EULER MARUYAMA APPROXIMATION FOR SDES WITH JUMPS AND NON-LIPSCHITZ COEFFICIENTS

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1 Qiao, H. Osaka J. Math. 51 (14), EULER MARUYAMA APPROXIMATION FOR SDES WITH JUMPS AND NON-LIPSCHITZ COEFFICIENTS HUIJIE QIAO (Received May 6, 11, revised May 1, 1) Abstract In this paper we show that stochastic differential equations with jumps and non-lipschitz coefficients have (, W, N p )-pathwise unique strong solutions by the Euler Maruyama approximation. Moreover, the Euler Maruyama discretisation has an optimal strong convergence rate. 1. Introduction Consider the following stochastic differential equation (SDE) with jumps in Ê d : t t t (1) Y t b(s, Y s ) ds (s, Y s ) dw s f (s, Y s, u) N É p (ds du), where bï Ê Ê d Ê d, Ï Ê Ê d Ê d m, f Ï Ê Ê d Í Ê d, are Borel measurable functions; {W t, t } is an m-dimensional standard Brownian motion defined on some complete filtered probability space (Đ, F, {F t } t, P) and {p t, t } is a stationary Poisson point process of the class (quasi left-continuous) defined on (Đ, F, {F t } t, P) with values in Í and characteristic measure, is an Ê d -valued F -measurable square integrable random variable. Here is a -finite measure defined on a measurable Ê normed space (Í, U) with the norm Í. Fix Í ¾ U with (Í Í ) ½ and Í u (du) ½. Let N Í p((, t], du) be the counting measure of p t such that N p ((, t], A) t(a) for A ¾ U. Denote ÉN p ((, t], du) Ï N p ((, t], du) t(du), the compensator of p t. In this paper, we study existence and uniqueness of solutions to Equation (1) under non-lipschitz conditions. Firstly, introduce some concepts. Given W and N p on a probability space, we recall that a strong solution to Equation (1) is a càdlàg process Y which is adapted to the filtration generated by W and N p and satisfies Equation (1). A weak solution of Equation (1) is a triple (Y, W, N p ) on a filtered probability space 1 Mathematics Subject Classification. 6H1, 6J75, 34A1. This work is ported by NSF (No ) of China. Í

2 48 H. QIAO (Đ, F, {F t } t, P) such that Y t is adapted to F t, W t and N p ((, t], du) are F t -Wiener process and F t -Poisson process, respectively, and (Y, W, N p ) solves Equation (1). We say that (, W, N p )-pathwise uniqueness holds for Equation (1) if for any filtered probability space (Đ ¼,F ¼,{Ft ¼} t, P ¼ ) carrying ¼, W ¼ and N p ¼ such that the joint distribution of ( ¼, W ¼, N p ¼) is the same as that of the given (, W, N p ), Equation (1) with ¼, W ¼ and N p ¼ instead of, W and N p cannot have more than one weak solution. Let us recall some recent results. By Euler Maruyama approximation, Skorokhod [1] proved that Equation (1) has a weak solution. And by smooth approximation of the coefficients, Situ [11, Theorem 175, p. 147] showed that Equation (1) has a weak solution. However, by successive approximation Cao, He and Zhang [] obtained that Equation (1) has a pathwise unique strong solution. Here we use the Skorokhod weak convergence technique and Lemma 1.1 in [3] to show that Equation (1) has a (, W, N p )-pathwise unique strong solution. Our approach is very close in spirit to the Yamada Watanabe theorem. As we know, it seems the first time to show solutions to stochastic differential equations with jumps by the method. Next assume the following non-lipschitz conditions for Equation (1): for all (t, x, y) ¾ Ê Ê d Ê d, (H b ) b(t, x) b(t, y) (t)x y 1 (x y), where stands for Euclidean norm in Ê d, (H ) (t, x) (t, y) (t)x y (x y), where is the Hilbert Schmidt norm from Ê m to Ê d, (H 1 f ) holds for p ¼ and 4, (H f ) T (H b,, f ) f (t, x, u) f (t, y, u) p¼ (du) (t)x y p¼ 3 (x y), Í lim h f (t h, x, u) f (t, x, u) (du), Í b(t, ) (t, ) Í f (t,, u) (du) dt ½, for any T, where (t) is locally square integrable and i is a positive continuous function, bounded on [1, ½) and satisfying () lim x i (x) log x 1 Æ i ½, i 1,, 3.

