Order-Preservation for Multidimensional Stochastic Functional Differential Equations with Jump

Transcription

1 arxiv: v1 [math.pr] 5 May 2013 Order-Preservation for Multidimensional Stochastic Functional Differential quations with Jump Xing Huang a) and Feng-Yu Wang a),b) a) School of Mathematical Sciences, Beijing Normal University, Beijing , China b) Department of Mathematics, Swansea University, Singleton Park, SA2 8PP, United Kingdom wangfy@bnu.edu.cn, F.-Y.Wang@swansea.ac.uk May 7, 2013 Abstract Sufficient and necessary conditions are presented for the order-preservation of stochastic functional differential equations on R d with non-lipschitzian coefficients driven by the Brownian motion and Poisson processes. The sufficiency of the conditions extends and improves some known comparison theorems derived recently for one-dimesional equations and multidimensional equations without delay, and the necessity is new even in these special situations. AMS subject Classification: 60J75, 47G20, 60G52. Keywords: Order-preservation, comparison theorem, stochastic functional differential equation. 1 Introduction The order-preservation of stochastic processes is crucial since it enables one to control complicated processes by using simpler ones. For a large class of diffusion-jump type Markov processes on R d, the order-preservation property has been well described in the distribution sense see [4, 15] and references therein), see also [14] for a study of super processes. To derive the pathwise order-preservation, one establishes the comparison theorem for stochastic Supported in part by Lab. Math. Com. Sys., NNSFC , ), SRFDP, and the Fundamental Research Funds for the Central Universities. 1

2 differential equations SDs), which goes back to [12, 16]. The study of comparison theorem for one-dimensional SDs is now very complete, see e.g. [3, 5, 6, 7, 8, 9, 10, 18] and references within. quations considered in these references include forward or backward SDs with jump and with delay. The aim of this note is to provide a sharp criterion on the comparison theorem for multidimensional stochastic functional stochastic differential equations SFDs), which is yet unknown in the literature. Throughout the paper, we fix a constant r 0 0 and a natural number d 1. Let C = { ξ = ξ 1,,ξ d ) : [ r 0,0] R d is cadlag }. Recall that a path is called cadlag, if it is right-continuous having finite left limits. For any ξ C, we have d ξ = ξ i s) <. i=1 sup s [ r 0,0] Then under the uniform norm the space C is complete but not separable. To make C a Polish space, we take the Skorohod metric rather than the uniform metric. For any cadlag f : [ r 0, ) R d and t 0, we let f t C be such that f t θ) = fθ+t) for θ [ r 0,0], and define f t C for t > 0 such that f t θ) = ft+θ) ) := lim s t+θ fs) for θ [ r 0,0]. We call f t ) t 0 the segment of ft)) t r0. Now, let Bt) be an m-dimensional Brownain motion, and let Nds,dz) be a Poisson counting process with characteristic measure ν on a measurable space, ), with respect to a complete filtered probability space Ω, F,{F t } t 0,P). We assume that B and N are independent. We will consider the order-preservation of SFDs driven by B and N. To characterize the non-lipshitz regularity of coefficients in the SDDs, we introduce the following class of control functions: U = { u C 1 0, );[1, )) : 1 0 ds =, lim sus) s 0 sus)2 = 0, } and s sus) is increasing and concave. Typical elements in this class are us) = 1 and us) = log1+s 1 ). Consider the following SFDs on R d : { dxt) = bt,x t )dt+σt,x t )dbt)+ 1.1) γt,x t,z)ndt,dz), d Xt) = bt, X t )dt+σt, X t )dbt)+ γt, X t,z)ndt,dz), where b, b : [0, ) C Ω R d, σ, σ : [0, ) C Ω R d R m, γ, γ : [0, ) C Ω R d are progressively measurable. 2

