Approximate Solutions of Set-Valued Stochastic Differential Equations

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1 Joral of Ucertai Systems Vol.7, No.1, pp.3-12, 213 Olie at: Approximate Soltios of Set-Valed Stochastic Differetial Eqatios Jfei Zhag, Shomei Li College of Applied Scieces, Beijig Uiversity of Techology Received 1 Jly 212; Revised 3 Agst 212 Abstract I this paper, we cosider the problem of approximate soltios of set-valed stochastic differetial eqatios. We firstly prove a ieqality of set-valed Itô itegrals, which is related to classical Itô isometry, ad a ieqality of set-valed Lebesge itegrals. Both of the ieqalities play a importat role to discss set-valed stochastic differetial eqatios. The we maily state the Carathodory s approximate method ad the Eler-Maryama s approximate method for set-valed stochastic differetial eqatios. We also ivestigate the errors betwee approximate soltios ad accrate soltios. c 213 World Academic Press, UK. All rights reserved. Keywords: set-valed radom variables, set-valed stochastic processes, set-valed stochastic differetial eqatios, the Carathodory s approximate soltios, the Eler-Maryama s approximate soltios 1 Itrodctio It is well-kow that the theory of classical stochastic differetial eqatios has bee developed deeply with wide applicatios i may fields (e.g. [3, 7, 13]). A stochastic differetial eqatio ca be writte as or its itegral form x(t) = x() + dx(t) = f(t, x(t))dt + g(t, x(t))db(t), (1.1) f(s, x(s))ds + g(s, x(s))db(s), where f : [, T ] R d R d ad g : [, T ] R d R d m are drift ad diffsio coefficiets respectively, {B(t) : t T } is a m-dimesioal Browia motio. Recetly, the discssio of set-valed stochastic differetial eqatios is becomig oe of importat topics sice the measremets of varios certaities arise ot oly from the radomess bt also from the impreciseess i some sitatios. For example, a geeral optio pricig model ca be writte as ds(t) = µ(t)s(t)dt + σ(t)s(t)db(t) with S() = s, S(t) is the stock price at time t, µ(t), σ(t) are give predictable processes, called expected retr rate ad volatility of the stock price respectively. However, both of µ(t) ad σ(t) are difficlt to obtai sice the factors of affectig the market are too complex. Bt it is relatively easy to estimate their lower ad pper bods. So we have to discss the set-valed model ds(t) = US(t)dt + V S(t)dB(t), where µ(t) [a, b] := U, σ(t) [c, d] := V (cf. [15]). This leads s cosider the followig geeral set-valed stochastic differetial eqatio dx(t) = F (t, X(t))dt + G(t, X(t))dB(t), Correspodig athor. lisma@bjt.ed.c (S. Li).

2 4 J. Zhag ad S. Li: Approximate Soltios of Set-Valed Stochastic Differetial Eqatios or set-valed stochastic itegral eqatio X(t) = X() + F (s, X(s))ds + G(s, X(s))dB(s), (1.2) where X(t), F (t, X(t)), G(t, X(t)) are set-valed stochastic processes. I the set-valed stochastic itegral eqatio (1.2), there are two kids of itegrals. The first oe is the set-valed Lebesge itegral of a set-valed stochastic process with respect to time t. For the defiitios ad properties of set-valed Lebesge itegrals of set-valed stochastic processes, readers may refer to [8, 1, 18] ad the refereces therei. The secod itegral is the set-valed Itô itegral of a set-valed