Download weak solutions of stochastic differential inclusions and their

Discussiones Mathematicae
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 Góra
Prof. Z. Szafrana 4a, 65–516 Zielona Góra, 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 solutions. 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 measures.
2000 Mathematics Subject Classification: 93E03, 93C30.
1. Introduction
The major contributions in the field of stochastic inclusions have been connected 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 continuous 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]
92
M. Michta
and [17]. We recall at first main definitions and known facts needed in the
paper. Let (Ω, F, {Ft }t∈[0,T ] , P ) be a complete filtered probability space
satisfying the usual hypothesis, i.e., {F t }t∈[0,T ] 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)) t∈[0,T ] be a set-valued stochastic process with values in Comp(IRd ⊗ IRm ), i.e., a family of F-measurable
set-valued mappings G(t) : Ω → Comp(IR m ⊗ IRd ), 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 t -adapted,
if G(t) is Ft -measurable for each t ∈ [0, T ]. We call G predictable, if it is
measurable with respect to predictable σ-field P(F t ) in [0, T ] × Ω. For a
stochastic process R we introduce the following notation: R t∗ = sups≤t |Rs |
and R∗ = sups≤T |Rs |. For a stopping time η, by R η we denote the stopped
process, i.e., Rtη = Rη∧t . Let S p [0, T ], (p ≥ 1) denote the space of all F t adapted and cádlág processes (Rt )t≤T , such that ||R||S p [0,T ] := ||R∗ ||Lp < ∞,
with Lp = Lp (Ω, R1 ). A semimartingale R = A + N is said to be a H p [0, T ]semimartingale (1 ≤ p ≤ ∞), if it has a finite H p [0, T ] − norm, defined
1
by: ||R||H p [0,T ] = inf x=n+a jp (N, A), where jp (N, A) = || [N, N ]T2 + 0T |dAs |
||Lp , ([N, N ]Rt ) is a quadratic variation process of local martingale part N,
and |At | = 0t |dAs | 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 = (Gt )t∈[0,T ] and a d dimensional semimartingale R
adapted to the filtration (Ft )t∈[0,T ] , R0 = 0, let us denote
R
SR (G) := {g ∈ P(Ft ) : 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 t )t∈[0,T ] is R-integrably
bounded if there exists a predictable and R-integrable process m such that
the Hausdorff distance δ(Gt , {0}) ≤ mt a.s., each t ∈ [0, T ].
2. Weak solutions
Let (Ω, F, (Ft )t∈[0,T ] , P ) be a given filtered probability space. For any random element R : Ω → Θ with values in a measurable space Θ, we denote by P R the measure on Θ being the distribution of R (under P ). Let
Weak solutions of stochastic differential inclusions ...
93
(AR , C R , ν R ) denote the local characteristics of a semimartingale R, with
respect to the fixed truncation function h : IR d → IRd (see e.g. [8] for details). For H : [0, T ] × Ω → IRm ⊗ IRd being any predictable andRbounded
(or locally bounded) mapping we will denote a stochastic integral HdR as
H ·R. Let h‘ :IRd+m → IRd+m be a fixed truncation function. For y ∈ IR d , let
P
Hy denote an m dimensional process with (Hy) i = j≤d H ij y j , for i ≤ m.
As in [8] let:
A
R,H,i
=





AR,i + [h‘i (y, Hy) − hi (y)] · ν R
P
(2) C R,H,ij
i≤d
i
H i−d,j ◦ AR,j + h‘i (y, Hy)− (Hh(y))i−d · ν R if d < i ≤ d+m
j≤d
(1)
if
h
 R,ij
C




i−d,k · C R,kj
 P
k≤d H
=
P

H j−d,k · C R,ik


 Pk≤d

k,l≤d (H
i−d,k H j−d,l )
,
if i, j ≤ d
if j ≤ d < i ≤ d + m
,
if i ≤ d < j ≤ d + m
· C R,kl if d < i, j ≤ d + m
and let ν R,H be defined by IG · ν R,H = IG (y, Hy) · ν R , for each Borel set G
in IRd+m .
By Propositions 5.3 and 5.6 Ch.IX [8] we have the following characterization for local characteristics of a stochastic integral.
