Set-valued Ito's formula with an application to the general set-valued backward stochastic differential equation
aa r X i v : . [ m a t h . P R ] F e b Set-valued Itˆo’s formula with an application to the generalset-valued backward stochastic differential equation ∗ Yao-jia Zhang, Zhun Gou and Nan-jing Huang † Department of Mathematics, Sichuan University, Chengdu, Sichuan 610064, P.R. China
Abstract.
The overarching goal of this paper is to establish a set-valued Itˆo’s formula. Asan application, we obtain the existence and uniqueness of solutions for the general set-valuedbackward stochastic differential equation which gives an answer to an open question proposedby Ararat et al. (C. Ararat, J. Ma and W.Q. Wu, Set-valued backward stochastic differentialequation, arXiv:2007.15073).
Keywords:
Set-valued stochastic integral; Set-valued Itˆo’s formula; Set-valued back-ward stochastic differential equation; Hukuhara difference; Picard iteration. : 26E25, 28B20, 60G07, 60H05, 60H10.
Set-valued differential equations, both deterministic and stochastic, have attracted the attention of manyscholars due to the wide applications of set-valued functions (mappings) in practical problems [6,7,13,24,35].Recently, many applications about set-valued deterministic/stochastic differential equations are found incomputer science [37], economics and finance [9, 17, 24, 30].Itˆo’s formula, established firstly by Itˆo [20], plays an important role in stochastic analysis [21, 22]. Inorder to study some new stochastic differential equations, the Itˆo’s formula has been extended in severaldirections. For example, Applebaum and Hudson [5] proved an Itˆo product formula for stochastic integralsagainst fermion Brownian motion. Al-Hussaini and Elliott [4] showed an Itˆo’s formula for a continuous semi-martingale X t with a local time L αt at α . Gradinaru et al. [15] gave an Itˆo’s formula for nonsemimartingales.Catuogno and Olivera [12] provided a fresh Itˆo’s formula for time dependent tempered generalized functions.Recently, a generalised Itˆo’s formula for L´evy-driven Volterra Processes was established by Bender et al. [10].However, to the best of our knowledge, there is no set-valued Itˆo’s formula. The first goal of this paper is togive a set-valued Itˆo’s formula.It is well known that backward stochastic differential equations (BSDEs), introduced by Pardoux andPeng [32], have been studied extensively in the literature. For instance, we refer the reader to [18,19,23,29,31].Recently, various examples have been given in the literature [3,11,14,36,39] to motivate the study of BSDEs.Very recently, Ararat et al. [8] provided some sufficient conditions to ensure the existence and uniqueness of ∗ This work was supported by the National Natural Science Foundation of China (11471230, 11671282). † Corresponding author. E-mail addresses: [email protected]; [email protected] Y t = ξ + Z Tt f ( t, Y s ) ds ⊖ Z Tt Z s dW s , where ( W s , s >
0) is a standard m -dimension Brownian movement with dW i dW j = 0( i = j ).However, as pointed out by Ararat et al. [8], the systematic study of the following general set-valuedbackward stochastic differential equation (GSVBSDE): Y t = ξ + Z Tt f ( t, Y s , Z s ) ds ⊖ Z Tt Z s dW s , (1.1)is still widely open. It is worth mentioning that (1.1) is a modelling tool used to capture the risk measureproblem arising in finance [8]. The second purpose of this paper is to study the existence and uniqueness of(1.1) by using the set-valued Itˆo’s formula.The rest of this paper is structured as follows. The second section recalls some necessary preliminaries,including some properties of Hukuhara difference, set-valued stochastic processes and stochastic integrals.After that in Section 3, we obtain the set-valued itˆo’s formula by employing some properties of set-valuedstochastic integrals. Finally, we show the existence and uniqueness of the solutions to (1.1) as the applicationof the set-valued Itˆo’s formula. In this section, we recall some necessary notations and definitions.
For a Hilbert space X , let P ( X ) be the set of all nonempty subsets of X . Let L ( X ) be the set of all closedsets in P ( X ) and K ( X ) be the set of all compact convex sets in P ( X ).For any A, B ∈ K ( X ), the Hausdorff distance between A and B is defined by h ( A, B ) := max (cid:26) sup x ∈ A inf y ∈ B d ( x, y ) , sup x ∈ B inf y ∈ A d ( x, y ) (cid:27) and the mapping k · k : K ( X ) → [0 , ∞ ) is defined by k A k := h ( A, { } ) = sup a ∈ A | a | , ∀ A ∈ K ( X ) . (2.1)Moreover, for any A, B ∈ K ( R n ) and α ∈ R , define A + B := { a + b : a ∈ A, b ∈ B } ; αA := { αa : a ∈ A } . Clearly, for
A, B, C ∈ K ( R n ), the following cancellation law holds: A + C = B + C ⇐⇒ A = B. Assume that (Ω , F , P ) is a probability measure space. A ∈ F is called an atomic set, if P ( A ) > P ( B ) = 0 or P ( A \ B ) = 0 for any Borel subset B ⊂ A . If there is no atomic sets under the measure P ,then P is called a nonatomic probability measure. In this paper, we always assume P is nonatomic. In thesequel, we assume that (Ω , F , F , P ) is a filtered probability space satisfying the usual conditions.Now we recall the Hukuhara difference, which defines the “subtraction” in the space K ( X ).2 efinition 2.1. [16] The Hukuhara difference is defined as follows: A ⊖ B = C ⇐⇒ A = B + C, A, B ∈ K ( X ) . For our main results, we need the following lemmas.
Lemma 2.1. [8] Let
A, B, A , A , B , B , C ∈ K ( R n ) . The following identities hold if all the Hukuharadifferences involved exist.( i ) A ⊖ A = { } , A ⊖ { } = A ;( ii ) ( A + B ) ⊖ ( A + B ) = ( A ⊖ A ) + ( B ⊖ B ) ;( iii ) ( A + B ) ⊖ B = A + ( B ⊖ B ) = ( A ⊖ B ) + B ;( iv ) A + ( B ⊖ B ) = ( A ⊖ B ) + B ;( v ) A = B + ( A ⊖ B ) . Lemma 2.2.
Let
A, B, C ∈ K ( R n ) . Then( i ) If A ⊖ C exists, then ( A + B ) ⊖ C exists for any B ∈ K ( R n ) ;( ii ) If A ⊖ C and C ⊖ B exist, then A ⊖ B exists and A ⊖ B = ( A ⊖ C ) + ( C ⊖ B ) .Proof. (i) Set M = A ⊖ C . Then A = C + M and so ( A + B ) ⊖ C = ( C + M + B ) ⊖ C = M + B . (ii) Let M = A ⊖ C , N = C ⊖ B , then A = C + M , C = B + N . Taking C = B + N in A = C + M , we have A = B + N + M , which implies A ⊖ B = M + N = ( A ⊖ C ) + ( C ⊖ B ). Lemma 2.3. [8]( i ) The mapping k · k : K ( X ) → [0 , ∞ ) defined by (2.1) satisfies the properties of a norm;( ii ) If A, B ∈ K ( X ) and A ⊖ B exists, then h ( A, B ) = k A ⊖ B k ;( iii ) For any A, B ∈ K ( R n ) , both A ⊖ B and B ⊖ A exist if and only if A is a translation of B , i.e., A = B + { c } , where c ∈ R n . In this subsection, we recall the set-valued stochastic processes and give a limit convergence theorem forset-valued processes.For a sub- σ -filed G ⊂ F , let L G (Ω , K ( R n )) denote the set of all G -measurable random variables valuedin K ( R n ). For any X ∈ K ( R n ), let S G ( X ) be the set of all G -measurable selection of X . Moreover, let L G (Ω , K ( R n )) denote the set of all G -measurable square-integrably bounded random variables valued in K ( R n ). For simplicity, let S G ( X ) = S G ( X ) T L G (Ω , K ( R n )). Definition 2.2. [24] • L ad (Ω , R n ) = L ad (Ω , F , P , R n ) , the set of all the n -dimensional measurable square-integrably randomvariables, is a Hilbert space equipped with the norm k · k a = ( E | · | ) . L ad ([0 , T ] × Ω , R n ) = n f ( t, ω ) | f ( t, ω ) is F t -adapted with E hR T | f ( t, ω ) | dt i < ∞ o is a Hilbert spacewith the norm k f ( t, ω ) k c = E hR T | f ( t, ω ) | dt i . • A set-valued mapping F : Ω → L ( X ) is called measurable if, for any closed set B ⊂ X , it holds that { ω ∈ Ω : F ( ω ) ∩ B = ∅} ∈ F . • A measurable set-valued mapping X : Ω → L ( R n ) is called a set-valued random variable. • A set-valued stochastic process f = { f t } t ∈ [0 ,T ] is a family of set-valued random variables taking valuesin L ( R n ) . • A set-valued stochastic process is called measurable if it is B ([0 , T ]) ⊗ F -measurable as a single functionon [0 , T ] × Ω . • L (Ω , L ( R n )) == L (Ω , F T , P , L ( R n )) , is the space of all measurable set-valued mappings F : E →L ( R n ) distinguished up to P -almost everywhere equality. • L t (Ω , L ( R n )) is the set of all F t -measurable random variables valued in L ( R n ) and L pt (Ω , L ( R n )) is theset of all F t -measurable p -integrably bounded random variables valued in L ( R n ) for any given t ∈ [0 , T ] . • A (Ω , L ( R d )) := (cid:8) F ∈ L (Ω , L ( R n )) : S ∗ ( F ) = ∅ (cid:9) and A t (Ω , L ( R d )) := (cid:8) F ∈ L t (Ω , L ( R n )) : S ∗ ( F ) = ∅ (cid:9) . • For any X ∈ A (Ω , L ( R d )) and any t ∈ [0 , T ] , S ∗ t ( X ) is the set of all F t -measurable selections of X . • For any X ∈ L (Ω , L ( R n )) the collection of all the selections of L ad (Ω , R n ) is denoted by S ∗ ( X ) = (cid:8) f ∈ L ad (Ω , R n ) : f ( ω ) ∈ F ( ω ) P - a.s. (cid:9) . • S ∗ T ( X ) = S ∗ T ( X ) T L ad (Ω , R n ) . • A set-valued random variable X ( ω ) is called p -integrably bounded if there exists m ∈ L ad (Ω , R + ) suchthat k X k m ( ω ) P -a.s. • L p ( E, L ( R n )) = (cid:8) F ∈ L ( E, L ( R n )) : k F k m ( ω ) P - a.s. (cid:9) , where m ( ω ) ∈ L pad (Ω , R + ) . • A set-valued stochastic process F is called p -integrably bounded if there exists m ∈ L ad ([0 , T ] × Ω , R + ) such that k F k m ( t, ω ) P -a.s.. • Denote the subtrajectory integrals of a set-valued stochastic process F : T × Ω → L ( R n ) by S ( F ) , whichis the set of all measurable and dt × P -integrable selectors of F . • Denote the subset S ( F ) := S ( F ) \ L ad ([0 , T ] × Ω , R n ) and S F ( F ) = (cid:8) f ∈ S ( F ) : f is F - measurable (cid:9) . Definition 2.3. [24] A set-valued process is called F -nonanticipative if it is Σ F -measurable, where Σ F = { A ∈ β T ⊗ F : A t ∈ F t ∀ t ∈ T } , and A t denotes the t -section of a set A ⊂ T × Ω , i.e., A t = { ω ∈ β T : A ( t, ω ) ∈ β T ⊗ F} . The set of allset-valued F -nonanticipative measurable square integrably bounded stochastic processes taking values in K ( R ) with S F ( · ) = ∅ is denoted by L ad ([0 , T ] × Ω , K ( R n )) . emma 2.4. [24] Let Φ t , Ψ t ∈ L ad ([0 , T ] × Ω , L ( R n )) . Then L ad ([0 , T ] × Ω , L ( R n )) is a complete metricspace with the metric d H (Φ t , Ψ t ) = [ E R T h (Φ t , Ψ t ) dt ] . Specially, L ad ([0 , T ] × Ω , K ( R n )) is a Banach spacewith the norm k Z ( t )( ω ) k s = [ E R T k Z k dt ] for Z ( t ) ∈ L ad ([0 , T ] × Ω , K ( R n )) . Lemma 2.5. (Kuratowski and Ryll-Nardzewski) [26] Assume that ( E, E ) is a P olish space and ( T, F ) is ameasurable space. If F : T → L ( E ) is measurable, then F admits a measurable selector. The set-valued conditional expectation and the set-valued martingale can be defined as follows.