process with respect to a Browia motio. Kisielewicz [4] first itrodced the cocept of set-valed stochastic itegral by sig selectio method i It is geeralized cocept of the classical stochastic Itô itegral. Followig his work, there are some reslts o set-valed differetial iclsios ad their applicatios i stochastic cotrol ad mathematical ecoomics (e.g. [5, 6, 15]). However, Kisielewicz s defiitio of set-valed stochastic itegrals is ot very satisfactory to discss setvaled stochastic differetial eqatios, becase oe ca ot prove that the set-valed stochastic itegral ξ(t, ω) =: G(s, ω)db(s) is a adapted set-valed stochastic process eve whe {G(t)} is a predictable setvaled stochastic process. Jg ad Kim [2] modified Kisielewicz s defiitio ad solved the above problem whe the basic space is R. Moreover, Zhag et. al [17] slightly modified the defiitio of [2] agai whe the basic space is a separable M-type 2 Baach space. I this paper, we shall give a ieqality of set-valed stochastic itegral (cf. Theorem 1), which is related to classical Itô isometry, ad prove a ieqality of set-valed Lebesge itegrals. Both of the ieqalities play a importat role to discss set-valed stochastic differetial eqatios. This is the first aim of this paper. Based o the defiitios ad properties of [9, 1, 18] ad the assmptios of set-valed drift coefficiet ad sigle-valed diffsio coefficiet i (1.2), Li ad Li [9], Li et al. [1] discssed set-valed Itô differetial (or itegral) eqatios i the Eclidea space R d ad Zhag et al. [18] ivestigated set-valed stochastic differetial eqatios i the M-type 2 Baach space. By sig the defiitios of set-valed Lebesge itegrals i [18] ad set-valed Itô stochastic itegrals i [17], we discssed the existece ad iqeess of soltios of set-valed stochastic itegral eqatios with set-valed drift ad diffsio coefficiets i [16]. We wold like to metio that Maliowski ad Michta [12] ad Michta [14] did some work o set-valed stochastic differetial eqatios whose drift ad diffsio coefficiets are set-valed mappigs, by sig Kisielewicz s defiitio of set-valed Itô itegral. Bt they cold ot prove that the soltio of their setvaled stochastic differetial eqatio is adapted, which is a very importat poit as we have poited ot before. Ever for classical stochastic differetial eqatios, however, there are oly a few types havig accrate soltios. So it is ecessary to cosider merical soltios of stochastic differetial eqatios. Ths it is importat to seek varios approximate methods i may models for practical ses (e.g. [13] ad its refereces). For the same reaso, i order to develop applicatios of set-valed stochastic differetial eqatios, we shall ivestigate the approximate soltios of set-valed stochastic differetial eqatios, which is the secod aim of this paper. We orgaize or paper as follows: i Sectio 2, we shall prove a ieqality of set-valed stochastic Itô itegrals ad a ieqality of set-valed Lebesge itegrals. Besides, we shall prove some other properties of set-valed Itô itegrals ad set-valed Lebesge itegrals, ad recall set-valed stochastic itegral eqatios. I Sectio 3, we shall itrodce the Caratheodory s approximate soltios ad ivestigate how to seek the size of iterative step ad iterative times by the error (distace betwee approximate soltio ad accrate soltio i some sese). Besides, we shall provide with the Eler-Maryama s approximate soltios ad have a discssio similarly. 