3 EULER MARUYAMA APPROXIMATION FOR SDES 49 (3) and REMARK 1.1. In [] b,, f satisfy for x, y ¾ D(Ê, Ê d ) b(t, x) b(t, y) (t, x) (t, y) (t) T rt x r y r, Í f (t, x, u) f (t, y, u) (du) b(t, ) (t, ) f (t,, u) (du) dt ½, Í for any T, where Ï Ê Ê is concave nondecreasing continuous function such that () and Ê (1 (u)) du ½, and D(Ê, Ê d ) stands for the space of all càdlàg mappings from Ê to Ê d. Comparing our conditions with (3), one will find that in our conditions the modulus of continuity for b in x is different from that for, which is convenient to control. In Section 4, we give an example to demonstrate it. Besides, b, and f do not depend on all the path but only on the value at t. Therefore, our conditions are more general. One of our aims is to prove the following result. Theorem 1.. Suppose that dim Í ½ and the coefficients b, and f satisfy (H b ), (H ), (H 1 f ), (H f ) and (H b,, f ). Then Equation (1) has a (, W, N p )-pathwise unique strong solution. To prove Theorem 1., we construct the Euler Maruyama approximation Y n for the solution of Equation (1). In [6], Higham and Kloeden also constructed the Euler Maruyama approximation of the solution to Equation (1) and showed the Euler Maruyama discretisation has a strong convergence order of one half. However, they required that b satisfies a one-sided Lipschitz condition and and f satisfy global Lipschitz conditions. Our other aim is to prove that under our non-lipschitz conditions the Euler Maruyama discretisation still has a strong convergence order of one half. We shall also consider the more general equation (4) X t t t b(s, X s ) ds t (s, X s ) dw s Í Í g(s, X s, u) N p (ds du), t Í f (s, X s, u) É N p (ds du) where g Ï Ê Ê d Í Ê d is a Borel measurable function. In Section 3 we show Equation (4) has a (, W, N p )-pathwise unique strong solution. In Section 4 an example is given to illustrate our result. Section 5 is Appendix to justify some conditions.

4 5 H. QIAO Throughout the paper, C with or without indices will denote different positive constants (depending on the indices) whose values are not important.. Proof of Theorem 1. Before proceeding to our proof, we first prepare a crucial lemma. Under (H b ), (H ) and (H 1 f ), pathwise uniqueness holds for Equa- Lemma.1. tion (1). Proof. Consider any filtered probability space ( ÉĐ, ÉF, { ÉF t } t, P) É carrying É, ÉW and N Ép such that the joint distribution of (É, ÉW, N Ép ) is the same as that of the given (, W, N p ). Assume that Yt 1, Y t are two weak solutions to Equation (1) with É, ÉW and N Ép instead of, W and N p. Set Z t Ï Yt 1 Yt. Then Z t t (b(s, Y 1 s ) b(s, Y t By Itô s formula we have that ( f (s, Ys 1 Í s )) ds t ( (s, Y 1 s ) (s, Y s )) d ÉW s, u) f (s, Y s, u)) É N Ép (ds du). where A 1 t i M c t i, j t t Z t Ï A 1 t M c t A t M d t A 3 t, Zs i (bi (s, Ys 1 ) bi (s, Ys )) ds, Z i s ( i j (s, Y 1 s ) i j (s, Y s )) d ÉW j s, A t M d t A 3 t t t (s, Y 1 s ) (s, Y s ) ds, t Í (Z s f (s, Ys 1 Í Z s f (s, Y 1 s, u) f (s, Y s, u) Z s ) É N Ép (ds du),, u) f (s, Y s, u) Z s i Z i s ( f i (s, Y 1 s, u) f i (s, Y s, u)) (du) ds.

