DISCRETE-TIME APPROXIMATIONS OF STOCHASTIC DELAY EQUATIONS: THE MILSTEIN SCHEME

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1 The Annals of Probability 24, Vol. 32, No. 1A, Institute of Mathematical Statistics, 24 DISCRETE-TIME APPROXIMATIONS OF STOCHASTIC DELAY EQUATIONS: THE MILSTEIN SCHEME BY YAOZHONG HU, 1 SALAH-ELDIN A. MOHAMMED 2 AND FENG YAN University of Kansas, Southern Illinois University and Williams Energy In this paper, we develop a strong Milstein approximation scheme for solving stochastic delay differential equations SDDEs). The scheme has convergence order 1. In order to establish the scheme, we prove an infinitedimensional Itô formula for tame functions acting on the segment process of the solution of an SDDE. It is interesting to note that the presence of the memory in the SDDE requires the use of the Malliavin calculus and the anticipating stochastic analysis of Nualart and Pardoux. Given the nonanticipating nature of the SDDE, the use of anticipating calculus methods in the context of strong approximation schemes appears to be novel. 1. Introduction. Discrete-time strong approximation schemes for stochastic ordinary differential equations SODEs) are well developed. For an extensive study of these numerical schemes, one may refer to [17], [18] and [19], Chapters 5 and 6. Some basic ideas of strong and weak orders of convergence are illustrated in [13]. If the rate of change of a physical system depends only on its present state and some noisy input, then the system can often be described by a stochastic ordinary differential equation SODE). However, in many physical situations the rate of change of the state depends not only on the present but also on the past states of the system. In such cases, stochastic delay differential equations SDDEs) or stochastic functional differential equations SFDEs) provide important tools to describe and analyze these systems. For various aspects of the qualitative theory of SFDEs the reader may refer to [2, 21] and the references therein. SDDEs and SFDEs arising in many applications cannot be solved explicitly. Hence, one needs to develop effective numerical techniques for such systems. Depending on the particular physical model, it may be necessary to design strong L p or almost sure) numerical schemes for pathwise solutions of the underlying SFDE. Strong approximation schemes for SFDEs may be used to Received April 21; revised December Supported in part by NSF Grants EPS , DMS and General Research Fund, University of Kansas. 2 Supported in part by NSF Grants DMS , DMS and DMS AMS 2 subject classifications. Primary 34K5, 6H7, 6H35; secondary 6C3, 6H1, 37H1, 34K28. Key words and phrases. Milstein scheme, Itô s formula, tame functions, anticipating calculus, Malliavin calculus, weak derivatives. 265

2 266 Y. HU, S.-E. A. MOHAMMED AND F. YAN simulate directly the a.s. stochastic dynamics of their trajectories or their random attractors. SFDEs are used to model population growth with incubation/gestation period [21]. In such models, one is often interested in estimating the actual population rather than its distribution and hence the need for strong approximation schemes. In this article, we will not consider the order of convergence of weak numerical schemes, although such schemes are useful for some applications of SODEs see [13, 17] and the references therein). In this connection, it is important to note that stochastic systems with memory do not correspond to deterministic PDEs in finitely many space variables) [2, 21]. Typically, a stochastic system with memory corresponds to an infinite-dimensional Feller diffusion whose principal coefficient degenerates on a hypersurface with finite-codimension [2], Chapter IV, Theorem 3.2 and [21], Theorem II.3). This aspect of SFDEs is in sharp contrast with the theory of SODEs where the latter theory has traditional ties to diffusions in Euclidean space. In a sense, the numerics of stochastic systems with memory resemble those of SPDEs in one space dimension. A strong Cauchy Maruyama scheme for a class of SFDEs with continuous memory, in the context of the Delfour Mitter state space R m L 2 [ τ,], R m ), was developed by Ahmed, Elsanousi and Mohammed [1]. See also [2], page 227, [15] and [4]. As in the case of SODEs, the Cauchy Maruyama scheme for SFDEs has order of convergence 1 2 [2], page 227, [15, 4, 8, 14]). In Sections 2 5, we establish the strong Milstein scheme for SDDEs with several delays. This scheme has a higher strong order of convergence 1 when compared with the Euler scheme which, as indicated above, has the strong order of convergence.5. Furthermore, when simulating the whole solution path {Xt),t [,a]}, the Milstein schemes for SDDEs and SODEs have the same complexity, even when one accounts for the simulation of the iterated stochastic integrals in the scheme. See Appendix B and the remarks therein.) Although the solution of the SDDE is adapted to the lagged) filtration of the driving noise, methods from anticipating stochastic analysis and the Malliavin calculus are necessary in order to derive an Itô formula for the segment of the solution process. The Itô formula is essential for the development of the Milstein scheme. In order to put our analysis in proper perspective, we highlight its essential features: a) The dynamics and the coefficients of the SDDEs are adapted, in fact, driven by Itô integrals; b) the formulation and implementation of the Milstein scheme do not require anticipating calculus ideas; c) the proof of convergence of the Milstein scheme as well as the Itô formula employ anticipating calculus techniques; d) anticipating calculus methods are used in the context of strong approximation schemes rather than weak ones where the Feynman Kac formula lends itself naturally to the use of Malliavin calculus methods); e) the application