2 Set-Valed Stochastic Differetial (or Itegral) Eqatios Throghot this paper, assme that R is the set of all real mbers ad R + = [, ), I = [, T ], R d is the d-dimesioal Eclidea space with sal orm, (X, X ) is a separable Baach space, B(E) is the Borel field of the space E, λ is the Lebesge measre o (I, B(I)). Let (Ω, A, P ) be a complete probability space, A be separable respect to P, the σ-field filtratio {A t : t I} satisfy the sal coditios (i.e. complete, o-decreasig ad right cotios), {B(t) : t I} be a Browia

3 Joral of Ucertai Systems, Vol.7, No.1, pp.3-12, motio, L p (Ω, A, P ; E) be the set of all E-valed radom variables with fiite p-order momets (p 1). Assme that K(X ) is the family of all oempty closed sbsets of X, ad K b(c) (X ) is the family of all oempty boded closed (covex) sbsets of X. For ay x X ad A K(X ), the metric betwee x ad A is defied by d(x, A) = if y A x y X. For ay A, B K(X ), Hasdorff metric betwee A ad B is defied as H X (A, B) = max{sp d(a, B), sp d(b, A)}. a A b B I particlar, deote A K = H X (A, {}). The K b (X ) ad K bc (X ) are complete spaces with respect to H X (Theorem of [11]). From ow o, we focs o the discssio i R d, which is easy to exted to a separable Baach space or a M-type 2 Baach space as athors have doe i [1, 11, 17, 18] respectively. A set-valed mappig F : Ω K(R d ) is called a set-valed radom variable if {ω Ω : F (ω) O } A, for ay ope set O of R d. For the eqivalet defiitios of a set-valed radom variable, readers may refer to Theorems ad of [11]. A R d -valed fctio f : Ω R d is called a almost everywhere selectio of F if f(ω) F (ω) for almost everywhere ω Ω, ad F is said to be L p -itegrably boded, if F K L p (Ω, A, P ; R + ). Let M be a set of A-measrable mappigs f : Ω R d, M is called decomposable, if for every f 1, f 2 M ad every A A sch that f 1 I A + f 2 I Ω\A M. We have that the p-order itegral selectio set of a L p -itegrably boded set-valed radom variable F S p F = {f Lp (Ω, A, P ; R d ) : f(ω) F (ω) a.s. (P )} is oempty, decomposable with respect to A ad closed i L p (Ω, A, P ; R d ). Coversely, if a give oempty sbset Γ L p (Ω, A, P ; R d ) is decomposable with respect to A ad closed, ca we fid a set-valed radom variable F sch that its selectio set S p F = Γ? The aswer is positive give by Hiai ad Umegaki i [1]. Readers also may refer to Theorem of [11]. Propositio 1 If Γ is a oempty closed sbset of L p (Ω, A, P ; R d ), the there exists a set-valed radom variable F sch that Γ = S p F if ad oly if Γ is decomposable with respect to A. Moreover, Γ is boded i L p (Ω, A, P ; R d ) if ad oly if F is L p -itegrably boded. as Let A be a sb-algebra of A. The the p-order itegral selectio set of F with respect to A is deoted S p F (A ) = {f L p (Ω, A, P ; R d ) : f(ω) F (ω) a.s.(p )}, where L p (Ω, A, P ; R d ) is the set of all R d -valed A -measrable radom variables with fiite p-order momets. Let N be the σ-algebra of the progressive evets i I Ω, i.e. N = {A I Ω : A ([, t] Ω) B A t, for every t I}. A d-dimesioal stochastic process f : I Ω R d is called progressive measrable if f is N -measrable. Let L 2 (R d ) = L 2 (I Ω, N, λ P ; R d ) be the set of all R d -valed progressive processes f = {f(t)} with T E[ f(t) 2 dt] <. Next, we shall recall some defiitios of set-valed stochastic processes ad set-valed stochastic Itô itegrals. The mappig F : I Ω K(R d ) is called a set-valed stochastic process, if for ay t I, F (t) : Ω K(R d ) is a set-valed radom variable. Similarly, we have the cocepts of set-valed progressive stochastic processes