3 For any s 0 and F s -measurable random variables ξ, ξ on C, a solution to 1.1) for t s with X s, X s ) = ξ, ξ) is a cadlag adapted process Xt), Xt)) t s such that P-a.s. for all t s t t Xt) = ξ0)+ br,x r )dr + σr,x r )dbr)+ γr,x r,z)ndr,dz), Xt) = ξ0)+ s t s br, Xr )dr + s t s σr, X r )dbr)+ [s,t] [s,t] γr, X r,z)ndr,dz), where, according to the initial condition X s, X s ) = ξ, ξ), X r and X r for r s are well defined. To ensure the existence and uniqueness of solutions, we make use of the following assumptions: A1) There exist some positive K C[0, )) and u U such that P-a.s. bt,ξ) bt,η) + bt,ξ) bt,η) + σt,ξ) σt,η) HS + σt,ξ) σt,η) HS ) + γt,ξ,z) γt,η,z) + γt,ξ,z) γt,η,z) νdz) Kt) ξ η u ξ η ), ξ,η C,t 0. A2) For any T > 0 there exists a constant CT) > 0 such that P-a.s. ) sup bt,0) + bt,0) + σt,0) HS + σt,0) HS t [0,T] + [0,T] γt,0,z) + γt,0,z) ) dtνdz) CT). When u 1, A1) reduces to the usual Lipschitz condition. In general, A1) allows the coefficients to be non-lipschitzian. According to Theorem 3.1 below, for any s 0 and F s -measurable random variables ξ, ξ on C, the equation 1.1) has a unique solution for t s with X s = ξ and X s = ξ, and the solution is non-explosive. We denote the solution by {Xs,ξ;t), Xs, ξ;t)} t s. To introduce the notion of order-preservation of the solutions, we take the usual partialorder on R d ; i.e. for x = x 1,,x d ),y = y 1,,y d ) R d, we write x y if x i y i holds for all 1 i d. Similarly, for ξ = ξ 1,,ξ d ),η = η 1,,η d ) C, we write ξ η if ξ i θ) η i θ) holds for all θ [ r 0,0] and 1 i d. Moreover, for any ξ,η C, let ξ η C be such that ξ η) i = min{ξ i,η i },1 i d; and let ξ η = { ξ) η)}. Definition 1.1. The solutions of 1.1) are called order-preserving, if for any s 0 and F s -measurable random variables ξ, ξ on C with P-a.s. ξ ξ, P-a.s. Xs,ξ;t) Xs, ξ;t) holds for all t s. Theorem 1.1. Assume A1) and A2). The solutions to 1.1) are order-preserving if the following conditions are satisfied: 3

4 1) P-a.s. b i t,ξ) b i t, ξ) holds for all 1 i d,t 0 and ξ, ξ C with ξ ξ and ξ i 0) = ξ i 0). 2) P-a.s. σ ij t,ξ) = σ ij t, ξ) holds for all t 0,1 i d,1 j m and ξ, ξ C with ξ i 0) = ξ i 0). 3) ν P-a.e. ξ i 0) + γ i t,ξ, ) ξ i 0) + γ i t, ξ, ) holds for all all 1 i d,t 0 and ξ, ξ C with ξ ξ. Comparing with existing comparison theorems derived in the above mentioned references for one-dimensional equations, Theorem 1.1 has a rather broad range of applications. Next, a multidimensional comparison theorem without delay has been presented in [13, Theorem 296]wherethecondition2 0 impliesthat b i t,x)dependsonlyonx i, andisthusmuchstronger than the condition 1) in Theorem 1.1 with r 0 = 0 i.e. the case without delay). Moreover, when r 0 = 0 i.e. without delay) Theorem 1.1 also covers the comparison theorem derived recently in [19] for ν) < and Lipschtizian coefficients. On the other hand, our next result shows that the conditions in Theorem 1.1 are also necessary for the order-preservation when the coefficients are continuous and either ν is finite or γt,ξ, ), γt,ξ, ) are integrable with respect to ν locally uniformly in t,ξ), where and in the sequel the continuity on C is with respect to the the Skorohod metric. This result is new even for the case without delay. Theorem 1.2. Assume A1), A2) and that the solutions to 1.1) are order-preserving. I) If P-a.s. b, b C[0, ) C;R d ) and for any n 1 1.2) lim sup ε γt,ξ,z) + γt,ξ,z) )νdz) = 0, ε 0 t [0,n], ξ n then condition 1) holds. II) If P-a.s. σ, σ C[0, ) C;R d R m ), then condition 2) holds. III) If P ν-a.e. γ, γ C[0, ) C;R d ), then condition 3) holds. Note that condition 1.2) holds if either ν is finite or γt,ξ, ), γt,ξ, ) are integrable with respect to ν locally uniformly in t,ξ). In the next section we present proofs of the above two theorems. In Section 3, we present a result on the existence and uniqueness of solutions to stochastic functional equations with jumps. 2 Proofs of Theorems 1.1 and 1.2 Proof of Theorem 1.1. Assume that 1) 3) hold. For any 0 and F t0 -measurable random variables ξ, ξ on C such that ξ ξ, we aim to prove that for any T >, 2.1) sup X i,ξ;r) X i, ξ;r)) + = 0, 1 i d. r [,T] 4