Theorem 1. Let H be any predictable and bounded (or locally bounded)
mapping H : [0, T ] × Ω → IRm ⊗ IRd and let (AR , C R , ν R ) be a local characteristics of a d dimensional semimartingale
R. Suppose (R, U ) is a d + m
R
dimensional semimartingale. Then, U = HdR if and only if (R, U ) admits
a local characteristics (AR,H , C R,H , ν R,H ).
Let D([0, T ], IRn ), (n ≥ 1) denote the space of right continuous functions
on [0, T ] with values in IRn , with left limits, endowed with the Skorokhod
topology. Let µ be a given probability measure on the space (IR m , β(IRm )) .
We consider the following stochastic inclusion:
dXt ∈ F (t−, X, Z)dZt ,
t ∈ [0, T ],
(SDI)
P X0 = µ,
where
F : [0, T ] × D([0, T ], IRm ) × D([0, T ], IRd ) → Comp(IRm ⊗ IRd )
94
M. Michta
is a set-valued mapping, Z is a d dimensional semimartingale defined on a
probability space (Ω, F, (Ft )t∈[0,T ] , 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 d ) with Zt (y) =
y(t) and DTd = σ{Zt : t ≤ T }, Dtd = σ{Zs : s ≤ t}, t ∈ [0, T ].
2. The canonical space of solutions: D([0, T ], IR m ) with Xt (x) = x(t),
and σ-fields FTX = σ{Xt : t ≤ T } and FtX = σ{Xs : s ≤ t}, t ∈ [0, T ].
3. The joint canonical path space: Ω ∼ = D([0, T ], IRm ) × D([0, T ], IRd )
with Yt (x, z) = (x(t), z(t)) and σ-fields FT∼ = σ{Yt : t ≤ T } and Ft∼ =
σ{Ys : s ≤ t}, t ∈ [0, T ]. Taking projections φ 1 : Ω∼ → D([0, T ], IRm ), with
φ1 (x, z) = x and φ2 : Ω∼ → D([0, T ], IRd ), with φ2 (x, z) = z, we introduce
on a measurable space (Ω∼ , FT∼ , (Ft∼ ) the following processes Z ∼ = Z ◦ φ2
and X ∼ = X ◦ φ1 .
Let (Ad , C d , ν d ) and (Am , C m , ν m ) be processes defined on D([0, T ], IR d ) and
D([0, T ], IRm ), respectively, satisfying the properties of local characteristics.
Let us consider also processes (Am ◦φ1 , C m ◦φ1 , ν m ◦φ1 ) and (Ad ◦φ2 , C d ◦φ2 ,
ν d ◦ φ2 ) on (Ω∼ , FT∼ , (Ft∼ ). Let Q be a probability measure on (Ω ∼ , FT∼ ,
(Ft∼ )t∈[0,T ] ). We introduce probability measures: P 1 = Qφ1 and P2 = Qφ2 on
(D([0, T ], IRm ) and (D([0, T ], IRd ), respectively. Let Z ∼ be a semimartingale
under Q with the local characteristics (A d ◦ φ2 , C d ◦ φ2 , ν d ◦ φ2 ).
Definition 1. By a weak solution or driving system to the stochastic inclusion (SDI) we mean a filtered probability space (Ω ∗ , F ∗ , (Ft∗ )t∈[0,T ] , P ∗ ) on
which there are defined:
(a) an Ft∗ -adapted, d dimensional semimartingale Z ∗ , with local characteristics (Ad ◦ ψ, C d ◦ ψ, ν d ◦ ψ), where ψ : Ω∗ → D([0, T ], IRd ), ψ(ω ∗ ) =
∗
Z ∗ (ω ∗ ) and P ∗Z = Qφ2 ,
(b) an m dimensional stochastic process X ∗ -called a solution process on
∗
(Ω∗ , F ∗ , (Ft∗ )t∈[0,T ] , P ∗ ), such that: P ∗X0 = µ and
Xt∗ = X0∗ +
for some Ft∗X
∗ ,Z ∗
Z
t
0
γ ∗ (s)dZs∗ ,
t ∈ [0, T ],
-predictable mapping
γ ∗ : [0, T ] × Ω∗ → IRm ⊗ IRd ,
γ ∗ (t, ω ∗ ) ∈ F (t, X ∗ (ω ∗ ), Z ∗ (ω ∗ )).