4 6 J. Zhag ad S. Li: Approximate Soltios of Set-Valed Stochastic Differetial Eqatios ad set-valed progressive L p -itegrably boded stochastic processes, deoted by L p (K(R d )). If a process F : I Ω K(R d ) is progressive L 2 -itegrably boded, the its selectio set is oempty ad closed. The we ca defie S 2 F (N ) = {f L 2 (R d ) : f(t, ω) F (t, ω), a.s.(λ P )} Γ t = { f(s)db(s) : f S 2 F (N )}, t I. We se L 2 ad L 2 t briefly to deote L 2 (Ω, A, P ; R d ) ad L 2 (Ω, A t, P ; R d ) respectively. Let deγ t be the decomposable closed hll of Γ t with respect to A t, where the closre is take i L 2 t. That is, for ay g deγ t ad ay give ε >, there exists a fiite A t -measrable partitio A 1,..., A m of Ω ad f 1,..., f m SF 2 (N ) sch that m g I Ai f i (s)db(s) L 2 < ε. By sig Propositio 1, we ca obtai the followig reslt (cf. [17]). i=1 Propositio 2 If F : I Ω K(R d ) is a progressive ad L 2 -itegrably boded set-valed process, the for ay t I, there exists a A t -measrable set-valed radom variable I t (F ) sch that S 2 I t (F )(A t) = deγ t. Based o Propositio 2, the followig cocept of set-valed Itô itegral was itrodced i [17]. Defiitio 1 For each t I, the elemet I(F t ) of K(R d ) is called Itô itegral of the set-valed process F = {F t : t I} with respect to a Browia motio {B(t) : t I}, if S 2 I t(f )(A t) = deγ t, ad deoted by I(F t ) = F (s)db(s). For < t T, F (s)db(s) = I [,t]f (s)db(s). The followig reslts of this sectio will play a importat role i Sectio 3. Before showig them, we state the Castaig Represetatio Theorem of set-valed Itô itegrals. Lemma 1 [17] For a progressive ad L 2 -itegrably boded set-valed process F : I Ω K(R d ), there exists a seqece {(ft i ) : i = 1, 2,...} SF 2 (N ) sch that for each t I, F (t, ω) = cl{f i t (ω) : i = 1, 2,...} a.s. ad { } I(F t )(ω) = cl fs(ω)db i s (w) : i = 1, 2,... a.s.. Theorem 1 If two set-valed stochastic processes F 1, F 2 L 2 (K(R d )), the (i) for every, t I, < t, the followig ieqality holds ( ) 1/2, H L 2(J(F t 1 ), J(F t 2 )) H R 2 d(f 1(s, ω), F 2 (s, ω))ds dp [,t] Ω where H L 2 is Hasdorff metric o K(L 2 ); (ii) for every, s, t I, s t, the followig eqality holds I t (F 1 ) = cl{i s (F 1 ) + I t s(f 1 )}, a.s.; (iii) the mappig I t (F 1 ) : [, T ] K(R d ) is cotios i t i the sese of L 2 for the set-valed case. Proof: (i) By Lemma 1, for j = 1, 2, there exists a seqece {(f ji t ) : i = 1, 2,...} SF 2 (N ) sch that for each, t I, < t, F j (t, ω) = cl{(f ji t (ω) : i = 1, 2,...} a.s. ad I t (F j )(ω) = cl{ f ji s (ω)db s (w) : i = 1, 2,...} a.s..

