ ^h h

(y, Hy)− (Hh(y))

ⁱ · ν

if d < i ≤ d+m , (1)

C

=



 

 



 

 



C

if i, j ≤ d

P

k≤d

H

· C

if j ≤ d < i ≤ d + m P

k≤d

H

· C

if i ≤ d < j ≤ d + m P

k,l≤d

(H

H

) · C

if d < i, j ≤ d + m (2) ,

and let ν

be defined by I

· ν

= I

(y, Hy) · ν

, for each Borel set G in IR

.

By Propositions 5.3 and 5.6 Ch.IX [8] we have the following character- ization for local characteristics of a stochastic integral.

Theorem 1. Let H be any predictable and bounded (or locally bounded) mapping H : [0, T ] × Ω → IR

⊗ IR

and let (A

, C

, ν

) be a local char- acteristics of a d dimensional semimartingale R. Suppose (R, U ) is a d + m dimensional semimartingale. Then, U = ^R HdR if and only if (R, U ) admits a local characteristics (A

, C

, ν

).

Let D([0, T ], IR

), (n ≥ 1) denote the space of right continuous functions on [0, T ] with values in IR

, with left limits, endowed with the Skorokhod topology. Let µ be a given probability measure on the space (IR

, β(IR

)) . We consider the following stochastic inclusion:

dX

∈ F (t−, X, Z)dZ

, t ∈ [0, T ], (SDI) P

= µ,

where

F : [0, T ] × D([0, T ], IR

) × D([0, T ], IR

) → Comp(IR

⊗ IR

)

is a set-valued mapping, Z is a d dimensional semimartingale defined on a probability space (Ω, F, (F

)

, P ).

To study weak solutions (or solution measures) to stochastic differential inclusion (SDI) we go to canonical path spaces. Similarly as in [7], let us introduce the following canonical path spaces:

1. The canonical space of driving processes: D([0, T ], IR

) with Z

(y) = y(t) and D

= σ{Z

: t ≤ T }, D

= σ{Z

: s ≤ t}, t ∈ [0, T ].

2. The canonical space of solutions: D([0, T ], IR

) with X

(x) = x(t), and σ-fields F

= σ{X

: t ≤ T } and F

= σ{X

: s ≤ t}, t ∈ [0, T ].

3. The joint canonical path space: Ω

= D([0, T ], IR

) × D([0, T ], IR

) with Y

(x, z) = (x(t), z(t)) and σ-fields F

= σ{Y

: t ≤ T } and F

= σ{Y

: s ≤ t}, t ∈ [0, T ]. Taking projections φ

: Ω

→ D([0, T ], IR

), with φ

(x, z) = x and φ

: Ω

→ D([0, T ], IR

), with φ

(x, z) = z, we introduce on a measurable space (Ω

, F

, (F

) the following processes Z

= Z ◦ φ

and X

= X ◦ φ

.

Let (A

, C

, ν

) and (A

, C

, ν

) be processes defined on D([0, T ], IR

) and D([0, T ], IR

), respectively, satisfying the properties of local characteristics.

Let us consider also processes (A

◦φ

, C

◦φ

, ν

◦φ

) and (A

◦φ

, C

◦φ

, ν

◦ φ

) on (Ω

, F

, (F

). Let Q be a probability measure on (Ω

, F

, (F

)

). We introduce probability measures: P

= Q

and P

= Q

on (D([0, T ], IR

) and (D([0, T ], IR

), respectively. Let Z

be a semimartingale under Q with the local characteristics (A

◦ φ

, C

◦ φ

, ν

◦ φ

).

Definition 1. By a weak solution or driving system to the stochastic inclu- sion (SDI) we mean a filtered probability space (Ω

, F

, (F

)

, P

) on which there are defined:

(a) an F

-adapted, d dimensional semimartingale Z

, with local character- istics (A

◦ ψ, C

◦ ψ, ν

◦ ψ), where ψ : Ω

→ D([0, T ], IR

), ψ(ω

) = Z

(ω

) and P

= Q

,

(b) an m dimensional stochastic process X

-called a solution process on (Ω

, F

, (F

)

, P

), such that: P

= µ and

X

= X

+ Z

0

γ

(s)dZ

, t ∈ [0, T ], for some F

-predictable mapping

γ

: [0, T ] × Ω

→ IR

⊗ IR

,

γ

(t, ω

) ∈ F (t, X

(ω

), Z

(ω

)).

We denote such solution by (Ω

, F

, (F

)

, P

, Z

, X

).