Weak solutions of stochastic differential inclusions ...
95
We denote such solution by (Ω∗ , F ∗ , (Ft∗ )t∈[0,T ] , P ∗ , Z ∗ , X ∗ ).
Remark 1. Let Q be a probability measure on (Ω ∼ , FT∼ , (Ft∼ )t∈[0,T ] ) such
∼
that QX0 = 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 ∗ ,
∗
∗
∗
(Ft∗ )t∈[0,T ] , P ∗ , Z ∗ , X ∗ ) such that Q = P ∗(X ,Z ) . Then, P ∗X = Qφ1 and
∗
P ∗Z = Qφ2 . 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 ∼ , FT∼ , (Ft∼ )t∈[0,T ] ) is a solution measure to (SDI) if and only if (Ω ∼ , FT∼ , (Ft∼ )t∈[0,T ] , 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 ] × IRm+d → Comp(IRm ⊗ IRd ) be a set-valued
function satisfying:
(i) F is integrably bounded (by some function m(·) ),
(ii) F is ([0, T ] × IRm+d )-Borel measurable,
(iii) F (t, ·) is lower semicontinuous for every fixed t ∈ [0, T ].
m ⊗IRd ) is defined by F ∗ (t, x, z) = F̃ (t, x(t−),
If F ∗ : [0, T ]×Ω∼ → Comp(IR
Rt
z(t−)), where F̃ (t, a, b) = 0 F (s, a, b)ds, then there exists a weak solution
to the stochastic differential inclusion:
dXt ∈ F ∗ (t, X, Z)dZt ,
P
X0
t ∈ [0, T ]
= 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, (Ft )t∈I ) 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 cádlág and F t -adapted processes.
96
M. Michta
Definition 2. A probability P on (Ω, F, (F t )t∈I ) is a solution to the martingale problem related to H, X and µ if
(i) P |H = µ,
(ii) each process belonging to X is a local martingale on (Ω, F, (F t )t∈I , 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, (Ft )t∈I ), 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 previous section, we have a given bounded and predictable set-valued mapping
F : [0, T ] × Ω∼ →Comp(IRm ⊗ IRd ), an initial probability measure µ, processes (Ad , C d , ν d ) defined on D([0, T ], IRd ), satisfying the properties of local
characteristics. As mentioned in the Introduction, one can take a truncation
function h as h(y) = yI{|y|≤1} . Below we use this function. Let us take:
(a) a filtered space (Ω, F, (Ft )t∈I ) as a joint canonical space (Ω∼ , FT∼ ,
(Ft∼ )t∈[0,T ] ),
(b) a sub σ-field H = σ(X0∼ ),
(c) a class: X = X 1 ∪ X2 , where
(i) X1 is a family consisting of processes:
f (Zt∼ )−f (Z0∼ )−
XZ
i≤d
−
Z
[0,t]×IRd
t
∼ ,i
1 X
∂
∼
f (Zs−
)dAZ
−
s
2 i,j≤d
0 ∂xi
∼
∼
f (Zs−
+ y) − f (Zs−
)−
X ∂
i≤d
∂xi
Z
∂2
∼
∼
f (Zs−
)dCsZ ,ij
0 ∂xi ∂xj
t
for each bounded function f ∈ C 2 (IRd ).
(b) X2 is a family consisting of processes:
f (Rt∼ ) − f (R0∼ ) −
−
1 X
2 i,j≤d+m
Z
t
0
X Z
i≤d+m 0
t
∼
∼
f (Zs−
)yi I{|y|≤1} ν Z (ds, dy),
∂
∼ ,γ,i
∼
f (Rs−
)dAZ
s
∂xi
∂2
∼
∼
f (Rs−
)dCsZ ,γ,ij
∂xi ∂xj
Weak solutions of stochastic differential inclusions ...
Z
−
97
∼
∂
∼
f (Rs−
)yi I{|y|≤1} ν Z ,γ (ds, dy),
∂xi
i≤d+m
∼
∼
f (Rs−
+y)−f (Rs−
)−
[0,t]×IRd+m
X
for each bounded function f ∈ C 2 (IRm+d ), where R∼ = (Z ∼ , X ∼ − X0∼ ),
and for some measurable and bounded function γ : [0, T ] × Ω ∼ → IRm ⊗ IRd .