Definition 2.4. [24] For a sub- σ -field G ⊂ F and X ∈ L (Ω , K ( R n )) , the conditional expectation of X withrespect G is defined as the unique set-valued random variable E [ X |G ] ∈ L G (Ω , K ( R n )) such that Z G E [ X |G ] d P = Z G Xd P , ∀ G ∈ G . Definition 2.5. [25] A set-valued process M = { M t } t ∈ [0 ,T ] is said to be a set-valued square-integrable F -martingale if( i ) M ∈ L ad ([0 , T ] × Ω , L ( R n )) ;( ii ) M t ∈ A t (Ω , L ( R n )) ;( iii ) M s = E [ M t |F s ] for all s t . Lemma 2.6. [8] Let F , F ∈ L ad ([0 , T ] × Ω , K ( R n )) . Then, F + F ∈ L ad ([0 , T ] × Ω , K ( R n )) and S F ( F + F ) = S F ( F ) + S F ( F ) . Furthermore, if F ⊖ F exists, then F ⊖ F ∈ L ad ([0 , T ] × Ω , K ( R n )) and S F ( F ⊖ F ) = S F ( F ) ⊖ S F ( F ) . Lemma 2.7.
Suppose that { X p ( t, ω ) } ∞ p =1 ⊂ L ad ([0 , T ] × Ω , K ( R n )) is a sequence of set-valued stochasticprocesses satisfying( i ) X q ⊖ X p exists for q > p ;( ii ) For each ε > , there exists an N ∈ N such that k X q ⊖ X p k s < ε when p, q > N .Then there exists a unique X ( t, ω ) ∈ L ad ([0 , T ] × Ω , K ( R n )) such that k X p ⊖ X k s → as p → + ∞ .Proof. It follows from the assumption (i) that the potential of the X p is increasing as p becoming larger.Thus, the assumption (ii) shows that, for each ε >
0, there exist an N ′ ∈ N and a ξ ∈ R p such that X q ⊖ X p = ξ with | ξ | < ε when n, m > N ′ . Since X q ⊖ X p exists, we have k X q ⊖ X p k s = " E Z T k X q ⊖ X p k ds = " E Z T h ( X q , X p ) ds when p, q > N ′ . By Lemma 2.4 and the fact { X p ( t, ω ) } ∞ p =1 in L ad ([0 , T ] × Ω , K ( R n )), there exists X ∗ ∈L ad ([0 , T ] × Ω , K ( R n )) such that k X ∗ ⊖ X p k s < ε for all t ∈ [0 , T ] and a.s ω ∈ Ω when p > N ′ .Next we show the uniqueness. If X , X ∈ L ad ([0 , T ] × Ω , K ( R n )) such that k X p ⊖ X k s → k X p ⊖ X k s → p → + ∞ , then it follows from Lemma 2.2 that k X ⊖ X k s = k X ⊖ X p + X p ⊖ X k s k X p ⊖ X k s + k X p ⊖ X k s → p → + ∞ . This ends the proof. 5 .3 Set-valued stochastic integrals In this subsection, we recall the the notions of set-valued stochastic integrals.
Definition 2.6. ( [8, 24]) • For F ∈ L ad ([0 , T ] × Ω , K ( R n )) , M F ([0 , T ] × Ω , K ( R n )) := (cid:8) F ∈ L ad ([0 , T ] × Ω , K ( R n )) : S F ( F ) = ∅ (cid:9) . • A set V ⊂ L ad (Ω , R n ) is said to be decomposable with respect to F if for any f , f ∈ V and D ∈ F ,it holds that D f + 1 D c f ∈ V . • Denote the decomposable hull of V ⊂ L ad (Ω , R n ) by dec ( V ) , which is the smallest decomposable subsetof L ad (Ω , R n ) contain V . Denote the closure of dec ( V ) in L ad (Ω , R n ) by dec ( V ) . • K w ( L ad ([0 , T ] × Ω , R n × m ) is the set of weakly compact convex subset of L ad ([0 , T ] × Ω , R n × m ) . Lemma 2.8. [34] Suppose f ∈ L ad ([0 , T ] × Ω , R m ) . Then the Itˆo integral I ( f ) = R T f ( t ) dW ( t ) is a randomvariable with E [ I ( f )] = 0 and E [ I ( t )] = E "Z T f ( t ) dt . Definition 2.7. [8, 24] • Denote the J : L ad ([0 , T ] × Ω , β T ⊗ F T , R n ) → L ad (Ω , F T , R n ) by J ( φ )( ω ) = Z T φ ( t ) dt ! ( ω ) P - a.s.. where φ ∈ L ad ([0 , T ] × Ω , β T ⊗ F T , R n ) , β T and F T are the σ -fields on T and Ω , respectively. • Denote the J : L ad ([0 , T ] × Ω , Σ F , R n × m ) → L ad (Ω , F T , R n ) by J ( ψ )( ω ) = Z T ψ ( t ) dW t ! ( ω ) P - a.s.. where ψ ∈ L ([0 , T ] × Ω , Σ F , R n × m ) . • Denote the sets J [1 [ s,t ] S ( F )] and J [1 [ s,t ] S ( F )] by J st [ S F ( F )] and J st [ S F ( G )] , respectively, which aresaid to be the functional set-valued stochastic integral of F and G on the interval [ s, t ] , respectively. Definition 2.8. [24] Let F ∈ L ad ([0 , T ] × Ω , L ( R n )) and G ∈ L ad ([0 , T ] × Ω , L ( R n × m )) such that S F ( F ) = ∅ and S F ( G ) = ∅ . Then there exist unique set-valued random variables R T F ( t ) dt and R T G ( t ) dW t in A T ( E, L ( R d )) such that S T ( R T F ( t ) dt ) = dec ( J [ S F ( F )]) and S T ( R T G ( t ) dW t ) = dec ( J [ S F ( G )]) . The set-valued random variables R T F ( t ) dt and R T G ( t ) dW t are, respectively, called the set-valued stochastic integralsof F ( t ) and G ( t ) with respect to t and W t on [0 , T ] . Moreover, for any t ∈ [0 , T ] , define Z t F ( s ) ds := Z T (0 ,t ] ( s ) F ( s ) ds, Z t G ( s ) dW s := Z T (0 ,t ] ( s ) G ( s ) dW s and Z Tt F ( s ) ds := Z T ( t,T ] ( s ) F ( s ) ds, Z Tt G ( s ) dW s := Z T ( t,T ] ( s ) G ( s ) dW s . In the sequel, we always suppose that the set-valued integrals exist.6 emma 2.9. [8] Let F , F ∈ L ad ([0 , T ] × Ω , K ( R n )) and G , G ∈ L ad ([0 , T ] × Ω , K ( R n × m )) . Then, forevery t ∈ [0 , T ] , Z t ( F ( s ) + F ( s )) ds = Z t F ( s ) ds + Z t F ( s ) ds and Z t ( G ( s ) + G ( s )) dW s = Z t G ( s ) dW s + Z t G ( s ) dW s hold P -a.s.. If F ⊖ F and G ⊖ G exist, then F ⊖ F ∈ L ad ([0 , T ] × Ω , K ( R n )) and G ⊖ G ∈ L ad ([0 , T ] × Ω , K ( R n × m )) for every t ∈ [0 , T ] . Moreover, Z t F ( s ) ⊖ F ( s ) ds = Z t F ( s ) ds ⊖ Z t F ( s ) ds and Z t G ( s ) ⊖ G ( s ) dW s = Z t G ( s ) dW s ⊖ Z t G ( s ) dW s hold P -a.s.. Lemma 2.10. [8] Let F ∈ L ad ([0 , T ] × Ω , K ( R n )) and G ∈ L ad ([0 , T ] × Ω , K ( R n × m )) . Then for each t ∈ [0 , T ] , Z T F ( s ) ds = Z t F ( s ) ds + Z Tt F ( s ) ds, Z T G ( s ) dW s = Z t G ( s ) dW s + Z Tt G ( s ) dW s and Z Tt F ( s ) ds = Z T F ( s ) ds ⊖ Z t F ( s ) ds, Z Tt G ( s ) dW s = Z T G ( s ) dW s ⊖ Z t G ( s ) dW s hold P -a.s.. Lemma 2.11. [8] For any
Z, Z , Z ∈ K w ( L ad ([0 , T ] × Ω , R n × m )) , the following statements hold:( i ) Z + Z ∈ K w ( L ad ([0 , T ] × Ω , R n × m )) and for every t ∈ [0 , T ] , Z t ( Z + Z ) dW s = Z t Z dW s + Z t Z dW s , P - a.s. ; ( ii ) If Z ⊖ Z exists, then Z ⊖ Z ∈ K w ( L ad ([0 , T ] × Ω , R n × m )) and for every t ∈ [0 , T ] , Z t ( Z ⊖ Z ) dW s = Z t Z dW s ⊖ Z t Z dW s , P - a.s. ; ( iii ) If Z ⊖ Z exists and R t Z dW s = R t Z dW s , P -a.s. for all t ∈ [0 , T ] , then Z = Z P -a.s.;( iv ) Z T ZdW s = Z t ZdW s + Z Tt ZdW s , Z Tt ZdW s = Z T ZdW s ⊖ Z t ZdW s . Lemma 2.12. [25] For every convex-valued square-integrable set-valued martingale M = { M t } t ∈ [0 ,T ] , thereexists G M ∈ P ( L ad ([0 , T ] × Ω , R n × m ) such that M t = M + R t G M dB s P -a.s. for all t ∈ [0 , T ] . Moreover, if M is uniformly square-integrably bounded, then G M is a convex weakly compact set, i.e., G M ∈ K w ( L ad ([0 , T ] × Ω , R n × m )) . Denote G by G := ( Z ( t ) ∈ K w ( L ad ([0 , T ] × Ω , R n × m )) : (cid:26)Z t Z ( s ) dW s (cid:27) t ∈ [0 ,T ] is an F t martingale ) . emark 2.1. Since K w ( L ad ([0 , T ] × Ω , R n × m )) is weakly closed and bounded, G is closed. Lemma 2.13. [8] Let X , X ∈ L T (Ω , K ( R n )) and G ⊂ F be a sub- σ -field. If X ⊖ X exists, then E [ X ⊖ X |G ] exists in L G (Ω , K ( R n )) and E [ X ⊖ X |G ] = E [ X |G ] ⊖ E [ X |G ] . In this section, we extend the classical Itˆo’s formula to the set-valued case. To this end, we need the followinglemma.