Remark 1. Let Q be a probability measure on (Ω

, F

, (F

)

) such that Q

= µ. Such a measure Q is called a joint solution measure a to stochastic inclusion (SDI), if there exists a weak solution to (SDI) (Ω

, F

, (F

)

, P

, Z

, X

) such that Q = P

. Then, P

= Q

and P

= Q

. Hence, we can see that in a canonical setting both notions coincide. Indeed, similarly as in [7] one can show:

Proposition 1. A probability measure Q on (Ω

, F

, (F

)

) is a so- lution measure to (SDI) if and only if (Ω

, F

, (F

)

, Q, Z

, X

) is a weak solution to (SDI).

In the case of a general driving semimartingale Z, the following existence result holds true (see [17] and [19]).

Theorem 2. Let F : [0, T ] × IR

→ Comp(IR

⊗ IR

) be a set-valued function satisfying:

(i) F is integrably bounded (by some function m(·) ), (ii) F is ([0, T ] × IR

)-Borel measurable,

(iii) F (t, ·) is lower semicontinuous for every fixed t ∈ [0, T ].

If F

: [0, T ]×Ω

→ Comp(IR

⊗IR

) is defined by F

(t, x, z) = ˜ F (t, x(t−), z(t−)), where ˜ F (t, a, b) = ^R

F (s, a, b)ds, then there exists a weak solution to the stochastic differential inclusion:

dX

∈ F

(t, X, Z)dZ

, t ∈ [0, T ] P

= µ.

3. Martingale problem related to (SDI)

Below we present the formulation of the multivalued martingale problem

related to the stochastic differential inclusion (SDI). The main results of this

part states the equivalence between the existence of solution measures and

solutions to the martingale problem. We start with a general formulation

(see [8]). Let (Ω, F, (F

)

) be a filtered measurable space and let H be

a sub-σ-field of F. Suppose that µ is a given initial probability. By X we

denote some family of c´adl´ag and F

-adapted processes.

Definition 2. A probability P on (Ω, F, (F

)

) is a solution to the mar- tingale problem related to H, X and µ if

(i) P |

= µ,

(ii) each process belonging to X is a local martingale on (Ω, F, (F

)

, P ).

We shall use notions and notations introduced in the Introduction, adapted to our canonical processes. Following a formulation in Definition 2, we will specify a filtered space (Ω, F, (F

)

), a sub σ-field H, an initial distribution and a class of processes X as elements of a martingale problem related to our (SDI). They are listed in points (a), (b), (c) below. As in the previ- ous section, we have a given bounded and predictable set-valued mapping F : [0, T ] × Ω

→Comp(IR

⊗ IR

), an initial probability measure µ, pro- cesses (A

, C

, ν

) defined on D([0, T ], IR

), satisfying the properties of local characteristics. As mentioned in the Introduction, one can take a truncation function h as h(y) = yI

. Below we use this function. Let us take:

(a) a filtered space (Ω, F, (F

)

) as a joint canonical space (Ω

, F

, (F

)

),

(b) a sub σ-field H = σ(X

), (c) a class: X = X

∪ X

, where

(i) X

is a family consisting of processes:

f (Z

)−f (Z

)− ^X

i≤d

Z

∂

∂x

f (Z

)dA

− 1 2

X

i,j≤d

Z

∂

∂x

∂x

f (Z

)dC

− Z

[0,t]×IR^d



f (Z

+ y) − f (Z

) − ^X

i≤d

∂

∂x

f (Z

)y

I



ν

(ds, dy),

for each bounded function f ∈ C

(IR

).

(b) X

is a family consisting of processes:

f(R

) − f (R

) − ^X

i≤d+m

Z

∂

∂x

f (R

)dA

− 1 2

X

i,j≤d+m

Z

∂

∂x

∂x

f(R

)dC

− Z

[0,t]×IR^d+m



f (R

+y)−f (R

) − ^X

i≤d+m

∂

∂x

f (R

)y

I



ν

(ds, dy),

for each bounded function f ∈ C

(IR

), where R

= (Z

, X

− X

), and for some measurable and bounded function γ : [0, T ] × Ω

→ IR

⊗ IR

. The relation between weak solutions (or solution measures) and solutions to the related martingale problem for SDI is described by the following result.

Theorem 3 ([17]). A probability measure Q on (Ω

, F

, (F

)) is a joint solution measure to the stochastic inclusion (SDI) if and only if it is a so- lution to the related martingale problem.