5 For A 1 t, A t t¾[,t ] T EULER MARUYAMA APPROXIMATION FOR SDES 51 and A 3 t, by (H b), (H ), (H 1 f ) and Taylor s formula it holds that A 1 t T T T T C T C t¾[,t ] A t Z s b(s, Y 1 s ) b(s, Y f (s, Ys 1 Í (s)z s 1 (Z s ) ds (s)z s 3 (Z s ) ds (s) (Z s ) ds (s) u¾[,s] t¾[,t ] A 3 t s ) ds T, u) f (s, Y s, u) (du) ds T Z u ds, (s, Y 1 s ) (s, Y s ) ds (s)z s (Z s ) ds where (x) Ï x log x 1, x, (log 1 1)x, x, for 1e. Using (H ), Burkholder s inequality and Young s inequality gives that t¾[,t ] M c t C C T s¾[,t ] Z s (s, Y 1 s ) (s, Y s ) ds Z s T Z s C s¾[,t ] Z s C s¾[,t ] 1 (s, Y 1 s ) (s, Y s ) ds T T (s) (Z s ) ds (s) u¾[,s] 1 Z u ds.

6 5 H. QIAO By (H 1 f ), Burkholder s inequality, Young s inequality and the mean value theorem we obtain that t¾[,t ] M d t T C T C T (Z s f (s, Ys 1 Í Z s f (s, Ys 1 Í f (s, Ys 1 Í 1, u) f (s, Ys, u) Z s ) N Ép (ds du), u) f (s, Y s, u) N Ép (ds du), u) f (s, Y s, u)4 N Ép (ds du) T 1 C Z s f (s, Y 1 s, u) f (s, Ys, u) N Ép (ds du) Í T 1 C f (s, Y 1 s, u) f (s, Ys, u)4 N Ép (ds du) Í T 1 C Z s f (s, Y 1 s, u) f (s, Ys, u) N Ép (ds du) s¾[,t ] Í T C Z s f (s, Ys s¾[,t ] Í 1, u) f (s, Ys, 1 u)4 I Z s {Zs } N Ép (ds du) T 4 Z s C (s) Z u ds. s¾[,t ] u¾[,s] The above estimates give that t¾[,t ] Z t C T (s) Applying Lemma 3.6 in [8] yields the conclusion. We now give u¾[,s] 1 Z u ds. Proof of Theorem 1.. We divide the proof into four steps. STEP 1. We define two processes. Let t n t n 1 t n t n i t n i1 be a sequence of partitions of Ê such that for every T (ti1 n ti n ) i Ï ti1t n

7 EULER MARUYAMA APPROXIMATION FOR SDES 53 as n ½. We define the Euler Maruyama approximation as the process {Y n t } satisfying (5) Y n t t t t b(s, Yk n n (s) ) ds (s, Yk n n (s) ) dw s Í f (s, Y n k n (s), u) É N p (ds du), where k n (s) Ï ti n for s ¾ [ti n, t i1 n ). Set t Ï t Í u É N p (ds du) t Í Í un p (ds du). STEP. We obtain some sequences. Now take two subsequences {Y l }, {Y m } of the approximation {Y n }. To four sequences of processes {Y l }, {Y m }, {W n Á W n W} and { n Á n } on (Đ, F, P), by Appendix and [1, Corollary, p. 13] there exist subsequences, m( j), a probability space ( ÇĐ, ÇF, P), Ç carrying B([, T ]) ÇF-measurable sequences Y Ç, Y Æm( j), W Ç j and Ç j, where B([, T ]) denotes Borel -algebra on [, T ], such that for every positive integer j finite dimensional distributions of ( Y Ç, Y Æm( j), W Ç j, Ç j ) and (Y, Y m( j), W, ) coincide, and for any t ¾ [, T ] ÇY t Yt Ç, W Ç j t Wt Ç, m( j) ÆY t Yt Æ, Ç j t Ç t, in probability as j ½, where Y Ç, Y Æ, W Ç and Ç are some B([, T ]) ÇF-measurable stochastic processes. By [1, Lemma 4, p. 65] and [11, Remark 18, p. 94] there exist Poisson point processes { Çp j } and Çp such that Set ÉN Çp j (ds du) N Çp j (ds du) (du) ds, ÉN Çp (ds du) N Çp (ds du) (du) ds, Ç j t Ï Ç t Ï t t Í u É N Çp j (ds du) Í u É N Çp (ds du) t t Í Í un Çp j (ds du), Í Í un Çp (ds du). ÇF j t Ï ( Y Ç, W Ç j s, N Çp j ((, s], du), s t), s ÇF t Ï ( Ç Ys, Ç Ws, N Çp ((, s], du), s t). Then for every j, ( Ç W j t, N Çp j ((, t], du), ÇF j t ) and ( Ç Wt, N Çp ((, t], du), ÇF t ) are Wiener processes and Poisson processes.