Lemma 3.1. [24]( i ) If (Ω , F , P ) is separable and Φ t ∈ L ad ([0 , T ] × Ω , L ( R n × m )) , then there exists a sequence { φ n } ∞ n =1 ⊂ S F (Φ) such that Z T Φ s dW s ! ( ω ) = cl L ( Z T φ nt dW s ! ( ω ) : n > ) P - a.s., where cl L means the closure taken in the norm topology of L ad ([0 , T ] × Ω , L ( R n )) .( ii ) If (Ω , F , P ) is separable and Ψ t ∈ L ad ([0 , T ] × Ω , L ( R n )) , then there exists a sequence { ψ n } ∞ n =1 ⊂ S F (Ψ) such that Z T Ψ s ds ! ( ω ) = cl L ( Z T ψ nt ds ! ( ω ) : n > ) P - a.s.. We define a set-valued Itˆo’s process as follows: X t = x + Z t f ( s ) dW s + Z t g ( s ) ds, t T, (3.1)where x = X ∈ L ad (Ω , R n ), f ( t ) ∈ L ad ([0 , T ] × Ω , K ( R n × m )) and g ( t ) ∈ L ad ([0 , T ] × Ω , K ( R n )).Clearly, Lemma 3.1 shows that the set-valued Itˆo’s process defined by (3.1) is a set-valued stochasticprocess and (3.1) can be rewritten as follows: X i ( t ) = x i + Z t f i ( s ) dW s + Z t g i ( s ) ds, i = 1 , · · · , n, (3.2)where f Ti ( t ) ∈ L ad ([0 , T ] × Ω , K ( R m )) and g i ( t ) ∈ L ad ([0 , T ] × Ω , K ( R )) are the i -th component of f ( t ) and g ( t ), respectively. Lemma 3.2.
Let A = n ( R t φ nt dW s )( ω ) : n > o and B = n ( R t ψ mt ds )( ω ) : m > o P -a.s., where { φ nt } ∞ n =1 ⊂ S F (Φ) and { ψ nt } ∞ n =1 ⊂ S F (Ψ) . Then cl L ( A ) + cl L ( B ) = cl L ( A + B ) .Proof. Let A = cl L ( A ) and B = cl L ( B ). Then we need to show that A + B = A + B .Step 1. For any given x ∈ A + B , there exists a ∈ A and b ∈ B such that x = a + b . Moreover, thereexist { a n } ⊂ A and { b n } ⊂ B such that a n → a and b n → b . Thus, a n + b n → a + b . Since a n + b n ∈ A + B ,we have x = a + b ∈ A + B , which implies A + B ⊂ A + B .8tep 2. For any given x ∈ A + B , there exists { x n } ⊂ A + B such that x n → x . This means that, forany given ε >
0, there exists a positive integer N > k x n − x m k c < ε when n, m > N . Since x n ∈ A + B , there exist a n = R t f n ( s ) dW s ∈ A and b n = R t g n ( s ) ds ∈ B such that x n = a n + b n such that Z t f n ( s ) dW s + Z t g n ( s ) ds → x. If there is no limit for { a n } , then that exists a ε > N >
1, there exists n, m > N satisfying k a n − a m k c > ε . This means that E "Z T (cid:18)Z t ( f n − f m ) dW s (cid:19) dt > ε . Thus, when n, m > N , it follows from Lemma 2.8 that k x n − x m k c = E "Z T (cid:18)Z t ( f n − f m ) dW s + Z t ( g n − g m ) ds (cid:19) dt = E " Z T (cid:18)Z t ( f n − f m ) ds (cid:19) + (cid:18)Z t ( g n − g m ) ds (cid:19) dt ! > ε , which is a contradiction with the fact k x n − x m k c < ε . Therefore, we know that there exists an a ∈ A suchthat a n → a . Similarly, there exists a b ∈ B such that b n → b . Thus, we have a + b = x ∈ A + B and so A + B ⊂ A + B .Combining Steps 1 and 2, we obtain A + B = A + B . Remark 3.1.
Let X be a Banach space. Then A + B ⊂ A + B always holds for any A, B ⊂ X . Especially, a + B = a + B holds for any a ∈ X and B ⊂ X . In order to obtain the set-valued Itˆo’s formula, we need the following assumption.
Assumption 3.1.
Assume that Y , Y are two topological spaces and ϕ : Y → Y is a continuous mappingsuch that, for any bounded subset A in Y , ϕ ( cl Y ( A )) = cl Y ( φ ( A )) , where ϕ ( A ) = ∪ a ∈ A ϕ ( a ) . Remark 3.2. ( i ) If ϕ : R n → R n is continuous, then it is easy to check that ϕ satisfies Assumption 3.1;( ii ) Any continuous closed mapping from a Banach space to another one satisfies Assumption 3.1;( iii ) The mapping ϕ : L ad ([0 , T ] × Ω , R n ) → L ad ([0 , T ] × Ω , R n ) defined by ϕ ( x ) = ( x , x , · · · , x n ) , ∀ x = ( x , x , · · · , x n ) ∈ L ad ([0 , T ] × Ω , R n ) satisfies Assumption 3.1. Now we are able to derive the set-valued Itˆo’s formula.
Theorem 3.1.