4. Weak compactness of the solution set

Let M(Ω

) denote the space of all probability measures on the canonical space (Ω

, F

, (F

)

), equipped with the topology of a weak conver- gence of probability measures (see [5]). By R

(F, µ) we denote the set of all probability measures Q ∈ M(Ω

) such that Q is a solution to the martingale problem related to the stochastic inclusion (SDI). By Theorem 3, if Q ∈ R

(F, µ), then Q is a joint solution measure and there exists a weak solution system (Ω

, F

, (F

)

, P

, Z

, X

). As noticed in Re- mark 1, the distribution law P

on D([0, T ], IR

) equals the measure Q

. Since φ

(X

, Z

) = X

, then P

= Q

. Hence, there is a convenient way to study the properties of the solution set. Namely, let R

(F, µ)

:=

{Q

: Q ∈ R

(F, µ)}. Clearly R

(F, µ)

⊂ M(D([0, T ], IR

)). Let (µ

) be a tight sequence of initial distributions. The compactness of the set S

k≥1

R

(F, µ

)

was established in Theorem 5 of [17] in the case of con- tinuous semimartingale satisfying the following condition:

Condition A: there exists the function h(t) = o(t), t → 0+, such that X

1≤j,l≤d

E

[Z

, Z

]

E

[Z

, Z

]

+ ^X

j≤d

||(A

)

||

≤ h(t), for t ∈ [0, T ].

In a similar way we can show the same property for the set ^S

R

(F, µ

).

Namely, the following result holds.

Theorem 4. Let Z be a continuous semimartingale satisfying Condition A with Z

= 0. Let (µ

) be a tight sequence of initial distributions and let F : [0, T ] × C([0, T ], IR

) → Comp(IR

⊗ IR

) be a measurable and bounded set-valued mapping such that the set ^S

R

(F, µ

) is nonempty.

Then, the set ^S

R

(F, µ

) is a nonempty and relatively compact subset of M(C([0, T ], IR

)).

P roof. Using Prokhorov‘s Theorem ([5]), it is enough to show that the set S

k≥1

R

(F, µ

) is tight. Let us remark first that

a→∞

lim sup

Q∈

S

k≥1R^loc(F,µ^k)

Q{||X

|| > a}

≤ lim

a→∞

sup

k≥1

µ

{x ∈ R

: ||x|| > a} = 0.

It is because the sequence (µ

) is tight. Hence by Theorem 8.2 [5], it is enough to use the following criterion: for every  > 0

n→∞

lim sup

Q∈

S

k≥1R^loc(F,µ^k)

Q{w ∈ C([0, T ], IR

) : ∆

( 1

n , w) > } = 0, (3)

where ∆

(δ, w) = sup{||w(t) − w(s)|| : s, t ∈ [0, T ], |s − t| < δ}. Let us take an arbitrary measure Q from the set ^S

R

(F, µ

). Then, there exist k ≥ 1, and measurable and bounded (say by a constant L > 0) mappings γ

: [0, T ] × Ω

→ IR

⊗ IR

, γ

(t, u) ∈ F (t, u) − dt × dQ − a.e and Q ∈ R

(γ

, µ

). Taking functions g : IR

→ R; g(x) = x

, i = 1, 2, . . . , m + d, we obtain, by the shape of the class X

and Theorem 1, the following continuous Q-loc. martingales (on (Ω

, F

, (F

)

)):

N

:=







Z

− A

if 1 ≤ i ≤ d X

− X

− A

∼,γ₋^k,i

t

if d < i ≤ d + m (4) .

Consequently, their second local characteristics are given by hN

, N

i

= C

, i, j = 1, 2, . . . , d + m. Let us take N

= (N

, . . . , N

).

For 0 ≤ t

< t

< T, let us introduce the stopping time τ (u) = inf{s >

0 : ||X

(u) − X

(u)|| >

} ∧ (t

− t

), where u ∈ Ω

. Then by Theorem

44 from [22], the process N

−N

is a continuous Q-local martingale, for

every fixed k ≥ 1. We let t

= 0 for simplicity. Then by (4) we obtain (X

− X

)

≤ 2(N

)

+ 2(A

)

,

and consequently

E

(X

− X

)

≤ 4E

(N

)

+ 4E

(A

)

. (5)

Since

E

(N

)

≤ m ^X

d+1≤i≤d+m

E

 sup

s≤τ

(N

)

 ,

then applying Burkholder-Davis-Gundy inequality (see e.g., [22]) to contin- uous Q− local martingales N

, we get:

E

(N

)