5 Joral of Ucertai Systems, Vol.7, No.1, pp.3-12, For fixed f 1,i, we ca choose a sbseqece {f 2i k : k 1} sch that d(f 1i, f 2i k ) d(f 1i, F 2 ), a.s.(λ P ), k. So we have d 2 (f 1i, f 2i k ) d 2 (f 1i, F 2 ), a.s.(λ P ), k. Sice d 2 (f 1i, f 2i k ) 2( F 1 2 K + F 2 2 K), a.s.(λ P ), ad F 1, F 2 is L 2 -itegrably boded, by sig domiated covergece theorem, we obtai d 2 (f 1i (s), f 2i k (s))ds dp d 2 (f 1i (s), F 2 (s))ds dp. Ths, [,t] Ω [,t] Ω if d 2 (f 1i (s), f 2i k (s))ds dp d 2 (f 1i (s), F 2 (s))ds dp. k 1 [,t] Ω [,t] Ω De to f 1i s (ω)db s (ω) J t (F 1 ), we have So, we get sp i 1 d 2 L 2( Similarly, we ca show that Hece, we have sp i 1 d 2 L 2( d 2 L ( 2 f s 1i (ω)db s (ω), J(F t 2 )) if k 1 d 2 L ( 2 f s 1i (ω)db s (ω), f 2i k s (ω)db s (ω)) = if k 1 1i (f s (ω) f 2i k s (ω))db s (ω) 2 L 2 = if k 1 [,t] Ω (f 1i (s) f 2i k (s)) 2 ds dp [,t] Ω d2 (f 1i (s), F 2 (s))ds dp. fs 1i (ω)db s (ω), J(F t 2 )) sp d 2 (f 1i (s), F 2 (s))ds dp. [,t] Ω i 1 f 2i s (ω)db s (ω), J t (F 1 )) HL 2 2(J (F t 1 ), J(F t 2 )) [,t] Ω [,t] Ω sp d 2 (f 2i (s), F 1 (s))ds dp. i 1 H 2 R d(f 1, F 2 )ds dp. (ii) By Lemma 1, there exists a seqece {(ft i ) : i = 1, 2,...} SF 2 (N ) sch that for each, s, t I, s < t, Ths { I(F t )(ω) = cl { s I(F s )(ω) = cl { Is(F t )(ω) = cl s } fr(ω)db i r (w) : i = 1, 2,... } fr(ω)db i r (w) : i = 1, 2,... } fr(ω)db i r (w) : i = 1, 2,... I t (F ) cl{i s (F ) + I t s(f )}, Coversely, for ay a I(F s ) + Is(F t ), it follows from that, for ay ε >, there exists m(ε), (ε) sch that s a ( fr m(ε) (ω)db r (w) + fr (ε) (ω)db r (w)) < ε. L 2 Or eqivaletly, = < ε. ( s a fr m(ε) (ω)db r (w) + ( t ( a I [,s] fr m(ε) s s a.s.. (ω) + I [s,t] fr (ε) (ω) a.s., a.s., a.s.. f (ε) r (ω)db r (w)) L 2 ) db r (w)) L 2

6 8 J. Zhag ad S. Li: Approximate Soltios of Set-Valed Stochastic Differetial Eqatios Sice ( I [,s] f m(ε) r ) (ω) + I [s,t] fr (ε) (ω) db r (w) I(F t ) ad I(F t ) is closed, we have a I(F t ). Hece I t (F ) = cl{i s (F ) + I t s(f )}, a.s.. (iii) Usig (i) ad (ii) we get that, for ay s, t I, s t, HL 2 2(It (F ), I(F s )) = HL 2 2(It s(f ) + I(F s ), I(F s )) H L 2 2(It s(f ), {}) F 2 Kds dp. [s,t] Ω Sice F is L 2 -itegrably boded, H 2 L 2 (I t (F ), I s (F )) coverges to as t s goes to, which yields the cotiity of I t (F ). The proof is completed. Now we recall the defiitio of set-valed Lebesge itegral give i [18]. Take a set-valed progressive L 2 - itegrably boded stochastic process F : I Ω K(R d ), L t (F ) := F (s)ds is called set-valed Lebesge itegral of F, if { } SL 2 t (F )(A t) = de f(s)ds : f SF 2 (N ),, t I, < t. Similar to Theorem 1 ad its proof, we have the followig reslts. Theorem 2 If two set-valed stochastic processes F 1, F 2 L 2 (K(R d )), the (i) for every, t I, < t, the followig ieqality holds HL 2 2(Lt (F 1 ), L t (F 2 )) (t ) H 2 R d(f 1(s, ω), F 2 (s, ω)ds dp ; [,t] Ω (ii) for every, s, t I with s t, it holds L t (F 1 ) = cl{l s (F 1 ) + L t s(f 1 )}, a.s.; (iii) L t (F 1 ) is cotios i t i the sese of L 2 for set-valed case. I the followig we state the existece ad iqeess of soltios of set-valed stochastic itegral eqatios. Let F, G : I K(R d ) K(R d ) be the elemets of L 2 (K(R d )). We cosider the followig set-valed stochastic itegral eqatio X(t) = X + L t (F (s, X(s))) + I t (G(s, X(s))), t I. (2.1) Theorem 3 [16] Assme that F (t, X) ad G(t, X) satisfy the followig coditios: (A1) Liear growth coditio: for every X K b (L 2 ), there exists costat K > sch that E(H 2 R d(f (t, X), {}) + H2 R d(g(t, X), {})) K(1 + X 2 K), t I where