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 ∼ , FT∼ , (Ft∼ )) is a joint
solution measure to the stochastic inclusion (SDI) if and only if it is a solution 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∼ , FT∼ , (Ft∼ )t∈[0,T ] ), equipped with the topology of a weak convergence of probability measures (see [5]). By R loc
Z (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 ∈ Rloc
Z (F, µ), then Q is a joint solution measure and there exists
a weak solution system (Ω∗ , F ∗ , (Ft∗ )t∈[0,T ] , P ∗ , Z ∗ , X ∗ ). As noticed in Re∗
mark 1, the distribution law P ∗X on D([0, T ], IRm ) equals the measure Qφ1 .
∗
∼
Since φ1 (X ∼ , Z ∼ ) = X ∼ , then P ∗X = QX . Hence, there is a convenient
1
way to study the properties of the solution set. Namely, let R loc
Z (F, µ) :=
∼
loc
1
m
k
{QX : Q ∈ Rloc
Z (F, µ)}. Clearly RZ (F, µ) ⊂ M(D([0, T ], IR )). Let (µ )
be a tight sequence of initial distributions. The compactness of the set
S
loc
k 1
k≥1 RZ (F, µ ) was established in Theorem 5 of [17] in the case of continuous semimartingale satisfying the following condition:
Condition A: there exists the function h(t) = o(t), t → 0+, such that
X
1≤j,l≤d
EP [Z j , Z j ]t EP [Z l , Z l ]t +
X
j
||(AZ )t ||4H 2 (P ) ≤ h(t), for t ∈ [0, T ].
j≤d
In a similar way we can show the same property for the set
Namely, the following result holds.
S
loc
k
k≥1 RZ (F, µ ).
98
M. Michta
Theorem 4. Let Z be a continuous semimartingale satisfying Condition
A with Z0 = 0. Let (µk ) be a tight sequence of initial distributions and
let F : [0, T ] × C([0, T ], IR m+d ) → Comp(IRm ⊗ IRd ) be a measurable and
S
k
bounded set-valued mapping such that the set k≥1 Rloc
Z (F, µ ) is nonempty.
S
k
Then, the set k≥1 Rloc
Z (F, µ ) is a nonempty and relatively compact subset
m+d
of M(C([0, T ], IR
)).
P roof. Using Prokhorov‘s Theorem ([5]), it is enough to show that the set
loc
k
k≥1 R (F, µ ) is tight. Let us remark first that
S
lim
a→∞
Q∈
S suploc
k≥1
R
(F,µk )
Q{||X0∼ || > a}
≤ lim sup µk {x ∈ Rm : ||x|| > a} = 0.
a→∞ k≥1
It is because the sequence (µk ) is tight. Hence by Theorem 8.2 [5], it is
enough to use the following criterion: for every > 0
(3)
lim
n→∞
Q∈
S suploc
k≥1
R
1
Q{w ∈ C([0, T ], IRm+d ) : ∆T ( , w) > } = 0,
n
(F,µk )
where ∆T (δ, w) = sup{||w(t) − w(s)|| : s, t ∈ [0, T ], |s − t| < δ}. Let us take
S
k
an arbitrary measure Q from the set k≥1 Rloc
Z (F, µ ). Then, there exist
k ≥ 1, and measurable and bounded (say by a constant L > 0) mappings
γ k : [0, T ] × Ω∼ → IRm ⊗ IRd , γ k (t, u) ∈ F (t, u) − dt × dQ − a.e and Q ∈
k
k
m+d → R; g(x) = x , i = 1, 2, . . . ,
Rloc
i
Z (γ , µ ). Taking functions g : IR
m + d, we obtain, by the shape of the class X 2 and Theorem 1, the following
continuous Q-loc. martingales (on (Ω ∼ , FT∼ , (Ft∼ )t∈[0,T ] )):
(4)
Ntk,i
:=



Zt∼,i − AZ
t
∼ ,i
k ,i
Z ∼ ,γ−
Xt∼, i−d − X0∼,i−d − At
if 1 ≤ i ≤ d
.
if d < i ≤ d + m
Consequently, their second local characteristics are given by hN k,i , N k,j it =
∼ k
C Z ,γ− ,ij , i, j = 1, 2, . . . , d + m. Let us take N k = (N k,d+1 , . . . , N k,d+m ).