5 For equations without delay, this can be done by using a Tanaka type formula for X i t) X i t)) + see [13, 153]). Below, we shall make use an approximation argument which plays the same role as the Tanaka type formula. For simplicity, we will simply denote Xt) = X,ξ;t) and Xt) = X, ξ;t) for t r 0. Recall that X,ξ;t) = ξt ) and X, ξ;t) = ξt ) for t [ r 0, ]. For any n 1 let ψ n : R [0, ) be constructed as follows: ψ n s) = ψ ns) = 0 for s,0], and 4n 2 s, s [0, 1 ], ψ n 2n s) = 4n 2 s 1), s [ 1, 1], n 2n n 0, otherwise. We have 2.2) 0 ψ n 1, and as n : 0 ψ ns) s +, sψ n s) 21 0, 1 )s) 0. n Let Since by 2) we have σ = σ and the Itô formula yields τ k = inf{t : Xt) Xt) Xt) k}, k 1. ψ n X i ) X i )) = ψ n ξ i 0) ξ i 0)) = 0, 2.3) ψ n X i t τ k ) X i t τ k )) = Mt τ k ) m j=1 t τk [,t τ k ] t τk b i s,x s ) b i s, X s ))ψ nx i s) X i s))ds σ ij s,x s ) σ ij s, X s )) 2 ψ n Xi s) X i s))ds {ψ n X i s ) X i s )+γ i s,x s,z) γ i s, X s,z) ) ψ n X i s ) X } i s )) Nds, dz) for any k,n 1, 1 i d and t, where Mt) := m j=1 t σ ij s,x s ) σ ij s, X s ))ψ nx i s) X i s))db j s). Noting tha ψ n Xi s) X i s)) 1 {X i s)> X i s)} and when X i s) > X i s) one has X s X s ) i 0) = X s ) i 0), it follows from 1) that P-a.s. b i s,x s X s ) b i s, X s ))ψ nx i s) X i s)) 0, n 1,s [,T]. 5

6 Combining this with A1) and 0 ψ n 1, we obtain P-a.s. 2.4) b i s,x s ) b i s, X s ))ψ nx i s) X i s)) = b i s,x s ) b i s,x s X s ))ψ n Xi s) X i s)) +b i s,x s X s ) b i s, X s ))ψ n Xi s) X i s)) CT) X s X s X s u X s X s X s ), n 1,s [,T] for some constant CT) > 0. Next, by A1), 2.2) and 2), we have P-a.s. 2.5) m i=1 σ ij s,x s ) σ ij s, X s ) 2 ψ n Xi s) X i s)) CT)1 {X i s) X i s) 0, 1 n )} Xi s) X i s) u x i s) X i s) ) 2 CT)εn), n 1,s [,T] for some constant CT) > 0, where since u U, εn) := sup s 0, 1 n ) sus) 2 0 as n. Moreover, A1), 2.2) and 2) also imply P-a.s. 2.6) m σ ij s,x s ) σ ij s, X s ) ψ n Xi s) X i s)) j=1 CT)X i s) X i s)) + ux i s) X i s)) + ), n 1,s [,T] for some constant CT) > 0. Finally, by 3) we have 2.7) X i s) X i s)+γ i s,x s X s, ) X i s)+ γ i s, X s, ), ν P-a.e. If X i s) X i s), then 2.7) becomes X i s)+γ i s,x s X s, ) X i s)+ γ i s, X s, ), ν P-a.e., so tha ψ n 1 and ψ ns) = 0 for s 0 imply ψ n X i s) X i s)+γ i s,x s, ) γ i s, X s, ) ) ψ n X i s) X i s)) = ψ n X i s) X i s)+γ i s,x s, ) γ i s, X s, ) ) = ψ n γ i s,x s, ) γ i s,x s X s, ) +X i s)+γ i s,x s X s, )) X i s)+ γ i s, X s, )) ) ψ n γ i s,x s, ) γ i s,x s X s, ) ) γ i s,x s, ) γ i s,x s X s, ), ν P-a.e. 6