X K = H L 2(X, {}). (A2) Lipschitz cotios coditio: for every X, Y K b (L 2 ), there exists costat K > sch that E(H 2 R d(f (t, X), F (t, Y )) + H2 R d(g(t, X), G(t, Y ))) KH2 L2(X, Y ), t I. The for ay X K b (L 2 ), there exists a iqe adapted set-valed soltio {X(t)} of eqatio (2.1) ad {X(t)} is cotios i t i the sese of L 2 for set-valed case. 3 Approximate Soltios I the previos sectio we have established existece ad iqeess of soltios of set-valed stochastic differetial eqatios (2.1). However, i geeral, the soltios do ot have explicit formlas. I practice, we therefore seek the approximate soltios. We call costat ε > is error, if the followig ieqality holds sp HL 2 2(X (t), X(t)) < ε,

7 Joral of Ucertai Systems, Vol.7, No.1, pp.3-12, where HL 2 (X 2 (t), X(t)) represets the L 2 distace betwee the approximate soltio X (t) ad accrate soltio X(t). I the proof of the existece ad iqeess of soltios for set-valed stochastic differetial eqatios, we se the Picard iterative i [16]. Hece, we ca se it to fid approximate soltio X (t). Bt it has a disadvatage that for ay t, for each, oe has to compte all X (t), X 1 (t),..., X 1 (t) i order to compte X (t), ad it ivolves a lot of calclatios. More efficiet ways are the Caratheodory s ad the Eler- Maryama s approximate methods becase they do ot eed to compte all X 1 (t),..., X 1 (t) bt compte X (t) directly by sig the reslts of previos calclatios before t. Now we will discss them i the followig. 3.1 The Caratheodory s Approximate Soltios Now we itrodce the Caratheodory s approximate soltios. For every iteger 1, defie X (t) = X for 1 t, ad t (, T ], let X (t) = X + F (s, X (s 1/))ds + Note that for t 1/, X (t) ca be compted by the for 1/ < t 2/, X (t) = X + F (s, X )ds + G(s, X (s 1/))dB(s). (3.1) G(s, X )db(s), X (t) = X (1/) + F (s, X (s 1/))ds + G(s, X (s 1/))dB(s) 1/ 1/ ad so o. I other words, X (t) ca be compted step-by-step o the itervals [, 1/], (1/, 2/],.... I calclatios, for ay give ε >, how to decide at which step we may stop the comptatios? That is how to fid the size of iterative steps ad iterative times so that the L 2 -distace betwee the approximate soltio ad accrate soltio is less tha ay give ε >. Or mai reslt will aswer this qestio. Firstly, we itrodce two Lemmas i order to establish the mai reslt. Lemma 2 Assme that liear growth coditio (A1) holds, the for all 1, we have sp X (t) 2 K C 1 := (1 + 3 X 2 K)e 3K(T +1)T. (3.2) Proof: By sig Theorems 1 ad 2, ad the fact that X (t) satisfies eqatio (3.1), we have X (t) 2 K 3 X 2 K + 3 F (s, X (s 1/))ds 2 K +3 G(s, X (s 1/))dB(s) 2 K 3 X 2 K + 3t H 2 R d(f (s, X (s 1/)), {})ds dp [,t] Ω +3 H 2 R d(g(s, X (s 1/)), {})ds dp [,t] Ω 3 X 2 K + 3K(t + 1) 3 X 2 K + 3K(T + 1) t (1 + X (s 1/) 2 K)ds (1 + sp r [,s] X (r) 2 K)ds for all t I. Coseqetly, we have 1 + sp X (r) 2 K X 2 K + 3K(T + 1) r [,t] (1 + sp X (r) 2 K)ds. r [,s]