3 DISCRETE-TIME APPROXIMATIONS OF SDDEs 267 of anticipating calculus methods seems unavoidable as soon as one seeks higherorder approximation results. In an essentially nonadapted setting, anticipating calculus methods have been used by Pardoux and Protter to study stochastic Volterra equations with anticipating coefficients. See [24] and the references therein. See also [7]. In order to describe our set-up, we need the following notation. Let R m be m-dimensional Euclidean space with the Euclidean norm x := x 2 1 xm 2, x = x 1,...,x m ) R m. Denote T := [,a], J := [ τ,], C := CJ; R m ),wherem is a positive integer, τ>isafixed delay [as in 1.6)] and a>. Furnish C with the supremum norm η C := sup τ s ηs) for all η C. Define the projection : C R mk associated with s 1,...,s k [ τ,] by 1.1) η) := ηs 1 ),..., ηs k ) ) R mk for all η C. DEFINITION 1.1. A function CT CJ; R m ); R) is tame if there exist φ CT R mk, R) and a projection : C R mk such that 1.2) t,η) = φ t, η) ) for all t T and η C. Let, F,P) be a probability space. For any continuous m-dimensional process X : [ τ,a] R m,definethesegment process X t, t [,a], by 1.3) X t u) := Xt u), t [,a], u [ τ,]. Observe that {X t } may be considered as a C-valued or L 2 J ; R m )-valued process. It is important that one should distinguish between the finite-dimensional current state Xt) and the infinite-dimensional segment X t, t [,a]. Assume that g : T R mk 1 LR d ; R m ) and h : T R mk 2 R m satisfy the following Lipschitz condition: 1.4) gt,x) gt,y) L x y, ht, z) ht, w) L z w for all t T, x,y R mk 1 and z, w R mk 2,whereL> is a constant, together with the boundedness condition [ ] 1.5) gt,) ht, ) <. sup t a Let 1 and 2 be two projections associated with two sets of points s 1,1,...,s 1,k1 [ τ,] and s 2,1,...,s 2,k2 [ τ,], respectively. Suppose

4 268 Y. HU, S.-E. A. MOHAMMED AND F. YAN {Wt):= W 1 t),..., W d t)) : t } is a d-dimensional standard Brownian motion defined on a filtered probability space, F,F t ) t,p)satisfying the usual conditions. Let η : C[ τ,]; R m ) be an F -measurable initial process. We will first consider the following class of Itô SDDEs: t η) g s, Xt) = 1 X s ) ) dws) h s, 2 X s ) ) ds, t, 1.6) ηt), τ t<. Under conditions 1.4) and 1.5), the SDDE 1.6) has a unique strong solution cf. [2], Theorem II.2.1, page 36, and Theorem V.4.3, pages 151 and 152). To see this, let Gt, η) := gt, 1 η)) and Ht,η) := ht, 2 η)) for t [,a], η C. It is easy to check that G and H satisfy the Lipschitz and local boundedness conditions with respect to the supremum norm on C) of Theorems II.2.1 and V.4.3 of [2]. Therefore, for each p 1, there exists a constant C = Cp,L,a) > such that 1.7) E X t 2p C C 1 E η 2p ) C for all η C,t [,a]. For any integers n, l 1, let π : t l <t l1 < < = t <t 1 <t 2 < <t n be a partition of [ τ,a]. Denote by π :=max l i n 1 t i1 t i ), the mesh of π. We now introduce the following Milstein scheme for the SDDE 1.6): 1.8) for t k <t t k1,and where 1.9) 1.1) X i,π t) = X i,π t k ) h i t k, 2 X π tk )) t tk ) g ij t k, 1 X π tk )) W j t) W j t k ) ) gij 1 j 1 tk, 1 X π tk )) u i 1 j 1,π t k s 1,j1 ) I j,j1 tk s 1,j1,t s 1,j1 ; s 1,j1 ) X π t) := η π t), t [ τ,], { u i 1j 1,π g i 1 j 1 t, 1 X t) := t π)), t,, τ t<, 1 s I j,j1 t s,t s; s) := t t s dw j t 2 ) dw j 1 t 1 ), t t, s [ τ,], and the starting path η π CJ,R m ) is prescribed e.g., a piecewise linear approximation of η using the partition points {t l,...,t }). In 1.8), X i, h i and g ij