7 Similarly, if X i s) X i s) then 2.7) becomes so tha ψ n 1 implies γ i s,x s X s, ) γ i s, X s, ), ν P-a.e., ψ n X i s) X i s)+γ i s,x s, ) γ i s, X s, ) ) ψ n X i s) X i s)) = ψ n X i s)+γ i s,x s, ) X i s) γ i s,x s X s, ) +γ i s,x s X s, ) γ i s, X s, ) ) ψ n X i s) X i s)) ψ n X i s) X i s)+γ i s,x s, ) γ i s,x s X s, )) ψ n X i s) X i s)) γ i s,x s, ) γ i s,x s X s, ), ν P-a.e. Combining these with A1) we obtain {ψ n X i s ) X i s )+γ i s,x s,z) γ i s, X s,z) ) 2.8) [,t τ k ] = [,t τ k ] t τk 2 CT) ds t τk ψ n X i s ) X } i s )) Nds, dz) {ψ n X i s) X i s)+γ i s,x s,z) γ i s, X s,z) ) ψ n X i s) X } i s)) dsνdz) γ i s,x s,z) γ i s,x s X s,z) νdz) X s X s X s u X s X s X s )ds for some constant CT) > 0 and all n 1,t [,T]. Let φ k s) = sup r [ r 0,s τ k ] Xr) Xr) Xr), s. By combining 2.3)-2.8) with X t0 X t0 and using the Burkholder-Davis-Gundy inequality, we obtain that for some constant CT) > 0 and all t [,T], d i=1 CT) sup r [ r 0,t τ k ] t ψ n X i r) X i r)) { φ k s)uφ k s)) } ds+ct)εn), k,n 1. Letting n using the Jensen inequality, we derive φ k t) CT) t {φ k s)}u φ k s) ) ds, t [,T],k 1. 7

8 Since 1 1 ds =, by the Bihari inequality this implies that see e.g. the end of the proof 0 sus) of Theorem 4.2 in [11]) φ k T) = 0, k 1. Letting k we prove 2.1). To prove Theorem 1.2, we need the following Lemma 2.1. For any h C 2 b Rd ), let Lh)t,ξ) = Lh)t,ξ) = d b i t,ξ) i hξ0))+ 1 2 i=1 + d σσ )t,ξ) i j hξ0)) i,j=1 { hξ0)+γt,ξ,z)) hξ0)) } νdz), d bi t,ξ) i hξ0))+ 1 2 i=1 + d σ σ )t,ξ) i j hξ0)) i,j=1 { hξ0)+ γt,ξ,z)) hξ0)) } νdz), t 0,ξ C, where i 1 i d) is the derivative with respect to the i-th component in R d. By A1) and A2), Lh and Lh are locally bounded with respect to the usual metric on [0, ) and the uniform norm on C. Let M be the class of all increasing functions on C, where a function h on C is called increasing if hξ) hη) holds for all ξ η. Lemma 2.1. Assume A1) and A2). If the solutions to 1.1) are order-preserving, then for any s 0 and F s -measurable random variables ξ, ξ on C with ξ ξ, and any h M Cb 2Rd ) with hξ0)) = h ξ0)), there holds P-a.s. lim inf t s ) Lh)t,X t s,ξ)) F s lim sup t s Lh)t, X t s, ξ)) F s ), whree X s,ξ) and X s, ξ) are the segment processes of Xs,ξ; ) and Xs; ξ, ) respectively. Proof. Simply denote Xt) = Xs,ξ;t), Xt) = Xs, ξ;t) for t s r 0. Let τ = inf{t s : Xt) + Xt) 1+ ξ + ξ }. Since hξ0)) = h ξ0)) and Xt) Xt) for all t s, we have 2.9) hxt τ)) Fs ) hξ0)) h Xt τ)) Fs ) h ξ0)), t s. By the Itô formula and the Fatou lemma, it is easy to see that 2.10) lim inf t s lim sup t s hxt τ)) ) Fs hξ0)) t s h Xt τ)) ) Fs h ξ0)) t s 8 lim inf t s lim sup t s Lh)t,X t ) F s ), Lh)t, X t ) F s ).