8 54 H. QIAO STEP 3. We show that Y Ç satisfies Equation (1). { Y Ç t } satisfies the following equation (6) By (15), ÇY t Y Ç Ç t t tt b(s, Ç Y k (s) ) ds t Í f (s, Ç Y k (s), u) É N Çp j (ds du). Y Ç t tt (s, Ç Y k (s) ) d Ç W j s Yt C T. Because Y Ç t converges to Yt Ç in probability for every t, we can choose a subsequence which is denoted by again, such that P-a.s. as j ½ ÇY t Yt Ç, t r k, k 1,,, where {r k, k 1,, } [, T ] are rational numbers in [, T ]. By Fatou s lemma, it holds that Ç Yt Ç Yrk Ç Ç ÇY tt Ç k lim inf j½ C T. Ç k lim j½ r k Y Ç r k lim inf k j½ Thus it follows from Chebyshev s inequality that lim N½ ÇP lim N½ st ÇP st Ç Y k (s) N lim N½ Ç Ys N lim N½ Therefore for any there exists a N such that ÇP st If we prove for any Æ (7) ÇP 1 st Y Ç ÇP ÇP st st Ç tt Y Ç t Y Ç s N, Ç Ys N Y Ç k (s) N P Ç Ys Ç N st k (s) N 1 st Ç Ys N t.. t b(s, Y Ç k (s) )ds b(s, Ys Ç )ds Æ 3 6,

9 (8) (9) ÇP 1 st Y Ç ÇP 1 st Y Ç EULER MARUYAMA APPROXIMATION FOR SDES 55 t t (s, Y Ç k (s) )d W Ç j s 6, k (s) N 1 st Ys Ç N t k (s) N 1 st Ç Ys N t Í f (s, Ç Y k (s),u) É N Çp j (dsdu) Í (s, Ç Ys )d Ç Ws Æ 3 f (s, Ys Ç,u) N É Çp (dsdu) Æ 3 6, then t b(s, Ç Y k (s) ) ds t t b(s, Ç Ys ) ds t (s, Y Ç k (s) ) d W Ç j s (s, Ç Ys ) d Ç Ws t t Í Í f (s, Ç Y k (s), u) É N Çp j (ds du) f (s, Ç Ys, u) É N Çp (ds du) in probability. Taking the limit in (6) as j ½, one sees that Y Ç satisfies Equation (1). Making use of (H b ) and (H ) and the convergence of Y Ç t and W Ç j t to Yt Ç and Wt Ç, respectively, we can, by the same way to the proof of the theorem of Section 3, Chapter in [1], show that (7) and (8) hold. Ê tê To (9), we insert a stochastic integral Í f (s, Ys Ç, u) N É Çp j (ds du) (adapted to the filtration ( ÇF t ) t ) and obtain t ÇP 1 st Y Ç k (s) N 1 st Ys Ç N f (s, Ç Y k (s), u) N É Çp j (ds du) Í Ç P 1 st Ç Y Ç P 1 st Ç Y Ï I 1 I. k (s) N 1 st Ç Ys N t k (s) N 1 st Ç Ys N Í t t f (s, Ys Ç, u) N É Çp (ds du) Æ 3 Í f (s, Ç Y k (s), u) É N Çp j (ds du) Í t Í t f (s, Ys Ç, u) N É Çp j (ds du) Æ 6 f (s, Ç Ys, u) É N Çp j (ds du) Í f (s, Y Ç s, u) N É Çp (ds du) Æ 6 For I 1, by Chebyshev s inequality and Burkholder s inequality we obtain that I 1 36 Æ Ç 1 st Ç Y k (s) N 1 st Ç Ys N t Í (s, Ç Y k (s), u) f (s, Ç Ys, u) (du) ds.