8 1 J. Zhag ad S. Li: Approximate Soltios of Set-Valed Stochastic Differetial Eqatios The the Growall ieqality (cf. Page 45 of [13]) implies sp X (r) 2 K (1 + 3 X 2 3K(T +1)t K)e r [,t] for all t I. I particlar, (3.2) follows whe t = T. Lemma 3 Assme that liear growth coditio (A1) holds, the for all 1, ad s < t T with t s 1, we have H 2 L 2(X (t), X (s)) C 2 (t s), where C 2 = 4K(1 + C 1 ) ad C 1 is defied i Lemma 2. Proof: By virte of Theorems 1 ad 2 ad ieqality H X (A + B, C + D) H X (A, C) + H X (B, D), for A, B, C, D K(X ) (cf. Lemma of [11]). we have HL 2 2(X (t), X (s)) 2HL 2 2( F (r, X (r 1/))dr, s +2HL 2 2( G(r, X (r 1/))dB(r), 2HL 2 2( F (r, X (r 1/))dr, {}) s F (r, X (r 1/))dr) s +2HL 2 2( G(r, X (r 1/))dB(r), {}) s 2(t s) H 2 R d(f (s, X (s 1/)), {})dr dp [s,t] Ω +2 H 2 R d(g(s, X (s 1/)), {})dr dp [s,t] Ω 2K(t s + 1) (1 + X (r 1/) 2 K)dr [s,t] 4K(1 + C 1 )(t s), G(r, X (r 1/))dB(r)) where i the last ieqality we sed the assmptio t s 1 ad Lemma 2. The proof is completed. Theorem 4 Assme that the liear growth coditio (A1) ad the Lipschitz coditio (A2) hold, X(t) is the iqe soltio of eqatio (2.1). The, for 1, sp HL 2 2(X (t), X(t)) C 3, where C 3 = 4C 2 KT (T + 1) exp[4kt (T + 1)] ad C 2 is defied i Lemma 3. Proof: By the Lipschitz coditio (A2), Theorems 1 ad 2, it is ot difficlt to obtai H 2 L 2(X (t), X(t)) 2(t + 1)K 4K(T + 1) where i last ieqality we sed t H 2 L 2(X (r 1/), X(r))dr (H 2 L 2(X (r 1/), X (r)) + H 2 L 2(X (r), X(r)))dr, H X (A, B) = H X (A + C, C + B) H X (A, C) + H X (B, C) for ay A, B, C K(X ). The sig Lemma 3, we have HL 2 (X 2 (r 1/), X (r)) C 2 / if r 1/, otherwise if r < 1/, HL 2 (X 2 (r 1/), X (r)) = HL 2 (X 2 (), X (r)) C 2 r C 2 /. Hece, we have sp r [,t] H 2 L 2 (X (r), X(r)) (4/)C 2 KT (T + 1) + 4K(T + 1) sp r [,s] H 2 L 2(X (r), X(r))ds.

9 Joral of Ucertai Systems, Vol.7, No.1, pp.3-12, Fially, by applyig the Growall ieqality, we have sp HL 2 2(X (t), X(t)) C 3, ad the proof is completed. Now we ca make sre, sch that is a iteger larger tha C 3 /ε. The we ca compte X (t) over the itervals [, 1/], (1/, 2/],..., step by step. Theorem 4 ca garatee that the approximate soltio X (t) is closed eogh to the accrate soltio X(t) iformly, that is sp HL 2 2(X (t), X(t)) < ε. 3.2 The Eler-Maryama s Approximate Soltios The Eler-Maryama s approximate soltio is defied as follows: for every iteger 1, defie X () = X, ad the for (k 1)/ < t k/ T, k = 1, 2,..., (k 1) t X (t) = X + k 1 F ( (k 1)) t s, X ds + k 1 G ( (k 1)) s, X db(s). The defie ˆX (t) = X I {} (t) + k 1 (k 1) X I((k 1)/,k/] (t) for t I, the we have X (t) = X () + F (s, ˆX (s))ds + G(s, ˆX (s))db(s), t I. Similar to the Caratheodory s approximate soltios, we ow ivestigate how to make sre the size of iterative step ad iterative times. We have the followig reslts. Lemma 4 Assme that liear growth coditio (A1) holds, the for all 1, the Eler-Maryama s approximate soltios X (t) satisfy sp X (t) K C 4 := (1 + 3 X 2 K)e 3K(T +1)T. Lemma 5 Assme that liear growth coditio (A1) holds, for all 1, ad s < t T with t s 1, the Eler-Maryama s approximate soltios X (t) satisfy H 2 L 2(X (t), X (s)) C 5 (t s), where C 5 = 8K(1 + C 4 ) ad C 4 is defied i Lemma 4. Theorem 5 Assme that the liear growth coditio (A1) ad the Lipschitz coditio (A2) hold, ad X(t) is the iqe set-valed soltio of eqatio (2.1). The, for 1, the Eler-Maryama s approximate soltios X (t) satisfy sp HL 2 2(X (t), X(t)) C 6, where C 6 = 4C 5 KT (T + 1) exp[4kt (T + 1)] ad C 5 is defied i Lemma 5. The proofs of above Lemmas 4 ad 5 ad Theorem 5 are omitted becase of the similarity of the proofs of Lemmas 2 ad 3 ad Theorem 4.