5 DISCRETE-TIME APPROXIMATIONS OF SDDEs 269 denote coordinate representations of X, h and g with respect to standard bases in the underlying Euclidean spaces, and the Einstein summation convention is used for repeated indices. In order to establish strong convergence of the above Milstein scheme for the SDDE 1.6), it turns out, surprisingly, that one requires the use of anticipating calculus techniques developed by Nualart and Pardoux [23]. In particular, one needs to develop an infinite-dimensional Itô formula for tame functions acting on the segment X t of the solution X of 1.6). Such an Itô formula is given in Section 2, Theorem 2.3. The formula is proved via anticipating calculus methods [23]. To understand the need for anticipating calculus in such an intrinsically adapted setting, it is instructive to look at the following simple one-dimensional SDDE: dxt) = g Xt 1), Xt) ) dwt), t, Xt) = Wt), 1 t<, where g : R 2 R is a smooth function and Wt),t 1, is a one-dimensional Brownian motion. For a second-order scheme, we formally seek a stochastic differential of the coefficient gxt 1), Xt)) on the right-hand side of the above SDDE. For t, 1], this gives formally d { g Xt 1), Xt) )} = d { g Wt 1), Xt) )} = g ) Wt 1), Xt) dwt 1) x g ) ) Wt 1), Xt) g Xt 1), Xt) dwt) y second-order terms. Note that although the coefficient gxt 1), Xt)) is F t -measurable, the first term g x Wt 1), Xt)) dwt 1) in the right-hand side of the last equality is an anticipating differential. Furthermore, it appears that the F t ) t 1 -adapted process [, 1] t Xt 1), Xt)) R 2 is not a semimartingale with respect to any natural filtration. In addition to this difficulty, the components Xt 1) and Xt) are not independent, so the existing anticipating versions of Itô s formula do not apply cf. [2, 3] and [23]); hence the need for a new Itô formula for tame functions in order to justify the above computation. In Theorem 2.1 in the next section we establish such a formula. Using the above-mentioned Itô formula and appropriate estimates on the weak Cameron Martin derivatives of X, it is shown in Section 5 Theorem 5.2) that, under suitable regularity conditions on the coefficients of 1.6), one gets the following global error estimate for the Milstein approximations: 1.11) E sup Xt π X t q C Cq) π q t a

6 27 Y. HU, S.-E. A. MOHAMMED AND F. YAN for any q 1. This says that the Milstein scheme has strong order of convergence Itô s formula for tame functions. In order to derive higher-order numerical schemes for SDDEs, we shall first prove an Itô formula for tame functions on CJ,R m ) Definition 1.1). Suppose that Wt) := W 1 t),...,w d t)), t, is d-dimensional standard Brownian motion on a filtered probability space, F,F t ) t,p). Denote by D = D 1,...,D d ) the Malliavin differentiation operator associated with {Wt): t }. Assume t η) us) dws) vs)ds, t >, 2.1) Xt) = ηt), τ t, where η belongs to C and is of bounded variation, u = u 1,...,u m ) T, u i L 2,4 d,loc, v = v 1,...,v m ) T,andv i L 1,4 loc. One can refer to [22], pages 61, 151 and 161), for the definition of L k,p d. Note that the processes u and v may not be adapted to the Brownian filtration F t ) t. For convenience, we define ut) = for t<ort>a, {, t >a, vt) = η t), τ t. 2.2) We also set Wt)= ift<ort>a, and denote Ut):= Vt):= us) dws), η) vs)ds, t >, ηt), τ t. If u L 2,p loc for some p>4, then the indefinite Skorohod integral us) dws) has a continuous version. Hence, we may assume that the process Xt),t τ, is sample continuous. Let T =[,a], J =[ τ,], C = CJ,R m ) be as before, and let be the projection associated with s 1,...,s k J. Although there is a multidimensional Itô formula for φt,xt)) [2, 3] and [22]), we cannot apply it to φt, X t )) because U t ) is of the form 2.3) us s 1 )dws s 1 ),..., ) us s k )dws s k ), t > and the components of the dk-dimensional process Wt s 1 ),..., Wt s k )) are not independent. However, the ideas in [23], Section 6, and in [22], page 161, can be used to derive an Itô formula for φt, X t )). See [28] for further details.