9 Combining this with 2.9) we finish the proof. Below we only prove the first formula in 2.10), as the proof of the second is completely similar. By the Itô formula, for t s,s+1] we have hxt τ)) ) Fs hξ0)) = t τ Lh)r,X s r ) ) Fs t s t s 1 {τ>t} inf Lh)r,X r) F s ) CPτ < t F s ), r s,t τ] where, due to A1) and A2), { } C := sup Lh r,η) : r [s,s+1], η 1+ ξ + ξ <. Ω Since τ > s due to the right-continuity of the solution, and since Lh is locally bounded, by letting t s we obtain the first formula in 2.10) from the Fatou lemma. Proof of Theorem 1.2. Let 1 i d and 0 be fixed. For any ξ, ξ C, let Xt) = X,ξ,;t), Xt) = X, ξ;t). a) Proof of III). Let ξ ξ. A1) and A2) imply that b,σ and γ,,z) + γ,,z) ) νdz) are locally bounded. We aim to prove 2.11) ξ i 0)+γ i,ξ, ) ξ i 0)+ γ i, ξ, ), ν P-a.e. Due to the continuity of γ and γ in the first two variables and the separability of [0, ) C recall that we use the Skorohod metric on C), this implies condition 3). Let τ = inf{t : X t ξ + X t ξ 1}. By the Itô formula and the local boundedness of the coefficients and the right-continuity of Xs), for any t > we have ψ n {X i t τ) X i t τ)} t τ { = b i s,x s ) b i s, X s ))ψ n {Xi s) X i s)} + + m j=1 σ ij s,x s ) σ ij s, X s )) 2 ψ nx i s) X i s)) [ ψn X i s) X i s)+γ i s,x s,z) γ i s, X s,z) ) ψ n X i s) X i s)) ] νdz)} ds. Since X i s) X i s),x i t τ) X i t τ) and ψ n s) = ψ n s) = 0 for s 0, this implies t τ { ψ n X i s) X i s)+γ i s,x s,z) γ i s, X s,z) ) } νdz) ds = 0 t. By τ >, the right-continuity of the solutions, and the joint-continuity of γ i and γ i in the first two variables, from this and the Fatou lemma we conclude that ψ n ξ i 0) ξ i 0)+γ i,ξ,z) γ i, ξ,z) ) νdz) = 0. 9