For 0 ≤ t0 < t1 < T, let us introduce the stopping time τ (u) = inf{s >
0 : ||Xt∼0 +u (u) − Xt∼0 (u)|| > 3 } ∧ (t1 − t0 ), where u ∈ Ω∼ . Then by Theorem
44 from [22], the process Ntk0 +t∧τ −Ntk0 is a continuous Q-local martingale, for
Weak solutions of stochastic differential inclusions ...
99
every fixed k ≥ 1. We let t0 = 0 for simplicity. Then by (4) we obtain
∼ ,γ k
−
)∗2
t∧τ ,
Z
k ∗4
EQ (X ∼ − X0∼ )∗4
τ ≤ 4EQ (N )τ + 4EQ (A
∼ ,γ k
−
Z
k ∗2
(X ∼ − X0∼ )∗2
t∧τ ≤ 2(N )t∧τ + 2(A
and consequently
(5)
Since
EQ (N k )∗4
τ
X
≤m
EQ
d+1≤i≤d+m
sup(Nsk,i )4
s≤τ
)∗4
τ .
,
then applying Burkholder-Davis-Gundy inequality (see e.g., [22]) to continuous Q− local martingales N k,i , we get:
EQ (N k )∗4
τ ≤ C4 m
X
k ,ii 2
Z ∼ ,γ−
EQ Cτ
d+1≤i≤d+m
,
with some universal constant C4 . Consequently by (2):
EQ (N k )∗4
τ ≤ C4 C(m)
X
X
EQ
d+1≤i≤d+m 1≤j,l≤d
Z
τ
0
,k,i−d,j
k,i−d,l
|γs−
||γs−
||dCsZ
∼ ,jl
2
| ,
with some constant C(m). From the boundedness of F we have |γ tk,i,l | ≤
supa∈F (t,X ∼ ,Z ∼ ) ||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:
(6)
EQ (N k )∗4
τ ≤ a(C4 , m, L)
X
EQ [Z ∼,j , Z ∼,j ]τ EQ [Z ∼,l , Z ∼,l ]τ ,
1≤j,l≤d
where a(C4 , m, L) is some constant not depending on Z ∼ and τ.
∼
k
Let us consider now the estimation of the
term E Q (AZ ,γ− )∗4
τ appearing in
R k
∼
(5). By Theorem 1, the semimartingale γs− dZs admits its first local charR k,i−d,j Z ∼,j
P
∼ k
∼ k
∼ k
acteristics AZ ,γ− = (AZ ,γ− ,i )i≤d , with AZ ,γ− ,i = j≤d γs−
dAs ,
i = d + 1, d + 2, . . . , d + m. Hence applying Emery‘s inequalities ([22]) and
boundedness of F , one can verify that
100
M. Michta
∼ k
EQ (AZ ,γ− )∗4
τ
3
XZ
X
≤ d m
d+1≤i≤d+m j≤d
≤ m2 d3 c44 L4
X
||(AZ
∼,j
·∧τ
0
4
∼,j k,i−d,j
γs−
dAsZ S 4 (Q)
)τ ||4H 2 (Q) ,
j≤d
where c4 is a universal constant. Using this inequality together with (6) we
finally obtain the following estimation in (5)
∼
EQ (X −
X0∼ )∗4
τ
≤ D
X
EQ [Z ∼,j , Z ∼,j ]τ EQ [Z ∼,j , Z ∼,j ]τ +
1≤j,l≤d
+
X
||(AZ
∼,j
)τ ||4H 2 (Q) ,
j≤d
for some constant D := a(C4 , d, c4 , m, L) depending only on indicated constants. Now, restoring t0 and setting t1 − t0 := α, we obtain:
EQ (Xt∼0 +· − Xt∼0 )∗4
α ≤ Dh(α),
where h is a function as in Condition A. By Tchebyshev‘s inequality we
have:
Dh(α)
Q sup ||Xt∼0 +s − Xt∼0 || > ≤
(7)
.