10 12 J. Zhag ad S. Li: Approximate Soltios of Set-Valed Stochastic Differetial Eqatios 4 Coclsios It is ecessary to cosider set-valed differetial (or itegral) eqatios becase the measremets of varios certaities arise ot oly from radomess bt also from the impreciseess i may sitatios. I this paper, we firstly prove a ieqality of set-valed Itô itegrals ad a ieqality of set-valed Lebesge itegrals. Both of the ieqalities play a importat role to discss set-valed stochastic differetial eqatios. I order to develop applicatios of set-valed stochastic differetial eqatios, it is ecessary to stdy approximate soltios of set-valed stochastic differetial eqatios. I or paper, we itrodced the Caratheodory s approximate soltios ad the Eler-Maryama s approximate soltios of set-valed stochastic differetial eqatios. Besides, we gave how to make sre their the size of iterative step ad iterative times. Ackowledgemets This research is spported by NSFC (No ), PHR (No ) ad Beijig Natral Sciece Fodatio (No ). Refereces [1] Hiai, F., ad H. Umegaki, Itegrals, coditioal expectatios ad martigales of mltivaled fctios, Joral of Mltivariate Aalysis, vol.7, pp , [2] Jg, E.J., ad J.H. Kim, O set-valed stochastic itegrals, Stochastic Aalysis ad Applicatios, vol.21, pp , 23. [3] Karatzas, I., ad S.E. Shreve, Browia Motio ad Stochastic Calcls, 2d Editio, Spriger-Verlag, [4] Kisielewicz, M., Set-valed stochastic itegrals ad stochastic iclsios, Discss. Math., vol.13, pp , [5] Kisielewicz, M., M. Michta, ad J. Motyl, Set-valed approach to stochastic cotrol, part I: existece ad reglarity properties, Dyamical Systems ad Applicatios, vol.12, pp , 23. [6] Kisielewicz, M., M. Michta, ad J. Motyl, Set-valed approach to stochastic cotrol, part II: viability ad semimartigale isses, Dyamical Systems ad Applicatios, vol.12, pp , 23. [7] Klebaer, F.C., Itrodctio to Stochastic Calcls with Applicatios, Imperial College Press, Lodo, [8] Li, J., ad S. Li, Set-valed stochastic Lebesge itegral ad represetatio theorems, Iteratioal Joral of Comptatioal Itelligece Systems, vol.12, pp , 28. [9] Li, J., ad S. Li, Itô type set-valed stochastic differetial eqatio, Joral of Ucertai Systems, vol.3, pp.52 63, 29. [1] Li, J., S. Li, ad Y. Ogra, Strog soltio of Itô type set-valed stochastic differetial eqatio, Acta Mathematica Siica (Eglish Series), vol.26, pp , 21. [11] Li, S., Y. Ogrra, ad V. Kreiovich, Limit Theorems ad Applicatios of Set-Valded ad Fzzy Set-Valed Radom Variables, Klwer Academic Pblishers, Netherlads, 22. [12] Maliowski, M.T., ad M. Michta, Fzzy stochastic itegal eqatio, Dyamic Systems ad Applicatios, vol.19, pp , 21. [13] Mao, X., Stochastic Differetial Eqatios ad Applicatios, 2d Editio, Horwood Pblishig, Chichester, 28. [14] Michta, M., O set-valed stochastic itegrals ad fzzy stochastic eqatios, Fzzy Sets ad Systems, vol.177, pp.1 19, 211. [15] Zhag, J., ad S. Li, Maximal (miimal) coditioal expectatio ad Eropea optio pricig with ambigos retr rate ad volatility, Iteratioal Joral of Approximate Reasoig, 212, [16] Zhag, J., ad S. Li, Some properties of set-valed Itô itegral ad the soltios of set-valed stochastic itegral eqatios, preprit. [17] Zhag, J., S. Li, I. Mitoma, ad Y. Okazaki, O set-valed stochastic itegrals i a M-type 2 Baach space, Joral of Mathematical Aalysis ad Applicatios, vol.35, pp , 29. [18] Zhag, J., S. Li, I. Mitoma, ad Y. Okazaki, O the soltio of set-valed stochastic differetial eqatio i M-type 2 Baach space, Tohok Mathematical Joral, vol.61, pp , 29.