10 Since ψ n s) s + as n, by letting n we arrive at ξ i 0) ξ i 0)+γ i,ξ,z) γ i, ξ,z) )+ νdz) = 0, and hence 2.11) holds. b) Proof of I). Let ξ ξ and ξ i 0) = ξ i 0). For any ε 0,1), let φ ε C0 R) such that 0 φ ε 1, φ ε [ ε,ε] = 1, φ ε [ 2ε,2ε] c = 0. Take h ε x) = x i ξ i 0) 0 φ ε s)ds, x R d. Then h ε M C 2 b Rd ) and by the continuity of b, b and the right continuity of the solutions, lim t t0 bt,x t ), h ε Xt)) = b i,ξ), lim t t0 bt, X t ), h ε Xt)) = b i, ξ), lim t t0 2 h ε Xt)) = lim t t0 2 h ε Xt)) = 0, h ε Xt)+γt,X t,z)) h ε Xt)) + h ε Xt)+ γt, X t,z)) h ε Xt)) 4ε) γt,x t,z) + γt, X t,z) ). Combining this with Lemma 2.1, we obtain P-a.s. b i,ξ) b i, ξ)+ sup t [, +1], η η 1+ ξ ξ { 4ε) γt,η,z) + γt, η,z) ) } νdz). Letting ε 0 and using 1.2) we prove b i,ξ) b i, ξ),p-a.s. This implies condition 1) by the continuity of b, b and the separability of [0, ) C. c) Proof II). If condition 2) does not hold, then there exist ξ, ξ C with ξ i 0) = ξ i 0) such that for some 1 j m one has Pσ ij,ξ) σ ij, ξ)) > 0. Since σ and σ are continuous, there exists a constant ε 0,1) such that PA ε ) > 0, where Let A ε := { σ ij t,η) σ ij t, η) ε for t [, +ε], η ξ + η ξ ε }. τ = inf{t : σ ij t,x t ) σ ij t, X t ) ε} τ = inf{t : X t ξ + X t ξ ε}, τ n = inf{t : X i t) X i t) n 1 }, n 1. Let g n s) = e ns 1. Since Xs i X s i, and due to a) X i s) X i s)+γ i s,x s,z) γ i s, X s,z) 0, s, 10

11 by the Itô formula we obtain 0 g n X i X i ) +ε) τ τ τ n )) t0 +ε) τ τ τ n { = g n Xi X i )s))b i s,x s ) b i s, X s )) + g n Xi s) X i s)) 2 + m σ ij s,x s ) σ ij s, X s )) 2 j=1 { gn X i s) X i s)+γ i s,x s,z) γ i s, X s,z)) g n X i s) X i s)) } νdz)} ds n 2 ε 2 2e Cne ) {ε τ ) τ ) τ n )}, n 1, where, according to A1) and A2), C := sup { b i t,η) b i t, η) + γ i t,η,z) γ i t, η,z) νdz) : t [, +ε], η ξ + η ξ ε } <. This implies τ ) τ ) τ n )) = 0 for large n, which is impossible since PA ε ) > 0 and due to the right-continuity of the solutions τ τ n τ > holds on the set A ε. 3 xistence and uniqueness of solutions When N = 0 and b,σ are deterministic, the following result is included in [11, Theorem 4.2]. The appearance of N makes the solution discontinuous, so that the argument in the proof of [11, Theorem 4.2] leading to the existence of weak solutions by proving the tightness of the approximating solutions is no longer valid. Moreover, since the coefficients are now random, the Yamada-Watanabe principle used there is invalid neither. Due to A1) and A2), the proof of the uniqueness and non-explosion is standard. To prove the existence, we approximate the original equation by those with Lipschitz coefficients, and construct a strong solution to the original equation by proving that the approximating solutions form a Cauchy sequence under the topology of locally uniform convergence. Theorem 3.1. Let b,σ,γ satisfy A1) and A2) with b = 0, σ = 0 and γ = 0. Then for any s 0 and F s -measurable random variable ξ on C, the equation dxt) = bt,x t )dt+σt,x t )dbt)+ γt,x t,z)ndt,dz), t s,x s = ξ has a unique solution which satisfies sup t [s r 0,T] ) Xt) F s <, T > s. 11