Let X t be the set-valued Itˆo process defined by (3.1) . If φ ( t, x ) is a continuous functionsatisfying Assumption 3.1 with continuous partial derivatives ∂φ∂t , ∂φ∂x and ∂ φ∂x for every x ∈ X it , then φ ( t, X t )9 s a set-valued Itˆo process and φ ( t, X ( t )) k = φ (0 , x ) k + Z t n X i =1 ∂φ k ∂x i ( s, X ( s )) f i ( s ) dW s + Z t " ∂φ k ∂t ( s, X ( s )) + n X i =1 ∂φ k ∂x i ( s, X ( s )) g i ( s ) + 12 n X i =1 ∂ φ k ∂x i ( s, X ( s )) f i ds,k = 1 , · · · , n P - a.s., where φ ( t, x ) : T × L ad ([0 , T ] × Ω , R n ) → L ad ([0 , T ] × Ω , R n ) , φ ( t, X ( t )) = { φ ( t, x ( t )) : x ( t ) ∈ X ( t ) } , ∂φ k ∂t ( s, X ( t )) = n ∂φ k ∂t ( s, x ( t )) : x ( t ) ∈ X ( t ) o , ∂φ k ∂x i ( s, X ( t )) = n ∂φ k ∂x i ( s, x ( t )) : x ( t ) ∈ X ( t ) o , ∂ φ k ∂x i ( s, X ( t )) = n ∂ φ k ∂x i ( s, x ( t )) : x ( t ) ∈ X ( t ) o and f i = n ˜ f Ti ˜ f i : ˜ f i ∈ f i o .Proof. Since f ( t ) and g ( t ) are convex-valued, square integrably bounded, by Lemma 3.1, there exist twosequences (cid:8) f d ( s ) (cid:9) ∞ d =1 ⊂ S F ( f ) and (cid:8) g d ( s ) (cid:9) ∞ d =1 ⊂ S F ( g ) such that Z t f ( s ) dW s = cl L (cid:26)(cid:18)Z t f d ( s ) dW s (cid:19) ( ω ) : d > (cid:27) a.e. P - a.s.. (3.3)and Z t g ( s ) ds = cl L (cid:26)(cid:18)Z t g d ( s ) ds (cid:19) ( ω ) : d > (cid:27) a.e. P - a.s.. It follows from (3.1) and Lemma 3.2 that X ( t ) = x + cl L (cid:26)(cid:18)Z t f d ( s ) dW s (cid:19) ( ω ) + (cid:18)Z t g d ( s ) ds (cid:19) ( ω ) : d , d > (cid:27) a.e. P - a.s.. For any given d , d >
1, let x d d ( t ) = x + Z t f d ( s ) dW s + Z t g d ( s ) ds a.e. P - a.s.. Then x d d ( t ) is a single-valued Itˆo’s process and so the classical Itˆo’s formula shows that φ ( t, x d d ( t )) k = φ (0 , x ) k + Z t n X i =1 ∂φ k ∂x i ( s, x d d ( s )) f d i ( s ) dW s + Z t " ∂φ k ∂t ( s, x d d ( s )) + n X i =1 ∂φ k ∂x i ( s, x d d ( s )) g d i ( s ) + 12 n X i =1 ∂ φ k ∂x i ( s, x d d ( s ))( f d i ) ds (3.4)for k = 1 , · · · , n and P - a.s. . It follows from Lemma 3.2 and Remark 3.1 that cl L (cid:8) x d d ( t ) : d > , d > (cid:9) = x + cl L (cid:26)Z t f d ( s ) dW s + Z t g d ( s ) ds : d > , d > (cid:27) = x + cl L (cid:26)Z t f d ( s ) dW s : d > , (cid:27) + cl L (cid:26)Z t g d ( s ) ds : d > (cid:27) = X. Let A = (cid:8) x d d ( t ) : d > , d > (cid:9) . Then A = cl L (cid:8) x d d ( t ) : d > , d > (cid:9) and so Assumption 3.1 yields φ ( t, X ( t )) = φ ( t, A ) = φ ( t, A ) . (3.5)10rom (3.4), Lemma 3.2 and Remark 3.1, one has φ ( t, X ( t )) k = x + cl L (Z t n X i =1 ∂φ k ∂x i ( s, x d d ( s )) f d i ( s ) dW s : d , d > ) + cl L (Z t " ∂φ k ∂t ( s, x d d ( s )) + n X i =1 ∂φ k ∂x i ( s, x d d ( s )) g d i ( s ) + 12 n X i =1 ∂ φ k ∂x i ( s, x d d ( s ))( f d i ) ds : d , d > ) . (3.6)For fixed f d ∈ S F ( f ), by Lemma 3.1, there exists a sequence (cid:8) x d (cid:9) ⊂ S F ( X ) such that Z t n X i =1 ∂φ k ∂x i ( s, X ( s )) f d i ( s ) dW s = cl L (Z t n X i =1 ∂φ k ∂x i ( s, x d ( s )) f d i ( s ) dW s , d > ) . (3.7)For fixed x d ∈ S F ( X ), it follows from Lemma 3.1 and (3.3) that there exists a sequence (cid:8) f d (cid:9) ⊂ S F ( f ) suchthat Z t n X i =1 ∂φ k ∂x i ( s, x d ) f i ( s ) dW s = cl L (Z t n X i =1 ∂φ k ∂x i ( s, x d ( s )) f d i ( s ) dW s , d > ) . (3.8)Combining (3.7) and (3.8), we have Z t n X i =1 ∂φ k ∂x i ( s, X ) f i ( s ) dW s = cl L (Z t n X i =1 ∂φ k ∂x i ( s, x d ( s )) f d i ( s ) dW s , d , d > ) . (3.9)Let M = (Z t n X i =1 ∂φ k ∂x i ( s, x d ( s )) f d i ( s ) dW s , d , d > ) and N = (Z t n X i =1 ∂φ k ∂x i ( s, x d d ( s )) f d i ( s ) dW s , d , d > ) . Since x d ∈ S F ( X ) = S F ( cl L A ), we can choose d = d d and so cl L N ⊂ cl L M . On the other hand,for any a ∈ cl L M , there exists a sequence { a q } ⊂ M such that a q → a . Since a q ∈ M , there exist x q ∈ (cid:8) x d (cid:9) ∞ d =1 ⊂ S F ( cl L A ) and f q ∈ (cid:8) f d (cid:9) ∞ d =1 ⊂ S F ( f ) such that a q = Z t n X i =1 ∂φ k ∂x i ( s, x q ( s ))( f q ) i ( s ) dW s . For fixed x q ∈ S F ( cl L A ), there exists a sequence { x ql } ⊂ A such that x ql → x q . Letting f ql = f q , we havelim ql → + ∞ Z t n X i =1 ∂φ k ∂x i ( s, x ql ( s ))( f ql ) i ( s ) dW s = a , which implies cl L M ⊂ cl L N . Thus, we have cl L M = cl L N and so Z t n X i =1 ∂φ k ∂x i ( s, X ) f i ( s ) dW s = cl L (Z t n X i =1 ∂φ k ∂x i ( s, x d d ( s )) f d i ( s ) dW s , d , d > ) . (3.10)Similarly, we can obtain the following equality Z t " ∂φ k ∂t ( s, X ( s )) + n X i =1 ∂φ k ∂x i ( s, X ( s )) g i ( s ) + 12 n X i =1 ∂ φ k ∂x i ( s, X ( s )) f i ds = cl L (Z t " ∂φ k ∂t ( s, x d d ) + n X i =1 ∂φ k ∂x i ( s, x d d )( g d ) i ( s ) + 12 n X i =1 ∂ φ k ∂x i ( s, x d d )( f d i ) ds, d , d > ) . (3.11)11ombining (3.5), (3.10), (3.11), we have φ ( t, X ( t )) k = φ (0 , X (0)) k + Z t n X i ∂φ k ∂x i ( s, X ( s )) f i ( s ) dW s + Z t " ∂φ k ∂t ( s, X ( s )) + n X i ∂φ k ∂x i ( s, X ( s )) g i ( s ) + 12 n X i =1 ∂ φ k ∂x i ( s, X ( s )) f i ds P - a.s. for k = 1 , · · · , n . This completes the proof. Remark 3.3.
Theorem 3.1 is a set-valued version of the classical Itˆo’s formula.
Theorem 3.2.
Let X ( t ) = x T + R Tt f ( s, Z ( s )) ds ⊖ R Tt Z ( s ) dW s and Z ∈ G . Then X k + Z Tt Z k ( s ) ds ⊂ ( x T ) k + 2 Z Tt X k f k ( s, Z ( s )) ds − Z Tt X k Z k ( s ) dW s , k = 1 , · · · , n P - a.s.. Proof.
From Lemma 3.1 and the property of Hukuhara difference, there exist (cid:8) z d (cid:9) ∈ S F ( Z ) such that Z Tt f ( s, Z ) ds ⊖ Z Tt Z s dW s ⊂ Z Tt f ( s, Z ( s )) ds − Z Tt Z ( s ) dW s = cl L (Z Tt f ( s, z d ) ds − Z Tt z d dW s : d > ) . Let X ∗ ( t ) = x T + Z Tt f ( s, Z ( s )) ds − Z Tt Z ( s ) dW s ,x d = x T + Z Tt f ( s, z d ) ds − Z Tt z d dW s . Then cl L (cid:8) x d : d > (cid:9) = X ∗ and x d ( t ) is a single-valued Itˆo’s process. Let φ ( t, x ) = ( x , x , · · · , x n ). Bythe classical Itˆo’s formula, we have( x d k ) = ( x T ) k + 2 Z Tt x d k f k ( s, z d ) − ( z d k ) ( s ) ds − Z Tt x d z d k ( s ) dW s . Similar to the proof of Theorem 3.1, we can obtain X ∗ k ( t ) = cl L n ( x d k ) : d > o =( x T ) k + cl L ( Z Tt x d k f k ( s, z d ) − ( z d k ) ( s ) ds : d > ) + cl L ( − Z Tt x d z d k ( s ) dW s : d > ) and so X k ( t ) ⊂ X ∗ k = x T + 2 Z Tt X k f k ( s, Z ( s )) − Z k ( s ) ds − Z Tt X k Z k ( s ) dW s , k = 1 , · · · , n P - a.s.. For any given y ∈ X k + R Tt Z k ( s ) ds , there exist m ∈ X k and n ∈ R Tt Z k ( s ) ds such that y = m + n . Since m ∈ X k ⊂ cl L n ( x d k ) : d > o and n ∈ R Tt Z k ( s ) ds = cl L nR Tt ( z d k ) dW s : d > o , we know that thereexist { m j } ⊂ n ( x d k ) : d > o and { n j } ⊂ nR Tt ( z d k ) dW s : d > o such that m j + n j → m + n . It followsLemma 3.2 and Remark 3.1 that m + n ∈ cl L ( ( x T ) k + 2 Z Tt x d k f k ( s, z d ) ds − Z Tt x d z d k ( s ) dW s : d > ) = ( x T ) k + cl L ( Z Tt x d k f k ( s, z d ) ds : d > ) + cl L ( Z Tt x d z d k ( s ) dW s : d > ) = ( x T ) k + 2 Z Tt X k f k ( s, Z ( s )) ds − Z Tt X k Z k ( s ) dW s . X k + Z Tt Z k ( s ) ds ⊂ ( x T ) k + 2 Z Tt X k f k ( s, Z ( s )) ds − Z Tt X k Z k ( s ) dW s , k = 1 , · · · , n P - a.s. and so the proof is completed.Similar as the proof of Theorem 3.2, we can get the following Corollary. Corollary 3.1.
Let X t = x T + R Tt f ( s, X s , Z ( s )) ds ⊖ R Tt Z ( s ) dW s and Z ∈ G . Then ( X t ) k + Z Tt Z k ( s ) ds ⊂ ( x T ) k + 2 Z Tt ( X s ) k f k ( s, X s , Z ( s )) ds − Z Tt ( X s ) k Z k ( s ) dW s , k = 1 , · · · , n, P - a.s.. In this section, we apply the set-valued Itˆo’s formula to obtain the existence and uniqueness of solutionsfor GSVBSDE (1.1). To this end, we first show the following lemma, which is a set-valued version of the Itˆoisometry.
Lemma 4.1.