10 56 H. QIAO We claim that for every s ¾ [, t] 1 st Ç Y k (s) N 1 st Ç Ys N Í (s, Ç Y k (s), u) f (s, Ç Ys, u) (du) in probability as j ½. In fact, for any ÇP 1 st Y Ç k (s) N 1 st Ys Ç N (s, Ç Y k (s), u) f (s, Ys Ç, u) (du) Í Ç P 1 Ç Y k (s) Ç Ys h 1 st Ç Y k (s) N 1 st Ç Ys N Í (s, Ç Y k (s), u) f (s, Ç Ys, u) (du) Ç P{ Ç Y k (s) Ç Ys h}. Take a small enough h and then by (H 1 f ) the claim is justified. Thus by dominated convergence theorem it holds that I 1 1. To I, we calculate. For any ±, I Ç P 1 st Ç Y Ç P 1 st Ç Y Ç P 1 st Ç Y Ï I 1 I I 3. k (s) N 1 st Ç Ys N k (s) N 1 st Ç Ys N k (s) N 1 st Ç Ys N t Í {u Í ±} t Í {u Í ±} t Í {u Í ±} t Í {u Í ±} And by Chebyshev s inequality and Burkholder s inequality I 1 I Because theorem 18 Æ Ç1st ÇY k (s) N 1 st Ç Ys N Ç 1st Ç Ys N Ê t Ê t Ç1 st Ys Ç N Í {u Í ±} t Ç Í {u Í ±} t f (s, Ys Ç,u) N É Çp j (dsdu) Æ 18 f (s, Ys Ç,u) N É Çp (dsdu) Æ 18 f (s, Ç Ys,u) É N Çp j (dsdu) Í {u Í ±} f (s, Ys Ç,u) N É Çp (dsdu) Æ 18 f (s, Ç Ys, u) (du) ds. Í {u Í ±} f (s, Ç Ys, u) (du) ds ½, by Fubini s f (s, Ç Ys, u) (du) ds 1 st Ç Ys N f (s, Ç Ys, u) ds (du).

11 EULER MARUYAMA APPROXIMATION FOR SDES 57 Noting { u Í ±} for ±, by absolute continuity of the Lebesgue integral one can take a small enough ± such that I 1 I 4. Finally, we treat I 3. Consider the partition sequence on [, t]: t Én t Én 1 t Én Én t, such that lim Én½ max k (t Én k1 t Én k ). I 3 Ç P 1 st Ç Y k (s) N 1 st Ç Ys N t Í {u Í ±} Én 1 t Én k1 k t Én k f (s, Ç Ys, u) É N Çp j (ds du) Í {u Í ±} f (k Én (s), Yk Ç Én (s), u) N É Çp j (ds du) Æ 54 µ Ç P 1 st Ç Y k (s) N 1 st Ç Ys N Én 1 k tk Én Én 1 t Én k1 k t Én k1 t Én k Í {u Í ±} Í {u Í ±} f (k Én (s), Ç Yk Én (s), u) É N Çp j (ds du) f (k Én (s), Yk Ç Én (s), u) N É Çp (ds du) Æ 54 µ Ç P 1 st Ç Y k (s) N 1 st Ç Ys N 54 Æ t Én 1 t Én k1 k tk Én t Í {u Í ±} Ç1st ÇY k (s) N 1 st Ç Ys N Í {u Í ±} Í {u Í ±} f (s, Ç Ys, u) f (k Én (s), Ç Yk Én (s), u) É N Çp (ds du) f (s, Ys Ç, u) N É Çp (ds du) Æ 54 Én 1 k f (k Én (s), Ç Yk Én (s), u)1 [t Én k,t Én k1 )(s) (du) ds µ Ç P 1 st Ç Y k (s) N 1 st Ç Ys N Én 1 t Én k1 k tk Én t Én k1 t Én k Í {u Í ±} Í {u Í ±} f (k Én (s), Ç Yk Én (s), u) N Çp j (ds du) f (k Én (s), Yk Ç Én (s), u) N Çp (ds du) Æ. 54 µ