12 Proof. Without loss of generality, we only prove for s = 0 and simply denote 0 = F 0 ). a) 0 sup t T Xt) < for any T > 0. Let Xt) be a solution to the equation. Let τ n = inf{t 0 : Xt) ξ +n}, n 1. By the Itô formula, the Burkholder inequality, A1), A2), and the Jensen inequality, for any T > 0 we may find a constant CT) > 0 such that for any n 1, the process φ n t) := 0 sup s t Xs τ n ) satisfies Let Gs) = s 1 t 0 φ n t) CT)+ ξ +CT) 0 φ n s)u 0 φ n s))ds, t [0,T]. 0 1 dr,s > 0. By the Bihari inequality we have rur) 0 φ n t) G 1 GCT)+ ξ )+CT)t ) <, t [0,T]. Letting n, we conclude that τ n and 0 sup t T Xt) <. b) The uniqueness of the solution. Let Xt) and Xt) be two solutions to the equation with the same initial data X 0. By the Itô formula, the Burkholder inequality, A1) and A2), and the Jensen inequality, for any T > 0 we may find a constant CT) > 0 such that the process φt) := sup s t Xs) Xs) satisfies 0 φt) CT) t 0 0 φs)u 0 φs))ds, t [0,T]. Since sus) 0φt) = 0 for t [0,T]. This implies that Xt) = Xt) for all t 0 since T > 0 is arbitrary. c) xistence of the solution for bounded b,σ and θ := γ,,z) νdz). If u 1, i.e. the coefficients are Lipschitz continuous in ξ C with respect to the uniform norm, then the existence and uniqueness of the solutions can be proved by a standard argument cf. [13]). To prove the existence of the solution, we approximate the coefficients by using Lipschitz ones as follows. Let µ be the distribution of the C-valued random variable B with Bs) := Br s),s [ r 0,0], where Bs) is a d-dimensional Brownian motion with B0) = 0. For any n 1, let b n t,ξ) = bt,ξ +n 1 η)µdη), σ n t,ξ) = σt,ξ +n 1 η)µdη), C C γ n t,ξ,z) = γt,ξ +n 1 η,z)µdz), t 0,ξ C,z. C Since b,σ and θ are bounded, applying [2, Corollary 1.3] for σ = 1I n d d,m = 0,Z = b = 0 and T = 1+r 0, we conclude that for any n 1, b n t,ξ) b n t,η) + σ n t,ξ) σ n t,η) HS + γ n t,ξ,z) γ n t,η,z) νdz) K n t) ξ η 12

13 holds for some positive K n C[0, )). Therefore, the equation 3.1) dx n) t) = b n t,x n) t )dt+σ n t,x n) t )dbt) + γ n t,x n) t, z)ndt, dz) for X n) 0 = ξ has a unique solution. Moreover, by the Jensen inequality, we see that A1) and A2) hold for b n,σ n and γ n uniformly in n 1. Next, by A1) we may find a positive function K C[0, )) such that b n t,ξ) b l t,ξ) + σ n t,ξ) σ l t,ξ) HS + γ n t,ξ,z) γ l t,ξ,z) νdz) Kt)ε n,l, where, according to µ ) <, sus) c1+s) for some constant c > 0, and sus) 0 as s 0, ε n,l := n 1 l 1 )η u n 1 l 1 )η )µdη) 0 as n,l. C Combining this with A1) we obtain b n t,ξ) b l t,η) + σ n t,ξ) σ l t,η) HS + 3.2) Kt)ε n,l + b n t,ξ) b n t,η) + σ n t,ξ) σ n t,η) HS + γ n t,ξ,z) γ n t,η,z) νdz) Kt)ε n,l +Kt) ξ η u ξ η ), t 0,ξ,η C γ n t,ξ,z) γ l t,η,z) νdz) for some positive K C[0, )). Moreover, since A1) and A2) hold for b n,σ n and γ n uniformly in n, by a) we have 3.3) sup n 1 0 sup X n) t) <, T > 0. t T Now, as in a) and b), by the Itô formula, the Burkholder inequality, A1) and A2) holding for b n,σ n and γ n uniformly in n, the Jensen inequality and 3.2), for any T > 0 we may find a constant CT) > 0 such that the process φ n,l t) := sup s t X n) s) X l) s) satisfies 0 φ n,l t) CT) t 0 0 φ n,l s)u 0 φ n,l s))ds+ct)εn,l), t [0,T]. Since εn,l) 0 as n,l, by the Bihari inequality and 1 0 lim 0sup X n) t) X l) t) = 0, T > 0. n,l t T 1 ds =, we obtain sus) Therefore, as n the process X n) converges locally uniformly to a process X, which solves the first equation in 1.1) according to A1), 3.3) and the facts that sus) c1+s) for some constant c > 0 and sus) 0 as s 0. 13

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