Suppose f ∈ L ad ([0 , T ] × Ω , P ( R m )) . Then the integral I ( f ) = R T f ( t ) dW ( t ) is a set-valuedrandom variable with E [ I ( f )] = 0 and E [ I ( t )] = E "Z T f ( t ) dt . (4.1) Proof.
By Lemma 3.1, there exists a sequence (cid:8) f d ( t ) (cid:9) ∞ d =1 ⊂ S F ( f ) such that Z T f ( s ) dW s = cl L (Z T f d ( s ) dW s : d > ) P - a.s. (4.2)and Z T f ( s ) ds = cl L (Z T ( f d ( s )) ds : d > ) P - a.s.. (4.3)It follows from (4.2) that E "Z T f ( s ) dW s = E " cl L (Z T f d ( s ) dW s : d > ) . For any given a ∈ cl L nR T f d ( s ) dW s : d > o , there exists a sequence { f n ( t ) } ∞ n =1 ⊂ (cid:8) f d ( t ) (cid:9) ∞ d =1 such that R T f n ( s ) ds → a . Thus, by Lemma 2.8, we know that E R T f n ( s ) ds = 0 → E a . This implies that E a = 0and so E [ I ( f )] = E "Z T f ( s ) dW s = E " cl L (Z T f d ( s ) dW s : d > ) = 0 . Next we show that (4.1) holds. In fact, by (4.2), we have E [ I ( f )] = E " cl L (Z T f d ( s ) dW s : d > ) P - a.s., cl L (Z T f d ( s ) dW s : d > ) = ( x : x ∈ cl L (Z T f d ( s ) dW s : d > )) . For any given b ∈ cl L nR T f d ( s ) dW s : d > o , there exists a sequence (cid:8) f k ( t ) (cid:9) ∞ k =1 ⊂ (cid:8) f d ( t ) (cid:9) ∞ d =1 such R T f k ( s ) dW s → b . It follows from Lemma 2.8 that E Z T f k ( s ) dW s ! = E "Z T ( f k ( s )) ds → E [ b ] . This leads to E [ I ( f )] = E " cl L (Z T ( f d ( s )) ds : d > ) . Thus, by (4.3), we know that (4.1) holds.Recall that, for A ∈ K ( X ) and a ∈ A , a is called an extreme point of A if it can not be written as a strictconvex combination of two points in A , that is for every y , y ∈ A and λ ∈ (0 , a = λy +(1 − λ ) y .Then, we denote ext ( A ) to be the set of all extreme points of A . Lemma 4.2. [8] Let
A, B ∈ K ( X ) . The Hukuhara differential A ⊖ B exists if and only if for every a ∈ ext ( A ) ,there exists x ∈ X such that a ∈ x + B and x + B ⊆ A . In this case, A ⊖ B is unique, closed, convex, and A ⊖ B = { x ∈ X | x + B ⊆ A } . Lemma 4.3.
Let X ( t ) , Y ( t ) ∈ L ad ([0 , T ] × Ω , L ( R n × m )) such that Z t X ( s ) dW s = Z t Y ( s ) dW s , ∀ t ∈ [0 , T ] . (4.4) Then X ( t ) ⊖ Y ( t ) exists and X ( t ) = Y ( t ) P -a.s..Proof. Assume that X ( t ) ⊖ Y ( t ) does not exist. Then Lemma 4.2 implies that there exists a ( t ) ∈ ext ( X ( t ))such that, for any x ( t ) ∈ L ad ([0 , T ] × Ω , R n × m ), a ( t ) / ∈ x ( t ) + Y ( t ) or x ( t ) + Y ( t ) * X ( t ). Taking x ( t ) = 0,one has a ( t ) / ∈ Y ( t ) or Y ( t ) * X ( t ). Suppose that a ( t ) / ∈ Y ( t ) holds. Then it follows from a ( t ) ∈ ext ( X ( t ))and (4.4) that Z t a ( s ) dW s ∈ Z t X ( s ) dW s = Z t Y ( s ) dW s , ∀ t ∈ [0 , T ] . From Lemma 3.1, there exists a sequence (cid:8) y d ( t ) (cid:9) ∞ d =1 ⊂ S F ( Y ( t )) such that Z T Y ( s ) dW s = cl L (Z T y d ( s ) dW s : d > ) P - a.s. (4.5)This leads that there exists a sequence { y n ( t ) } ∞ n =1 ⊂ (cid:8) y d ( t ) (cid:9) ∞ d =1 such that R t ( y n − a ) dW s →
0. It followsfrom Lemma 2.8 that E [ R T ( y n − a ) ds ] = k y n ( s ) − a k c →
0, which implies y n ( s ) → a P -a.s.. Since S F ( Y ( t )) is closed, we have a ∈ S F ( Y ( t )) ⊂ Y ( t ). This in contradiction with a ( t ) / ∈ Y ( t ). Similarly, when Y ( t ) * X ( t ), we can obtain a contradiction. Thus, we know that X ( t ) ⊖ Y ( t ) exists and so (4.4) implies that Z t X ( s ) ⊖ Y ( s ) dW s = 0 , ∀ t ∈ [0 , T ] . By Lemma 4.1, one has E (cid:20)Z t ( X ( s ) ⊖ Y ( s )) ds (cid:21) = 0 , ∀ t ∈ [0 , T ] . X ( t ) , Y ( t ) ∈ L ad ([0 , T ] × Ω , L ( R n × m )), let x ( t ) ∈ X ( t ) ⊖ Y ( t ) such that x ( t ) = k X ( t ) ⊖ Y ( t ) k ,by taking t = T , we have E "Z T k X ( s ) ⊖ Y ( s ) k ds = E "Z T x ( t ) ds ∈ E "Z T ( X ( s ) ⊖ Y ( s )) ds = 0 , which implies that E "Z T k X ( s ) ⊖ Y ( s ) k ds = k X ( s ) ⊖ Y ( s ) k s = 0and so X ( t ) = Y ( t ) P -a.s..We also need the following lemma. Lemma 4.4. [8] Let ξ ∈ L T (Ω , K ( R n )) and f ( t, ω ) ∈ L ad ([0 , T ] × Ω , K ( R n )) . Then there exists a uniquepair ( X, Z ) ∈ L ad ([0 , T ] × Ω , K ( R n )) × G such that X ( t ) = ξ + Z Tt f ( s, ω ) ds ⊖ Z Tt Z ( s ) dW s , P - a.s. t ∈ [0 .T ] . We first consider the following set-valued equation X ( t ) + Z Tt Z ( s ) dW s = ξ + Z Tt f ( s, Z ( s )) ds, P - a.s. t ∈ [0 .T ] , where ξ ∈ L T (Ω , K ( R n )), Z t ∈ G and f : [0 , T ] × Ω × P ( R n × m ) → K ( R n ). Assumption 4.1.
Assume that f : [0 , T ] × Ω × P ( R n × m ) → K ( R n ) satisfies the following conditions:( i ) for any given A ∈ P ( R n × m ) , f ( · , · , A ) ∈ L ad ([0 , T ] × Ω , K ( R n )) ;( ii ) for any fixed ( t, ω ) ∈ [0 , T ] × Ω and A, B ∈ P ( R n × m ) , the Hukuhara difference f ( t, ω, A ) ⊖ f ( t, ω, B ) exists whenever A ⊖ B exists;( iii ) for any given A, B ∈ P ( R n × m ) with A ⊖ B existing, there is a constant c > such that k f ( t, ω, A ) ⊖ f ( t, ω, B ) k c k A ⊖ B k , ∀ ( t, ω ) ∈ [0 , T ] × Ω . Theorem 4.1.