4
s≤α
Let T ∗ = [T ] + 1. For an arbitrary n ∈ N , let us divide the interval [0, T ∗ ]
by points { ni }, i = 0, 1, 2, . . . , T ∗ n. Then,
Q ∆T
1
n
,X
∼
> ≤Q
( T ∗ n−1 [
i=0
sup
1
0≤s≤ n
||Xt∼0 +s
− Xt∼0 ||
Hence and by (7) with α = n1 , we get:
Q ∆T
1
n
,X
∼
> ≤
34 T ∗ Dnh( n1 )
.
4
>
3
)
.
Weak solutions of stochastic differential inclusions ...
101
Hence by Condition A, we have:
lim
n→∞
Q∈
S suploc
k≥1
R
(F,µk )
Q ∆T
1
1
n
,X
∼
> = 0.
In a similar way one obtain:
lim
n→∞
Q∈
S suploc
k≥1
R
Q ∆T
(F,µk )
n
, Z ∼ > = 0,
which completes the proof.
Remark 2. Let us put in particular Zt := (t, Wt ), 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 ], IRm ) → Comp(IRm ⊗ IR1 ) and G : [0, T ] × C([0, T ], IRm ) →
Comp(IRm ⊗ IRd −1 ). Then the stochastic inclusion (SDI) has the form
dXt ∈ F (t, X)dt + G(t, X)dWt ,
P X0 = µ,
In this case one can choose h(t) = d2 t2 + t4 . Thus Theorem 4 extends earlier
results obtained in [13, 16] and [18].
For the case of a noncontinuous integrator we consider the stochastic inclusion driven by the Levy process L on the interval [0, T ]. Namely, we consider
the following inclusion
dXt ∈ F ∗ (t−, X, L)dLt ,
P
X0
t ∈ [0, T ]
= µ
1 ) defined
with a set-valued mapping F ∗ : [0, T ] × D([0, T ], IR2 ) → Comp(IR
Rt
∗
by F (t, x, z) = F̃ (t, x(t−), z(t−)), where F̃ (t, a, b) = 0 F (s, a, b)ds and
F : [0, T ]×IR2 → Comp(IR1 ) 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: At = bt, Ct = σ 2 t, ν(dt, dx) = dtm(dx), where b = E(L1 ), σ > 0
and m(dx) is a measure on IR1 \{0} that integrates the function min(1, x 2 )
(see [8] for details). We assume also that the integrator L has a finite
second moment. Then, Lt = Mt + tEL1 , where M is a square integrable
102
M. Michta
martingale. Since the integrator is a cádlág 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 n } be a sequence of semimartingales (defined possibly
on different probability spaces (Ωn , F n , (Ftn )t∈[0,T ] , P n )). We will use the
following:
Definition 3 ([24]). The sequence {Z n } of semimartingales satisfies the
uniform tightness condition (UT) if for every q ∈ IR + the family of random
variables
Z
q
0
Usn dZsn : U n ∈ Uqn , n ∈ IN
is tight in IR, where Uqn denotes the family of predictable processes of the
form
Usn
=
U0n
+
k
X
Uin I{ti <s≤ti+1 } ,
i=0
for 0 = t0 < . . . < tk = q and every Uin being an Ftni measurable random
variable such that |Uin | ≤ 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 n } be a sequence of semimartingales satisfying (UT).
Then the following statements hold true
(i) for every q ∈ IR+ the sequences {supt≤q |Ztn |} and {[Z n ]q } are tight in
IR1 ,
(ii) if {U n } is a sequence of predictable processes such that for every q ∈
IR+ the sequence {sup
|Utn |} is tight in IR1 , then the sequence of
R · t≤q
n
stochastic integrals { 0 Us dZsn } 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 ] ×
IR2 → Comp(IR1 ) is a set-valued function satisfying the assumptions of
Theorem 2. Let (µk ) be a tight sequence of initial distributions. Then,
S
∗
k 1
the set k≥1 Rloc
L (F , µ ) is nonempty and relatively compact in the space
M(D([0, T ], IR1 )).
Weak solutions of stochastic differential inclusions ...
103
∗
k 1
P roof. The nonemptiness of the set k≥1 Rloc
L (F , µ ) follows from Theorem 2 and Theorem 3. Let us take an arbitrary sequence of measures
{Rn } :
[
∗
k 1
{Rn } ⊂
Rloc
L (F , µ ) .