≤ C

m ^X

d+1≤i≤d+m

E

^ C

∼,γ^k₋,ii τ



,

with some universal constant C

. Consequently by (2):

E

(N

)

≤ C

C(m) ^X

d+1≤i≤d+m

X

1≤j,l≤d

E

Z

0

|γ

||γ

||dC

|



,

with some constant C(m). From the boundedness of F we have |γ

| ≤ sup

||a|| ≤ L dt×dQ−a.e. Then applying the Kunita-Watanabe inequality (Theorem 25 Ch.II [22]) and Cauchy-Schwarz inequality to the right hand side above, we obtain:

E

(N

)

≤ a(C

, m, L) ^X

1≤j,l≤d

E

[Z

, Z

]

E

[Z

, Z

]

, (6)

where a(C

, m, L) is some constant not depending on Z

and τ.

Let us consider now the estimation of the term E

(A

)

appearing in

(5). By Theorem 1, the semimartingale ^R γ

dZ

admits its first local char-

acteristics A

= (A

)

, with A

= ^P

^R γ

dA

,

i = d + 1, d + 2, . . . , d + m. Hence applying Emery‘s inequalities ([22]) and

boundedness of F , one can verify that

E

(A

)

≤ d

m ^X

d+1≤i≤d+m

X

j≤d

Z

γ

dA

4

S⁴(Q)

≤ m

d

c

L

^X

j≤d

||(A

)

||

,

where c

is a universal constant. Using this inequality together with (6) we finally obtain the following estimation in (5)

E

(X

− X

)

≤ D

 X

1≤j,l≤d

E

[Z

, Z

]

E

[Z

, Z

]

+

+ ^X

j≤d

||(A

)

||

 ,

for some constant D := a(C

, d, c

, m, L) depending only on indicated con- stants. Now, restoring t

and setting t

− t

:= α, we obtain:

E

(X

− X

)

≤ Dh(α),

where h is a function as in Condition A. By Tchebyshev‘s inequality we have:

Q

 sup

s≤α

||X

− X

|| > 



≤ Dh(α)



. (7)

Let T

= [T ] + 1. For an arbitrary n ∈ N , let us divide the interval [0, T

] by points {

}, i = 0, 1, 2, . . . , T

n. Then,

Q



∆

^ 1

n , X

^ > 



≤ Q

(

[

i=0

 sup

0≤s≤¹_n

||X

− X

|| >  3

 ) .

Hence and by (7) with α =

, we get:

Q



∆

^ 1

n , X

^ > 



≤ 3

T

Dnh(

)



.

Hence by Condition A, we have:

n→∞

lim sup

Q∈

S

k≥1R^loc(F,µ^k)

Q



∆

^ 1

n , X

^ > 



= 0.

In a similar way one obtain:

n→∞

lim sup

Q∈

S

k≥1R^loc(F,µ^k)

Q



∆

 1

n , Z

^ > 



= 0,

which completes the proof.

Remark 2. Let us put in particular Z

:= (t, W

), where W is a d − 1 dimensional Wiener process and F (t, x, z) := (F (t, x), G(t, x)), with F : [0, T ] × C([0, T ], IR

) → Comp(IR

⊗ IR

) and G : [0, T ] × C([0, T ], IR

) → Comp(IR

⊗ IR

). Then the stochastic inclusion (SDI) has the form

dX

∈ F (t, X)dt + G(t, X)dW

, P

= µ,

In this case one can choose h(t) = d

t

+ t

. Thus Theorem 4 extends earlier results obtained in [13, 16] and [18].

For the case of a noncontinuous integrator we consider the stochastic inclu- sion driven by the Levy process L on the interval [0, T ]. Namely, we consider the following inclusion

dX

∈ F

(t−, X, L)dL

, t ∈ [0, T ] P

= µ

with a set-valued mapping F

: [0, T ] × D([0, T ], IR

) → Comp(IR

) defined by F

(t, x, z) = ˜ F (t, x(t−), z(t−)), where ˜ F (t, a, b) = ^R

F (s, a, b)ds and F : [0, T ]×IR

→ Comp(IR

) are given. We assume m = d = 1 for simplicity.