Let ξ ∈ L T (Ω , K ( R n )) and f ( t, ω, Z ( t )) satisfy Assumption 4.1. Then there exists a pair ( X, Z ) ∈ L ad ([0 , T ] × Ω , K ( R n )) × G such that X ( t ) + Z Tt Z ( s ) dW s = ξ + Z Tt f ( s, ω, Z ( s )) ds, ∀ t ∈ [0 , T ] , P - a.s.. (4.6) Moreover, if ( X ( t ) , Z ( t )) and ( X ( t ) , Z ( t )) are two solutions of (4.6) and Z ( t ) ⊖ Z ( t ) exists, then X ( t ) = X ( t ) and Z ( t ) = Z ( t ) P -a.s..Proof. We first show the existence of solutions for (4.6). Let X = { } and Z = { } . By Assumption4.1(i), we know that f ( t, Z ) ∈ L ad ([0 , T ] × Ω , K ( R n )). It follows from Lemma 4.4 that there is a pair( X ( t ) , Z ( t )) ∈ L ad ([0 , T ] × Ω , K ( R n )) × G such that X ( t ) + Z Tt Z ( s ) dW s = ξ + Z Tt f ( s, ω, Z ( s )) ds, ∀ t ∈ [0 , T ] , P - a.s.. In the same way, we can obtain a sequence { ( X p ( t ) , Z p ( t )) } ⊂ L ad ([0 , T ] × Ω , K ( R n )) × G such that X p ( t ) + Z Tt Z p ( s ) dW s = ξ + Z Tt f ( s, ω, Z p − ( s )) ds, ∀ t ∈ [0 , T ] , P - a.s., p = 1 , , · · · . (4.7)15ow we conclude that for each t ∈ [0 , T ], X p ( t ) ⊖ X p − ( t ) and Z p ( t ) ⊖ Z p − ( t ) exist P -a.s.. In fact,for p = 1, it is clear that X ( t ) ⊖ X and Z ( t ) ⊖ Z exist P -a.s. since X = { } and Z = { } . Assumethat the assertion is true for p − p > X p − ( t ) ⊖ X p − ( t ) and Z p − ( t ) ⊖ Z p − ( t ) exist P -a.s.. Itfollows from Assumption 4.1 (ii) that f ( t, ω, Z p − ( t )) ⊖ f ( t, ω, Z p − ( t )) exists for all t ∈ [0 , T ] and P -a.s..Since f ( t, ω, Z p − ( t )) , f ( t, ω, Z p − ( t )) ∈ L ad ([0 , T ] × Ω , K ( R n )), Lemma 2.10 shows that f ( t, ω, Z p − ( t )) ⊖ f ( t, ω, Z p − ( t )) ∈ L ad ([0 , T ] × Ω , K ( R n )) and Z Tt f ( s, ω, Z p − ( s )) ds ⊖ Z Tt f ( s, ω, Z p − ( s )) ds = Z Tt (cid:2) f ( s, ω, Z p − ( s )) ⊖ f ( s, ω, Z p − ( s )) (cid:3) ds, ∀ t ∈ [0 , T ] , P - a.s.. For fixed t ∈ [0 , T ], we know that R Tt f ( s, ω, Z p − ( s )) ds, R Tt f ( s, ω, Z p − ( s )) ds ∈ L T (Ω , L ( R n )) and soLemma 2.13 implies that E "Z Tt f ( s, ω, Z p − ( s )) ds (cid:12)(cid:12)(cid:12)(cid:12) F t ⊖ E "Z Tt f ( s, ω, Z p − ( s )) ds (cid:12)(cid:12)(cid:12)(cid:12) F t = E "Z Tt f ( s, ω, Z p − ( s )) ⊖ f ( s, ω, Z n − ( s )) ds (cid:12)(cid:12)(cid:12)(cid:12) F t . (4.8)By (4.7), we have E " X p ( t ) + Z Tt Z p ( s ) dW s (cid:12)(cid:12)(cid:12)(cid:12) F t = E " ξ + Z Tt f ( s, ω, Z p − ( s )) ds (cid:12)(cid:12)(cid:12)(cid:12) F t . It follows from Definition 2.5 and Lemma 4.1 that, p = 1 , , · · · , X p ( t ) = X p ( t ) + E "Z Tt Z p ( s ) dW s = E " ξ + Z Tt f ( s, ω, Z p − ( s )) ds (cid:12)(cid:12)(cid:12)(cid:12) F t , ∀ t ∈ [0 , T ] , P - a.s.. (4.9)Now from (4.8) and (4.9), one has X p ( t ) ⊖ X p − ( t ) = E "Z Tt f ( s, ω, Z p − ( s )) ds (cid:12)(cid:12)(cid:12)(cid:12) F t ⊖ E "Z Tt f ( s, ω, Z p − ( s )) ds (cid:12)(cid:12)(cid:12)(cid:12) F t , ∀ t ∈ [0 , T ] , P - a.s. with p = 1 , , · · · . This means X p ⊖ X p − exists P -a.s.. Next we show that Z p ⊖ Z p − exists P -a.s.. To thisend, let M p ( t ) = E " ξ + Z T f ( s, ω, Z p − ( s )) ds (cid:12)(cid:12)(cid:12)(cid:12) F t , ∀ t ∈ [0 , T ] , p = 1 , , · · · . Then, by Lemma 2.1 (ii) and Lemma 2.13, we have M p ( t ) ⊖ M p − ( t ) = E "Z T [ f ( s, ω, Z p − ( s )) ⊖ f ( s, ω, Z p − ( s ))] ds (cid:12)(cid:12)(cid:12)(cid:12) F t , ∀ t ∈ [0 , T ] , p = 1 , , · · · . By (4.7) with t = 0, one has E " X p (0) + Z T Z p ( s ) dW s (cid:12)(cid:12)(cid:12)(cid:12) F t = E " ξ + Z T f ( s, ω, Z p − ( s )) ds (cid:12)(cid:12)(cid:12)(cid:12) F t and so Definition 2.5 shows that X p (0) + Z t Z p ( s ) dW s = E " ξ + Z T f ( s, ω, Z p − ( s )) ds (cid:12)(cid:12)(cid:12)(cid:12) F t = M n ( t ) . (4.10)16ince M p ( t ) ⊖ M p − ( t ) is a uniformly square-integrable set-valued martingale, it follows from Lemma 2.12that there exists ˜ Z p ( t ) ∈ G such that M p ( t ) ⊖ M p − ( t ) = M p (0) ⊖ M p − (0) + Z t ˜ Z p ( s ) dW s = X p (0) ⊖ X p − (0) + Z t ˜ Z p ( s ) dW s . Moreover, by (4.10) and Lemma 2.11, one has X p (0) + Z t Z p ( s ) dW s = M p − ( t ) + M p ( t ) ⊖ M p − ( t )= X p − (0) + Z t Z p − ( s ) dW s + X n (0) ⊖ X p − (0) + Z t ˜ Z p ( s ) dW s = X p − (0) + X p (0) ⊖ X p − (0) + Z t [ Z p − ( s ) + ˜ Z p ( s )] dW s , ∀ t ∈ [0 , T ] , P - a.s. with p = 1 , , · · · . From Lemma 4.3, this yields that Z p ( t ) = Z p − ( t ) + ˜ Z p ( t ) ∈ G and so Z p ( t ) ⊖ Z p − ( t ) =˜ Z p ( t ) exists P -a.s.. Thus, by the induction, we know that for each t ∈ [0 , T ], X p ( t ) ⊖ X p − ( t ) and Z p ( t ) ⊖ Z p − ( t ) exist P -a.s..Next we show that there is a pair ( X ( t ) , Z ( t )) ∈ L ad ([0 , T ] × Ω , K ( R n )) × G such that k X ( s ) ⊖ X p ( s ) k s → , k Z ( s ) ⊖ Z p ( s ) k s → . Indeed, it follows from (4.7) that X p +1 ( t ) ⊖ X p ( t ) + Z Tt Z p +1 ( s ) ⊖ Z p ( s ) dW s = Z Tt f ( s, ω, Z p ( s )) ⊖ f ( s, ω, Z p − ( s )) ds, ∀ t ∈ [0 , T ] , P - a.s., p = 1 , , · · · , (4.11)which is equal to X p +1 i ( t ) ⊖ X pi ( t ) + Z Tt Z p +1 i ( s ) ⊖ Z pi ( s ) dW s = Z Tt f i ( s, Z p ( s )) ⊖ f i ( s, Z p − ( s )) ds, t ∈ [0 , T ] , i = 1 , · · · , n, where X p +1 i ( t ) ⊖ X pi ( t ) , f i ( t, Z p ( t )) ⊖ f i ( t, Z p − ( t )) ∈ L ad ([0 , T ] × Ω , K ( R ))and ( Z p +1 i ( t ) ⊖ Z pi ( t )) T ∈ K w ( L ad ([0 , T ] × Ω , K ( R m )))are the components of X p +1 ( t ) ⊖ X p ( t ), f ( s, Z p ( s )) ⊖ f ( t, Z p − ( t )) and Z p +1 ( t ) ⊖ Z p ( t ), respectively. Itfollows from Theorem 3.2 that( X p +1 i ( t ) ⊖ X pi ( t )) + Z Tt ( Z p +1 i ( s ) ⊖ Z pi ( s )) ds ⊂ Z Tt ( X p +1 i ( s ) ⊖ X pi ( s ))( f i ( s, Z p ( s )) ⊖ f i ( s, Z p − ( s ))) ds − Z Tt ( X p +1 i ( s ) ⊖ X pi ( s ))( Z p +1 i ( s ) ⊖ Z pi ( s )) dW s ,i = 1 , · · · , n. From Lemma 4.1, Assumption 4.1 and the basic inequality ρ a + ρb > ab with ρ = 2 c , we have E k X p +1 ( t ) ⊖ X p ( t ) k + E Z Tt k Z p +1 ( s ) ⊖ Z p ( s ) k ds E Z Tt k Z p ( s ) ⊖ Z p − ( s ) k ds + 2 c E Z Tt k X p +1 ( s ) ⊖ X p ( s ) k ds. (4.12)17enote u p ( t ) = E Z Tt k X p ( s ) ⊖ X p − ( s ) k ds, v p ( t ) = E Z Tt k Z p ( s ) ⊖ Z p − ( s ) k ds, p = 1 , , · · · . Then (4.12) leads to − ddt (cid:16) u p +1 ( t ) e c t (cid:17) + e c t v p +1 ( t ) e c t v p ( t ) . (4.13)Integrating from t to T for two sides of (4.13), we obtain u p +1 ( t ) + Z Tt e c ( s − t ) v p +1 ( s ) ds Z Tt e c ( s − t ) v p ( s ) ds. (4.14)Noting u p +1 ( t ) , v p +1 ( t ) >