12 58 H. QIAO From (H 1 f ), (H f ) and [1, Lemma 4, p. 65] it follows I 3 4. STEP 4. We show that Equation (1) has a (, W, N p )-pathwise unique strong solution. In the same way as Step 3 we can prove that Y Æ satisfies Equation (1). Since the initial values in both cases are the same ( Y Ç Y Æm( j) m( j) because Y Y ) and the joint distribution of the initial value, W Ç and N Çp coincides with distribution of, W and N p, by Lemma.1 we conclude that Yt Ç Yt Æ for all t (a.s.). Hence, by applying Lemma 1.1 in [3] we obtain that Yt n converges in probability to Y t in (Đ, F, P). By the same way as Step 3 it holds that Y satisfies Equation (1). 3. The convergence rate for the Euler Maruyama approximation In the section we consider the convergence rate for the Euler Maruyama approximation {Yt n} defined in (5), that is, for a fixed timestep ½t and t i n i½t, Y n t t t where k n (s) t n i for s ¾ [t n i, t n i1 ). t b(s, Yk n n (s) ) ds (s, Yk n n (s) ) dw s Í f (s, Y n k n (s), u) É N p (ds du), Theorem 3.1. Suppose Í Ê d and b, and f satisfy those conditions in Theorem 1.. Moreover, b, are independent of t and f (t, x, u) f (x)u, where f (x) is a real function in x. Then there exists a T such that Yt n Y t O(½t), tt where O(½t) means that O(½t)½t is bounded. Proof. Set H t Ï Yt n Y t and then H t satisfies the following equation H t t (b(y n k n (s) ) b(y s)) ds t t (Y n k n (s) ) (Y s) dw s Í ( f (Y n k n (s) ) f (Y s ))u É N p (ds du). By Itô s formula we obtain that H t Ï J 1 J J 3 J 4 J 5,

13 EULER MARUYAMA APPROXIMATION FOR SDES 59 where J 1 i J i, j t t H i s (bi (Y n k n (s) ) bi (Y s )) ds, H i s ( i j (Y n k n (s) ) i j (Y s )) dw j s, J 3 J 4 J 5 t t (Y n k n (s) ) (Y s) ds, t Í Í (H s f (Y n k n (s) )u f (Y s )u H s ) É N p (ds du), H s f (Y n k n (s) )u f (Y s )u H s Hs i ( f (Yk n n (s) )ui f (Y s )u i ) (du) ds. i For T and J 1, J 3 and J 5, by the same technique as that of dealing with A 1 t, A t and A 3 t in Lemma.1, one can get (1) J 5 J 1 J 3 tt tt tt T C (H s ) ds 1 T T H s ds T C (Yk n n (s) Ys n ) ds T C H r ds 1 T H r ds rs rs T 1 Yk n n (r) Yr n ds rs T C Yk n n (r) Yr n ds, rs where Jensen s inequality is used in the last inequality. For J, by the same means as that of dealing with M 1 t (11) tt J 14 C st T H s C rs T (Y n k n (s) Y n s ) ds in Lemma.1, we have rs Yk n n (r) Yr n ds. H r ds

14 6 H. QIAO For J 4, by the similar method to that of dealing with Mt can obtain (1) Next, for t n i Y n s Y n t n i Y n t n i tt J 4 4 H s C st st C T rs s ti1 n, it follows from (5) that s t n i b(y n t n i s b(yk n n (r) ) dr (Yk n n (r) ) dw r )(s t n i ) (Y n t n i t n i H r ds. s t n i )(W s W t n i ) f (Yt n ) i n (15) and Burkholder s inequality admit us to get (13) t n i st n i1 Y n k n (s) Y n s 3Ct n i1 t n i 3C 3C C½t, t n i st n i1 where the last constant C is independent of ½t. Combining (1), (11), (1) and (13), we have tt H r s t n i in Lemma.1, one Y n k n (s) Y n s Í s t n i f (Y n k n (r) )u É N p (dr du) t n i st n i1 T H t C ds 1 T rs T Yk n n (r) Y n rs T C Yk n n (r) Yr n ds rs C Yk n n (s) Ys n st T C rs CT (C(½t) ) C½t. Í u É N p (dr du). Í u É N p (drdu) W s W t n i rs r 1 ds H r ds T (C(½t)1 ) H r ds