Since L is a semimartingale with independent increments then the local

characteristics (A, C, ν) of the integrator are deterministic and they have

the form: A

= bt, C

= σ

t, ν(dt, dx) = dtm(dx), where b = E(L

), σ > 0

and m(dx) is a measure on IR

\{0} that integrates the function min(1, x

)

(see [8] for details). We assume also that the integrator L has a finite

second moment. Then, L

= M

+ tEL

, where M is a square integrable

martingale. Since the integrator is a c´adl´ag process we cannot proceed as earlier. We shall use the Aldous Criterion of Tightness (see e.g., Theorem 4.5 Ch.VI in [8]). Let {Z

} be a sequence of semimartingales (defined possibly on different probability spaces (Ω

, F

, (F

)

, P

)). We will use the following:

Definition 3 ([24]). The sequence {Z

} of semimartingales satisfies the uniform tightness condition (UT) if for every q ∈ IR

the family of random variables

 Z

0

U

dZ

: U

∈ U

, n ∈ IN



is tight in IR, where U

denotes the family of predictable processes of the form

U

= U

+

k

X

i=0

U

I

,

for 0 = t

< . . . < t

= q and every U

being an F

measurable random variable such that |U

| ≤ 1, for every i ∈ IN ∪ {0}, k, n ∈ IN.

The main properties of uniformly tight sequences of semimartingales are presented below (see [23] for details).

Theorem 5. Let {Z

} be a sequence of semimartingales satisfying (UT).

Then the following statements hold true

(i) for every q ∈ IR

the sequences {sup

|Z

|} and {[Z

]

} are tight in IR

,

(ii) if {U

} is a sequence of predictable processes such that for every q ∈ IR

the sequence {sup

|U

|} is tight in IR

, then the sequence of stochastic integrals { ^R

U

dZ

} satisfies (UT).

Under the same notations as before the following theorem holds true.

Theorem 6. Let L be a Levy process as above. We assume that F : [0, T ] ×

IR

→ Comp(IR

) is a set-valued function satisfying the assumptions of

Theorem 2. Let (µ

) be a tight sequence of initial distributions. Then,

the set ^S

R

(F

, µ

)

is nonempty and relatively compact in the space

M(D([0, T ], IR

)).

P roof. The nonemptiness of the set ^S

R

(F

, µ

)

follows from The- orem 2 and Theorem 3. Let us take an arbitrary sequence of measures {R

} :

{R

} ⊂ ^[

k≥1

R

(F

, µ

)

.

Then, for every n ≥ 1 there exist k

≥ 1, the joint solution measure Q

∈ R

(F

, µ

) and (by Theorem 3 and Remark 1) a weak solution system (Ω

, F

, (F

)

, P

, L

, X

) with the following properties:

(i) Q

= (P

)

, R

= (P

)

(ii) L

is an F

-adapted square integrable Levy process, with local char- acteristics A

= bt, C

= σ

t, ν(dt, dx) = dtm(dx) and X

is a solution process on (Ω

, F

, (F

)

, P

) such that (P

)

= µ

and

X

= X

+ Z

0

γ

(s)dL

, t ∈ [0, T ],

for some (F

)

-predictable and bounded (say by the constant C) process

γ

: [0, T ] × Ω

→ IR

,

γ

(t) ∈ F (t, X

, L

) dt ⊗ dP

-a.s.

Since the sequence of processes {γ

} is uniformly bounded it follows that the sequence {sup

|γ

|} is tight in IR

for every q ∈ IR

. For every q ∈ IR

let us consider the family U

described in Definition 3. Then using first Khinthine’s inequality and next Emery‘s inequality [22], we get the following estimation:

P

   

Z

U

dL

> K



≤ 1

K

E

 sup

0≤t≤q

Z

U

dL



≤ c

K

Z

0

U

dL

H_[0,q]²

≤ c

K

 σ

+

Z

x

m(dx) + qM

 Z

0

E

(U

)

ds

≤ c

q K

 σ

+

Z

x

m(dx) + qM



,

where M := E

[(L

)

] < ∞ (we have assumed that the Levy process has the finite second moment). Hence, the sequence {L

} satisfies (UT). By Theorem 5 we claim that the sequence { ^R

γ

dL

} satisfies (UT) as well.

Consequently, we infer the tightness of the sequence {sup

| ^R

γ

dL

|}

for every q ∈ IR

. For n, N ≥ 1 let T

denote the set of (F

)

- stopping times that are bounded by N . Then, similarly as above one can show

sup

S,T∈T_Nⁿ:S≤T ≤S+θ

P

ⁿ |X

− X

| > ε ^o

≤ 1

ε

sup

S,T∈T_Nⁿ:S≤T ≤S+θ

E

 sup

S≤q≤T

Z

γ

L



≤ c

C

θ ε

 σ

+

Z

x

m(dx) + θM



for every ε, θ > 0 and n ∈ IN. Thus we have

θ→0+

lim lim sup

n

sup

S,T∈T_Nⁿ:S≤T ≤S+θ

P

{|X

− X

| > ε} = 0,

and by the Aldous Criterion of Tightness (see Theorem 4.5 Ch.VI in [8]) we claim the tightness of the sequence {X

}, which implies the some property for the sequence {R

}. Hence by Prohorov’s Theorem we infer that the set ^S

R

(F

, µ

) is relatively compact in the space M(D([0, T ], IR

)) equipped with the topology of weak convergence.