0, it follows from (4.14) that Z Tt e c ( s − t ) v p +1 ( s ) ds Z Tt e c ( s − t ) v p ( s ) ds. Iterating the inequality and taking t = 0 in above inequality, we have Z T e c t v p +1 ( t ) dt − p ¯ ce c , where ¯ c = sup t T v ( t ) = E Z T k Z ( t ) k dt. Moreover, it follows from (4.14) that u p +1 ( t ) Z Tt e c ( s − t ) v p ( s ) ds and so u p +1 (0) − p ¯ ce c . (4.15)However, from (4.13), (4.15) and the fact that ddt u p +1 ( t )
0, we have v p +1 (0) c u p +1 (0) + 12 v p (0) − p +1 ¯ cc e c + 12 v p (0) . It is easy to check that v p +1 (0) − p (cid:16) p ¯ c c e c + v (0) (cid:17) . (4.16)For any q > p , from Lemma 2.2, we know that X q ( t ) ⊖ X p ( t ) and Z q ( t ) ⊖ Z p ( t ) exist and X q ( t ) ⊖ X p ( t ) = X q ( t ) ⊖ X q − ( t ) + X q − ( t ) ⊖ X q − ( t ) + · · · + X p +1 ( t ) ⊖ X p ( t ) ,Z q ( t ) ⊖ Z q ( t ) = Z q ( t ) ⊖ Z q − ( t ) + Z q − ( t ) ⊖ Z q − ( t ) + · · · + Z p +1 ( t ) ⊖ Z p ( t ) . It follows from (4.15), (4.16) and the triangle inequality that k X q ( t ) ⊖ X p ( t ) k s ( q − p )2 − p ¯ ce c and k Z q ( t ) ⊖ Z p ( t ) k s ( q − p )2 − p (cid:16) p ¯ c c e c + v (0) (cid:17) . X, Z ) ∈ L ad ([0 , T ] × Ω , K ( R n )) × G suchthat k X ( t ) ⊖ X p ( t ) k s → , k Z ( t ) ⊖ Z p ( t ) k s → . Now it follows from (4.7) that X ( t ) + Z Tt Z ( s ) dW s = ξ + Z Tt f ( s, ω, Z ( s )) ds, ∀ t ∈ [0 , T ] , P - a.s.. Next we show the uniqueness. Assume that ( X ( t ) , Z ( t )) and ( X ( t ) , Z ( t )) are two solutions of (4.6)and Z ( t ) ⊖ Z ( t ) exists. Then X ( t ) + Z Tt Z ( s ) dW s = ξ + Z Tt f ( s, Z ( s )) ds, X ( t ) + Z Tt Z ( s ) dW s = ξ + Z Tt f ( s, Z ( s )) ds. It follows from Assumption 4.1 (ii) that f ( s, Z ( s )) ⊖ f ( s, Z ( s )) exists. By Lemma 2.10 and Lemma 2.13,we have X ( t ) ⊖ X ( t ) = E " ξ + Z Tt f ( s, Z ( s )) ds (cid:12)(cid:12)(cid:12)(cid:12) F t ⊖ E " ξ + Z Tt f ( s, Z ( s )) ds (cid:12)(cid:12)(cid:12)(cid:12) F t = E "Z Tt f ( s, Z ( s )) ⊖ f ( s, Z ( s )) ds (cid:12)(cid:12)(cid:12)(cid:12) F t , which implies that X ( t ) ⊖ X ( t ) exists. This yields that X ( t ) ⊖ X ( t ) + Z Tt Z ( s ) ⊖ Z ( s ) dW s = Z Tt f ( s, Z ( s )) ⊖ f ( s, Z ( s )) ds, which is equal to X i ( t ) ⊖ X i ( t ) + Z Tt Z i ( s ) ⊖ Z i ( s ) dW s = Z Tt f i ( s, Z ( s )) ⊖ f i ( s, Z ( s )) ds, i = 1 , · · · , n, where X i ( t ) ⊖ X i ( t ) , f i ( t, Z ( t )) ⊖ f i ( t, Z ( t )) ∈ L ad ([0 , T ] × Ω , K ( R ), ( Z i ( t ) ⊖ Z i ( t )) T ∈ K w ( L ad ([0 , T ] × Ω , K ( R m ))) are the component of X ( t ) ⊖ X ( t ) , f ( s, Z ( t )) ⊖ f ( s, Z ( t )) and Z ( t ) ⊖ Z ( t ) respectively. Itfollows from Corollary 3.2 that E k X ( t ) ⊖ X ( t ) k + E Z Tt k Z ( s ) ⊖ Z ( s ) k ds E Z Tt k X ( s ) ⊖ X ( s ) kk f i ( s, Z ( s )) ⊖ f i ( s, Z ( s )) k ds E Z Tt k Z ( s ) ⊖ Z ( s ) k ds + 2 c E Z Tt k X ( s ) ⊖ X ( s ) k ds. (4.17)Denote u ( t ) = E Z Tt k X ( s ) ⊖ X ( s ) k ds, v ( t ) = E Z Tt k Z ( s ) ⊖ Z ( s ) k ds. Then (4.17) leads to − ddt (cid:16) u ( t ) e c t (cid:17) + e c t v ( t ) e c t v ( t ) . (4.18)Integrating from t to T for two sides of (4.18), we obtain u ( t ) + Z Tt e c ( s − t ) v ( s ) ds Z Tt e c ( s − t ) v ( s ) ds. This implies that u ( t ) Z Tt e c ( s − t ) v ( s ) ds, v ( t ) v ( t ) . Thus, v (0) = 0 and u (0) = 0. 19e now consider the following set-valued equation X ( t ) + Z Tt Z ( s ) dW s = ξ + Z Tt f ( s, X ( s ) , Z ( s )) ds, P - a.s. t ∈ [0 .T ] . where ξ ∈ L T (Ω , K ( R n )), Z t ∈ G and f : [0 , T ] × Ω × K ( R n ) × P ( R n × m ) → K ( R n ). Assumption 4.2.
Assume f : [0 , T ] × Ω × K ( R n ) × P ( R n × m ) → K ( R n ) satisfies the following conditions:( i ) for any given A ∈ K ( R n ) and B ∈ P ( R n × m ) , f ( · , · , A, B ) ∈ L ad ([0 , T ] × Ω , K ( R n )) ;( ii ) for any fixed ( t, ω ) ∈ [0 , T ] × Ω , A, B ∈ K ( R n ) and C, D ∈ P ( R n × m ) , the Hukuhara difference f ( t, ω, A, C ) ⊖ f ( t, ω, B, C ) exists whenever A ⊖ B exists and the Hukuhara difference f ( t, ω, B, C ) ⊖ f ( t, ω, B, D ) exists whenever C ⊖ D exists;( iii ) there exists a constant c > such that, for any A, B ∈ K ( R n ) and C, D ∈ P ( R n × m ) with A ⊖ B and C ⊖ D existing, k f ( t, ω, A, C ) ⊖ f ( t, ω, B, D ) k c ( k A ⊖ B k + k C ⊖ D k ) , ∀ t ∈ [0 , T ] . Theorem 4.2.
Let ξ ∈ L T (Ω , K ( R n )) and f satisfy the Assumption 4.2. Then, there exists a pair ( X, Z ) ∈L ad ([0 , T ] × Ω , K ( R n )) × G such that X ( t ) + Z Tt Z ( s ) dW s = ξ + Z Tt f ( s, X ( s ) , Z ( s )) ds, P - a.s. t ∈ [0 .T ] . (4.19) Moreover, if ( X ( t ) , Z ( t )) and ( X ( t ) , Z ( t )) are two solutions to (4.19) with X ( t ) ⊖ X ( t ) and Z ( t ) ⊖ Z ( t ) existing, then X ( t ) = X ( t ) , Z ( t ) = Z ( t ) P -a.s..Proof. We first show the existence of solutions for (4.19). For any given X = { } and Z = { } , by Lemma4.4, we can obtain a sequence { ( X p ( t ) , Z p ( t )) } satisfying X p ( t ) + Z Tt Z p ( s ) dW s = ξ + Z Tt f ( s, X p − ( s ) , Z p − ( s )) ds, t ∈ [0 , T ] , p = 1 , , · · · . (4.20)We conclude that X p ( t ) ⊖ X p − ( t ), Z p ( t ) ⊖ Z p − ( t ), f ( t, X p ( t ) , Z p ( t )) ⊖ f ( t, X p − ( t ) , Z p − ( t )) exist. In fact,for p = 1, it is trivial since X = { } and Z = { } . Assume that the assertion is true for p − p > X p − ( t ) ⊖ X p − ( t ) and Z p − ( t ) ⊖ Z p − ( t ) exist. From Assumption 4.2 (ii), f ( t, X p − ( t ) , Z p − ( t )) ⊖ f ( t, X p − ( t ) , Z p − ( t )) and f ( t, X p − ( t ) , Z p − ( t )) ⊖ f ( t, X p − ( t ) , Z p − ( t )) exist for any t ∈ [0 , T ]. It followsfrom Lemma 2.2 that f ( t, X p − ( t ) , Z p − ( t )) ⊖ f ( t, X p − ( t ) , Z p − ( t )) exists and f ( t, X p − ( t ) , Z p − ( t )) ⊖ f ( t, X p − ( t ) , Z p − ( t ))= f ( t, X p − ( t ) , Z p − ( t )) ⊖ f ( t, X p − ( t ) , Z p − ( t )) + f ( t, X p − ( t ) , Z p − ( t )) ⊖ f ( t, X p − ( t ) , Z p − ( t )) . Thus, Lemma 2.13 yields that E "Z Tt f ( s, X p − ( s ) , Z p − ( s )) ⊖ f ( s, X p − ( s ) , Z p − ( s )) ds (cid:12)(cid:12)(cid:12)(cid:12) F t = E "Z Tt f ( s, X p − ( s ) , Z p − ( s )) (cid:12)(cid:12)(cid:12)(cid:12) F t ⊖ E "Z Tt f ( s, X p − ( s ) , Z p − ( s )) ds (cid:12)(cid:12)(cid:12)(cid:12) F t . (4.21)By (4.20), one has E " X p ( t ) + Z Tt Z p ( s ) dW s (cid:12)(cid:12)(cid:12)(cid:12) F t = E " ξ + Z Tt f ( s, ω, X p − ( s ) , Z p − ( s )) ds (cid:12)(cid:12)(cid:12)(cid:12) F t .