15 By Lemma.1 in [13] it holds that EULER MARUYAMA APPROXIMATION FOR SDES 61 tt H t A exp{ CT}, where A T (C(½t)1 ) CT (C(½t) ) C½t. Thus, there exists a T such that The proof is completed. Yt n Y t O(½t). tt 4. Existence of a (, W, N p )-pathwise unique strong solution for Equation (4) Theorem 4.1. Suppose dim Í ½. Under (H b ), (H ), (H 1 f ), (H f ) and (H b,, f ), then Equation (4) has a (, W, N p )-pathwise unique strong solution. Proof. let D p be the domain of p t and D {s ¾ D p Ï p s ¾ Í Í }. Since (Í Í ) ½, D is a discrete set in (, ½) a.s. Set 1 n be the enumeration of all elements in D. It is easy to see that n is a stopping time for each n and lim n½ n ½ a.s. (We disregard the trivial case of (Í Í ).) Set Xt 1 Yt, t ¾ [, 1 ), Y 1 g( 1, Y 1, p 1 ), t 1. The process {X 1 t, t ¾ [, 1]} is clearly the unique solution of Equation (4) in the time interval [, 1 ]. Next, set ÉF t F t1, É X X 1 1, ÉW t W t1 W 1, Ép t p t1 and É 1 1. We can determine the process X É t on [, É 1 ] with respect to ÉF t, X É, ÉW t, Ép t in the same way as Xt 1. Define X t by X 1 X t t, t ¾ [, 1 ], ÉX t 1, t ¾ [ 1, ]. It is easy to see that {X t, t ¾ [, ]} is the unique solution of Equation (4) in the time interval [, ]. Continuing this process, X t is determined uniquely in the time interval [, n ] for every n and hence X t is determined globally. Thus the proof is complete.

16 6 H. QIAO 5. An example Let b(t, x) Ï (t) k1 sin(kx) k, (t, x) Ï Ô (t)(1 3 sin x, 3 sin x,, m 3 sin mx), f (t, x, u) Ô (t) k1 Í {u ¾ Í, u Í 1}, g(t, x, u), sin (kx) k 3 u Í, where (t) is continuous, bounded on (, 1] and locally square integrable. Then by Lemma 3.1 and 4.1 in [1] and the Hölder inequality b(t, x) b(t, y) (t) k1 sin(kx) sin(ky) k (t) k1 sin(k(x y)) k (t, x) (t, y) (t) C(t)x yé 1 (x y), m sin(kx) sin(ky) k 3 k1 sin(kx) sin(ky) (t) k 3 4(t) k1 k1 sin(k(x y)) k 3 and C(t)x y É (x y), Í f (t, x, u) f (t, y, u) (du) C(t) C(t) k1 C(t) k1 sin (kx) sin (ky) k 3 k1 k 3 sin(kx) sin(ky) sin(kx) sin(ky) k 3 C(t)x y É (x y), k1 sin(kx) sin(ky) k 3

17 EULER MARUYAMA APPROXIMATION FOR SDES 63 where É 1 (x) Ï log x 1, x, log 1 1 x, x, and É (x) Ï log x 1, x, (((log 1 ) 1 (1)(log 1 ) 1 )x (1)(log 1 ) 1 ) x, x. We take 1 (x) Ï C É 1 (x) and (x) Ï C É (x). It is easily justified that 1 (x) and (x) satisfy (). Note that b(t, x) does not satisfy the condition (3) because log x 1 (log x 1 ) for x. Thus our result generalizes one in [] in some sense. 6. Appendix We show that {Y n }, {W n Á W n W} and { n Á n } satisfy conditions in [1, Corollary, p. 13], i.e. (14) (i) (ii) tt lim lim N½ n½ tt Æ, lim lim h n½ P{ n t N}, s th T P{ n s n t Æ}. Firstly we deal with {Y n }. It holds by Burkholder s inequality and (H b,, f ) that T Yt n 4 C 1 Yk n n (s) dt Gronwall s inequality gives that (15) 4 C tt 1 Y n t C T, st st Y n s dt. where C T is independent of n. Then applying Chebyshev s inequality yields that lim lim P{Y n N½ n½ t N} lim lim P Y n tt N½ n½ t N tt lim lim 1 N½ n½ N Yt n lim N½ 1 N n tt tt Y n t.