S
k≥1
Then, for every n ≥ 1 there exist kn ≥ 1, the joint solution measure Qkn ∈
∗
kn
Rloc
L (F , µ ) and (by Theorem 3 and Remark 1) a weak solution system
k
k
(Ω n , F n , (Ftkn )t∈[0,T ] , P kn , Lkn , X kn ) with the following properties:
(i) Qkn = (P kn )(X
kn ,Lkn )
, Rn = (P kn )X
kn
(ii) Lkn is an Ftkn -adapted square integrable Levy process, with local characteristics At = bt, Ct = σ 2 t, ν(dt, dx) = dtm(dx) and X kn is a solution
kn
process on (Ωkn , F kn , (Ftkn )t∈[0,T ] , P kn ) such that (P kn )X0 = µkn and
Xtkn = X0kn +
for some (Ftkn )X
process
kn ,Lkn
Z
t
0
γ kn (s)dLks n ,
t ∈ [0, T ],
-predictable and bounded (say by the constant C)
γ kn : [0, T ] × Ωkn → IR1 ,
γ kn (t) ∈ F (t, X kn , Lkn ) dt ⊗ dP kn -a.s.
Since the sequence of processes {γ kn } is uniformly bounded it follows that
the sequence {supt≤q |γtkn |} is tight in IR1 for every q ∈ IR+ . For every
q ∈ IR+ let us consider the family Uqn described in Definition 3. Then using
first Khinthine’s inequality and next Emery‘s inequality [22], we get the
following estimation:
P
kn
Z
q
0
Usn dLks n >K
≤
1 kn
E
K2
Z
sup 0≤t≤q
t
0
Z
c22 · n kn 2
≤ 2
Us dLs K
c22 q
K2
≤
2
H[0,q]
0
c2
≤ 22
K
σ +
Z
σ2 +
Z
2
2 Usn dLks n 2
x m(dx) + qM
Z
x2 m(dx) + qM ,
q
0
E kn (Usn )2 ds
104
M. Michta
where M := E kn [(Lk1n )2 ] < ∞ (we have assumed that the Levy process has
the finite second moment). Hence, the sequence
{L kn } satisfies (UT). By
R· k
k
Theorem 5 we claim that the sequence { 0 γs n dLs n } satisfies (UT)
as well.
Rt k
n
Consequently, we infer the tightness of the sequence {sup t≤q | 0 γs dLks n |}
kn
kn
for every q ∈ IR+ . For n, N ≥ 1 let TNn denote the set of (Ftkn )X ,L stopping times that are bounded by N . Then, similarly as above one can
show
sup
S,T ∈TNn :S≤T ≤S+θ
1
≤ 2
ε
n
P kn |XTkn − XSkn | > ε
sup
S,T ∈TNn :S≤T ≤S+θ
c2 C 2 θ
σ2 +
≤ 2 2
ε
Z
E
kn
Z
sup S≤q≤T
2
x m(dx) + θM
q
S
o
2
γτkn Lkτ n for every ε, θ > 0 and n ∈ IN. Thus we have
lim lim sup
θ→0+
sup
n S,T ∈T n :S≤T ≤S+θ
N
P kn {|XTkn − XSkn | > 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 kn }, which implies the some property
for the sequence {Rn }. Hence by Prohorov’s Theorem we infer that the
S
∗
k
1
set k≥1 Rloc
L (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 C0 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 systems of Zakai type, SIAM J. Contr. Optim. 34 (5) (1996), 1592–1615.
[5] P. Billingsley, Convergence of Probability Measures, Wiley, New York, 1968.
Weak solutions of stochastic differential inclusions ...
105
[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, Applicationes Math. 29 (4) (2002), 387–398.
[19] M. Michta and J. Motyl, High order stochastic inclusions and their applications, 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 Itô equation, Bull. Acad. Pol. Sci.
46 (1998), 419–430.
[22] P. Protter, Stochastic Integration and Differential Equations: A New Approach, Springer, New York, 1990.
106
M. Michta
[23] L. Slomiński, Stability of stochastic differential equations driven by general
semimartingales, Dissertationes Math. 349 (1996), 1–109.
[24] C. Stricker, Loi de semimartingales et critéres de compacité, 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