20t follows from Definition 2.5 and Lemma 4.1 that X p ( t ) = X p ( t ) + E "Z Tt Z p ( s ) dW s = E " ξ + Z Tt f ( s, ω, X p − ( s ) , Z p − ( s )) ds (cid:12)(cid:12)(cid:12)(cid:12) F t , ∀ t ∈ [0 , T ] , P - a.s., p = 1 , , · · · . (4.22)By (4.21) and (4.22), X n ( t ) ⊖ X n − ( t ) exists and X p ( t ) ⊖ X p − ( t ) = E " ξ + Z Tt f ( s, X p − ( s ) , Z p − ( s )) ds (cid:12)(cid:12)(cid:12)(cid:12) F t ⊖ E " ξ + Z Tt f ( s, X p − ( s ) , Z p − ( s )) ds (cid:12)(cid:12)(cid:12)(cid:12) F t . Let M p ( t ) = E " ξ + Z T f ( s, X p − ( s ) , Z p − ( s )) ds (cid:12)(cid:12)(cid:12)(cid:12) F t . Then by the same argument above, we can shows that M p ( t ) ⊖ M p − ( t ) exists and M p ( t ) ⊖ M p − ( t ) = E "Z T f ( s, X p − ( s ) , Z p − ( s )) ⊖ f ( s, X p − ( s ) , Z p − ( s )) ds (cid:12)(cid:12)(cid:12)(cid:12) F t . By (4.20) with t = 0, we have E " X p (0) + Z T Z p ( s ) dW s (cid:12)(cid:12)(cid:12)(cid:12) F t = E " ξ + Z T f ( s, ω, X p − ( s ) , Z p − ( s )) ds (cid:12)(cid:12)(cid:12)(cid:12) F t . It follows from Definition 2.5 that M p ( t ) = X p (0) + Z t Z p ( s ) dW s = E " ξ + Z T f ( s, ω, X p − ( s ) , Z p − ( s )) ds (cid:12)(cid:12)(cid:12)(cid:12) F t . (4.23)Noting that M p ( t ) ⊖ M p − ( t ) is a uniformly square-integrable set-valued martingale, by Lemma 2.12, thereexists ˜ Z p ( t ) ∈ G such that M p ( t ) ⊖ M p − ( t ) = M p (0) ⊖ M p − (0) + Z T ˜ Z p ( s ) dW s = X p (0) ⊖ X p − (0) + Z T ˜ Z p ( s ) dW s . Moreover, by (4.23) and Lemma 2.11, one has X p (0) + Z t Z p ( s ) dW s = M p − ( t ) + M p ( t ) ⊖ M p − ( t )= X p − (0) + X p (0) ⊖ X p − (0) + Z t Z p − ( s ) dW s + Z t ˜ Z p ( s ) dW s = X p − (0) + X p (0) ⊖ X p − (0) + Z t [ Z p − ( s ) + ˜ Z p ( s )] dW s , ∀ t ∈ [0 , T ] , P - a.s. with p = 1 , , · · · . From Lemma 4.3, this implies Z p ( t ) = Z p − ( t )+ ˜ Z p ( t ) ∈ G and so Z p ( t ) ⊖ Z p − ( t ) = ˜ Z p ( t )exists. Thus, it follows from (4.22) that X p +1 ( t ) ⊖ X p ( t ) + Z Tt Z p +1 ( s ) ⊖ Z p ( s ) dW s = Z Tt f ( s, X p +1 ( s ) , Z p +1 ( s )) ⊖ f ( s, X p ( s ) , Z p ( s )) ds, t ∈ [0 , T ] , which is equal to X p +1 i ( t ) ⊖ X pi ( t )+ Z Tt Z p +1 i ( s ) ⊖ Z pi ( s ) dW s = Z Tt f i ( s, X p +1 ( s ) , Z p +1 ( s )) ⊖ f i ( s, X p ( s ) , Z p ( s )) ds, t ∈ [0 , T ] , X p +1 i ( t ) ⊖ X pi ( t ) , f i ( t, X p ( t ) , Z n +1 ( t )) ⊖ f i ( t, X p − ( t ) , Z p ( t )) ∈ L ad ([0 , T ] × Ω , K ( R ))and ( Z p +1 i ( t ) ⊖ Z pi ( t )) T ∈ K w ( L ad ([0 , T ] × Ω , K ( R m )))are the components of X p +1 ( t ) ⊖ X p ( t ), f ( t, X p ( t ) , Z p +1 ( t )) ⊖ f ( t, X p − ( t ) , Z p ( t )) and Z p +1 ( t ) ⊖ Z p ( t ),respectively, i = 1 · · · , n .Similar to the proof of Theorem 4.1, from Corollary 3.1, Lemma 4.1 and Assumption 4.2 (ii), we have E k X p +1 ( t ) ⊖ X p ( t ) k + E Z Tt k Z p +1 ( s ) ⊖ Z p ( s ) k ds c E Z Tt k X p +1 ( s ) ⊖ X p ( s ) k ds + 12 E Z Tt k X p ( s ) ⊖ X p − ( s ) k ds + 12 E Z Tt k Z p ( s ) ⊖ Z p − ( s ) k ds, i = 1 , · · · , n. Denote u p ( t ) = E Z Tt k X p ( s ) ⊖ X p − ( s ) k ds, v p ( t ) = E Z Tt k Z p ( s ) ⊖ Z p − ( s ) k ds, p = 1 , , · · · . Then it follows that − d ( u p +1 ( t ) e c t ) dt + e c t v p +1 ( t ) e c t ( u p ( t ) + v p ( t )) , ∀ t ∈ [0 , T ] , u n +1 ( T ) = 0 , p = 1 , , · · · . Integrating from t to T for the two sides of the above inequality, one has u p +1 ( t ) + Z Tt e c ( s − t ) v p +1 ( s ) ds Z Tt e c ( s − t ) u p ( s ) ds + 12 Z Tt e c ( s − t ) v p ( s ) ds, which implies that u p +1 ( t ) Z Tt e c ( s − t ) u p ( s ) ds + 12 Z Tt e c ( s − t ) v p ( s ) ds (4.24)and Z Tt e c ( s − t ) v p +1 ( s ) ds Z Tt e c ( s − t ) u p ( s ) ds + 12 Z Tt e c ( s − t ) v p ( s ) ds. (4.25)Let c = Z T k Z k ds = sup t T v ( t ) , c = Z T k X k ds = sup t T u ( t ) . Iterating the inequalities (4.24) and (4.25), and taking t = 0, we have u p +1 (0) T ( c + c )2 p − p X k =1 ( e c T ) k k ! , Z T e c s v p +1 ( s ) ds T ( c + c )2 p − p X k =1 ( e c T ) k k ! . Thus, u p (0) → v p (0) →
0. For q > p , from Lemma 2.2, X q ( t ) ⊖ X p ( t ) and Z q ( t ) ⊖ Z p ( t ) exist and X q ( t ) ⊖ X p ( t ) = X q ( t ) ⊖ X q − ( t ) + X q − ( t ) ⊖ X q − ( t ) + · · · + X p +1 ( t ) ⊖ X p ( t ) ,Z q ( t ) ⊖ Z p ( t ) = Z q ( t ) ⊖ Z q − ( t ) + Z q − ( t ) ⊖ Z q − ( t ) + · · · + Z p +1 ( t ) ⊖ Z p ( t ) . From (4.24), (4.25) and the triangle inequality, we have k X q ( t ) ⊖ X p ( t ) k s ( q − p ) T ( c + c )2 p − p X k =1 ( e c T ) k k !22nd k Z q ( t ) ⊖ Z p ( t ) k s ( q − p ) ( c + c )2 p − p X k =1 ( e c T ) k k !Thus, Theorem 2.7 and Remark 2.1 show that there exists a pair ( X, Z ) ∈ L ad ([0 , T ] × Ω , K ( R n )) × G suchthat k X ( t ) ⊖ X p ( t ) k s → , k Z ( t ) ⊖ Z p ( t ) k s → . Now (4.20) leads to X ( t ) + Z Tt Z ( s ) dW s = ξ + Z Tt f ( s, ω, X ( s ) , Z ( s )) ds, ∀ t ∈ [0 , T ] , P - a.s.. Next we prove the uniqueness of solutions for (4.19). Let ( X ( t ) , Z ( t )) and ( X ( t ) , Z ( t )) be twosolutions. Then X ( t ) + Z Tt Z ( s ) dW s = ξ + Z Tt f ( s, X ( s ) , Z ( s )) ds and X ( t ) + Z Tt Z ( s ) dW s = ξ + Z Tt f ( s, X ( s ) , Z ( s )) ds. Since X ( t ) ⊖ X ( t ), Z ( t ) ⊖ Z ( t ) exist, by Assumption 4.2 (ii) and Lemma 2.2, we know that f ( t, X ( t ) , Z ( t )) ⊖ f ( t, X ( t ) , Z ( t )) exists and f ( t, X ( t ) , Z ( t )) ⊖ f ( t, X ( t ) , Z ( t ))= f ( t, X ( t ) , Z ( t )) ⊖ f ( t, X ( t ) , Z ( t )) + f ( t, X ( t ) , Z ( t )) ⊖ f ( t, X ( t ) , Z ( t ))This implies that X ( t ) ⊖ X ( t ) + Z Tt Z ( s ) ⊖ Z ( s ) dW s = Z Tt f ( s, X ( s ) , Z ( s )) ⊖ f ( s, X ( s ) , Z ( s )) ds, which equal to X i ( t ) ⊖ X i ( t ) + Z Tt Z i ( s ) ⊖ Z i ( s ) dW s = Z Tt f i ( s, X ( s ) , Z ( s )) ⊖ f i ( s, X ( s ) , Z ( s )) ds, i = 1 , · · · , n, where X i ( t ) ⊖ X i ( t ) , f i ( t, X ( t ) , Z ( t )) ⊖ f i ( t, X ( t ) , Z ( t )) ∈ L ad ([0 , T ] × Ω , K ( R )and ( Z i ( t ) ⊖ Z i ( t )) T ∈ K w ( L ad ([0 , T ] × Ω , K ( R m )))are the component of X ( t ) ⊖ X ( t ) , f ( t, X ( t ) , Z ( t )) ⊖ f ( t, X ( t ) , Z ( t )) and Z ( t ) ⊖ Z ( t ), respectively. Itfollows from Corollary 3.1, Lemma 2.8 and Assumption 4.2 (ii) that E k X ( t ) ⊖ X ( t ) k + E Z Tt k Z ( s ) ⊖ Z ( s ) k ds E Z Tt k Z ( s ) ⊖ Z ( s ) k ds + (4 c + 12 ) E Z Tt k X ( s ) ⊖ X ( s ) k ds, Denote u ( t ) = E Z Tt k X ( s ) ⊖ X ( s ) k ds, v ( t ) = E Z Tt k Z ( s ) ⊖ Z ( s ) k ds. Then it follows that − d ( u ( t ) e (4 c + ) t ) dt + e c t v ( t ) e (4 c + ) t v ( t ) , ∀ t ∈ [0 , T ] . t to T for the two sides of the above inequality, we have u ( t ) + Z Tt e (4 c + )( s − t ) v ( s ) ds Z Tt e (4 c + )( s − t ) v ( s ) ds This implies that v ( t ) v ( t ) and u ( t ) R Tt e (4 c + )( s − t ) v ( s ) ds . Thus, we have v (0) = 0 and u (0) = 0. Remark 4.1. ( i ) Theorem 4.2 reduces to Theorem 5.9 of [8] when f ( t, X ( t ) , Z ( t )) ≡ f ( t, X ( t )) ;( ii ) Theorem 4.2 gives an answer to an open problem proposed by Ararat et al. [8]. References [1] B. Ahmad, S. Sivasundaram. Dynamics and stability of impulsive hybrid setvalued integro-differentialequations with delay.
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