References

[1] N.U. Ahmed, Nonlinear stochastic differential inclusions on Banach space, Stoch. Anal. Appl. 12 (1) (1994), 1–10.

[2] N.U. Ahmed, Impulsive perturbation of C

semigroups and stochastic evolution inclusions, Discuss. Math. DICO 22 (1) (2002), 125–149.

[3] N.U. Ahmed, Optimal relaxed controls for nonlinear infinite dimensional stochastic differential inclusions, Optimal Control of Differential Equations, M. Dekker Lect. Notes. 160 (1994), 1–19.

[4] N.U. Ahmed, Optimal relaxed controls for infinite dimensional stochastic sys- tems of Zakai type, SIAM J. Contr. Optim. 34 (5) (1996), 1592–1615.

[5] P. Billingsley, Convergence of Probability Measures, Wiley, New York, 1968.

[6] S. Hu and N. Papageorgiou, Handbook of Multivalued Analysis, Vol. 1, Theory, Kluwer, Boston, 1997.

[7] J. Jacod, Weak and strong solutions of stochastic differential equations, Stochastics 3 (1980), 171–191.

[8] J. Jacod, A.N. Shiryaev, Limit Theorems for Stochastic Processes. Springer, New York, 1987.

[9] M. Kisielewicz, M. Michta, J. Motyl, Set-valued approach to stochastic control.

Parts I, II, Dynamic. Syst. Appl. 12 (3&4) (2003), 405–466.

[10] M. Kisielewicz, Quasi-retractive representation of solution set to stochastic inclusions, J.Appl. Math. Stochastic Anal. 10 (3) (1997), 227–238.

[11] M. Kisielewicz, Set-valued stochastic integrals and stochastic inclusions, Stoch.

Anal. Appl. 15 (5) (1997), 783–800.

[12] M. Kisielewicz, Stochastic differential inclusions, Discuss. Math. Differential Incl. 17 (1–2) (1997), 51–65.

[13] M. Kisielewicz, Weak compactness of solution sets to stochastic differential inclusions with convex right-hand side, Topol. Meth. Nonlin. Anal. 18 (2003), 149–169.

[14] M. Kisielewicz, Weak compactness of solution sets to stochastic differential inclusions with non-convex right-hand sides, Stoch. Anal. Appl. 23 (5) (2005), 871–901.

[15] M. Kisielewicz, Stochastic differential inclusions and diffusion processes, J.

Math. Anal. Appl. 334 (2) (2007), 1039–1054.

[16] A.A. Levakov, Stochastic differential inclusions, J. Differ. Eq. 2 (33) (2003), 212–221.

[17] M. Michta, On weak solutions to stochastic differential inclusions driven by semimartingales, Stoch. Anal. Appl. 22 (5) (2004), 1341–1361.

[18] M. Michta, Optimal solutions to stochastic differential inclusions, Applica- tiones Math. 29 (4) (2002), 387–398.

[19] M. Michta and J. Motyl, High order stochastic inclusions and their applica- tions, Stoch. Anal. Appl. 23 (2005), 401–420.

[20] J. Motyl, Stochastic functional inclusion driven by semimartingale, Stoch.

Anal. Appl. 16 (3) (1998), 517–532.

[21] J. Motyl, Existence of solutions of set-valued Itˆ o equation, Bull. Acad. Pol. Sci.

46 (1998), 419–430.

[22] P. Protter, Stochastic Integration and Differential Equations: A New Ap-

proach, Springer, New York, 1990.

[23] L. S lomi´ nski, Stability of stochastic differential equations driven by general semimartingales, Dissertationes Math. 349 (1996), 1–109.

[24] C. Stricker, Loi de semimartingales et crit´eres de compacit´e, Sem. de Probab.

XIX Lecture Notes in Math. 1123 (1985), Springer Berlin.

[25] D. Stroock and S.R. Varadhan, Multidimensional Diffusion Processes, Springer, 1975.

Received 5 June 2009