18 64 H. QIAO Assume t s and then Yt n Ys n t s t b(r, Yk n n (r) ) dr (r, Yk n n (r) ) dw r t s Í s f (r, Y n k n (r), u) É N p (dr du). By (H b,, f ), (15) and Burkholder s inequality we obtain that Y s (n) Y t (n) C T s t, where C T is independent of n. Then for any Æ, lim lim h n½ s th P{Ys n Yt n Æ} lim lim h n½ lim s th h n s th 1 Æ Y n s Y n t 1 Æ Y n s Y n t. By tt W t 4T and (W s W t ) s t we know that {W t, t } satisfies (14). Because (Í Í ) ½, by Chebyshev s inequality and Burkholder s inequality it holds that lim P{ t Í N} lim P t Í N N½ tt N½ tt t lim P N½ u N É p (ds du) tt Í N Í lim P t N½ u N p (ds du) Í Í N lim N½ tt 4 N P tt N½ lim lim N½ 16 T N tt t Í u É N p (ds du) Í p(t) Í 1 Í Í (p(t)) N u Í(du) ds, Í Í µ

19 and lim h s th EULER MARUYAMA APPROXIMATION FOR SDES 65 P{ s t Í Æ} lim h s th lim lim P h s th h s th lim lim h s th h s th lim lim h s th h s th t P s t 4 t Æ s P 4 t Æ s From this we obtain that { t, t } satisfies (14). s Í u É N p (dr du) Í Æ Í Í u N p (dr du) Í u É N p (dr du) Í Í srtp(r) Í 1 Í Í (p(r)) Æ u Í(du) dr Í P{N p ((s, t] (Í Í )) 1} (1 e s t(í Í ) ). ACKNOWLEDGEMENTS. The author would like to thank Professor Xicheng Zhang for his valuable discussions and also wish to thank referees for suggestions and improvements. Æ µ References [1] H. Airault and J. Ren: Modulus of continuity of the canonic Brownian motion on the group of diffeomorphisms of the circle, J. Funct. Anal. 196 (), [] G. Cao, K. He and X. Zhang: Successive approximations of infinite dimensional SDEs with jump, Stoch. Dyn. 5 (5), [3] I. Gyöngy and N. Krylov: Existence of strong solutions for Itô s stochastic equations via approximations, Probab. Theory Related Fields 15 (1996), [4] D.J. Higham and P.E. Kloeden: Numerical methods for nonlinear stochastic differential equations with jumps, Numer. Math. 11 (5), [5] D.J. Higham and P.E. Kloeden: Convergence and stability of implicit methods for jump-diffusion systems, Int. J. Numer. Anal. Model. 3 (6), [6] D.J. Higham and P.E. Kloeden: Strong convergence rates for backward Euler on a class of nonlinear jump-diffusion problems, J. Comput. Appl. Math. 5 (7), [7] N. Ikeda and S. Watanabe: Stochastic Differential Equations and Diffusion Processes, second edition, North-Holland, Tokyo, [8] X. Mao: Adapted solutions of backward stochastic differential equations with non-lipschitz coefficients, Stochastic Process. Appl. 58 (1995), [9] P.E. Protter: Stochastic Integration and Differential Equations, second edition, Springer, Berlin, 4.

20 66 H. QIAO [1] K. Sato: Lévy Processes and Infinitely Divisible Distributions, Cambridge Univ. Press, Cambridge, [11] R. Situ: Theory of Stochastic Differential Equations with Jumps and Applications, Springer, New York, 5. [1] A.V. Skorokhod: Studies in the Theory of Random Processes, Addison-Wesley Publishing Co., Inc., Reading, MA, [13] X. Zhang: Euler-Maruyama approximations for SDEs with non-lipschitz coefficients and applications, J. Math. Anal. Appl. 316 (6), Department of Mathematics Southeast University Nanjing, Jiangsu P.R. China