7 DISCRETE-TIME APPROXIMATIONS OF SDDEs 271 We denote by 2.4) { 1, i = j, δ ij =, i j, the Kronecker delta. For any process Xt), t [ τ,a], denote its delayed) increments by 2.5) li X := Xt l s i ) Xt l 1 s i ), 1 l n, i = 1, 2,...,k. Assume that φ C 1,2 T R mk, R), andwrite 2.6) φt, x) := φt, x 1,..., x m ) where x := x 1,..., x m ), x i := x i1,...,x ik ) R k,1 i m. We now state an Itô formula for tame functions. THEOREM 2.1. Assume that X is a continuous process defined by 2.1), where η : J R m is of bounded variation, u = u 1,...,u m ) T, u i L 2,4 d,loc, v = v 1,...,v m ) T and v i L 1,4 loc. Suppose φ C1,2 T R mk, R). Then φ t, X t ) ) φ, X ) ) 2.7) = 1 2 φ s, Xs ) ) ds s k m i, i 1,j 1 =1 φ s, Xs ) ) d X s )) x 2 φ s, Xs ) ) u i 1 s s i ) 1 i 1 j D ssi X j 1 s s j )ds. REMARKS. 1. The Itô formula 2.7) may also be expressed in the form φ t, X t ) ) φ, X ) ) 2.8) where = 1 2 φ s, Xs ) ) ds s k i, φ s, Xs ) ) d X s )) x ] [ 2 φ Tr s, Xs ) ) s s i,s j ) ) x i x j s α, β) := 1 2{ u )s X s α, β) u ) s X s β, α) }, α,β [ τ,] ds,

8 272 Y. HU, S.-E. A. MOHAMMED AND F. YAN and the two-parameter process u ) s X s : J 2 LR m ; R m ) is defined by u ) s X s α, β) := I { sα β} us α) [ u T s α)i { sα sβ} sβ sβ ] D sα ur) dwr) D sα vr)dr for all α, β [ τ,]. 2. Suppose d = m = 1. Let us define a trace operator. For1 i, j k, define ± 2.9) s i,s j Xs) := lim Dssi Xs s j ε) ± D ssi Xs s j ε) ) R ε and ± s i Xs) := ± s i,s 1 Xs),..., ± s i,s k Xs)) R k. Then the Itô formula for tame functions can be written as φ t, X t ) ) φ, X ) ) φ = s, Xs ) ) φ 2.1) ds s, Xs ) ) d W s ) s x 1 k 2 φ s, Xs 2 xi 2 ) ) s i Xs), s i Xs) ds R d a.s. for all t T,where x := x 1,...,x k ) and, R d denotes the Euclidean inner product on R d. See [23], Remark The Itô formula 2.7) still holds if the initial path is an F -measurable process η : C with a.a. sample paths of bounded variation. A similar remark also holds for Theorem 5.2 of Section 5. For simplicity, we shall prove the Itô formula for the case d = m = 1. We thus assume in what follows that d = m = 1. PROOF OF THEOREM 2.1. For any integer n 1, let {π n := t <t 1 < < t n = a} be a partition of [,a]. Then by Taylor s theorem, we may write φ t, X t ) ) φ, X ) ) = [ φ tl, )) X tl φ tl 1, ))] X tl [ φ t l 1, X tl )) φ tl 1, X tl 1 ))]

9 DISCRETE-TIME APPROXIMATIONS OF SDDEs 273 where = { k φˆt l, )) X tl tl s 1 2 φ tl 1, X tl 1 )) li X k i, 2 φ tl 1, } )) X tl li X lj X, t T, t l := t l t l 1, X tl = X tl 1 α l Xtl X tl 1 ), ˆt l = t l 1 γ l t l for some random variables α l,γ l 1, l = 1,...,n. The Itô formula 2.1) will then follow from Propositions 2.3 and 2.4. The rest of this section is devoted to the proofs of Propositions PROPOSITION 2.2. Suppose that Wt) is a one-dimensional Brownian motion. Let u L 1,2 loc be such that ut) = if t>aor t<. Assume that τ s 1, s 2 and let π n := t <t 1 < <t n = a be a family of partitions of T =[,a], with π n as n. Then 2.11) [ n l s 1 lim n in probability. If s 1 s 2, then t l 1 s 1 us) dws) ] 2 as1 = u 2 s) ds 2.12) lim n l s 1 l s 2 us) dws) us) dws) = t l 1 s 1 t l 1 s 2 in probability. Furthermore, if u L 1,2, then the above convergences are in L 1, R). PROOF. We prove the proposition for u L 1,2. The general case u L 1,2 loc follows by a standard localization argument [22]. If u i,u j,v i,v j L 1,2 with u i t) = v i t) = ift< or t>a s i and u j t) = v j t) = ift<ort>a s j.set 2.13) U i t) := U j t) := u i s) dws), V i t) := u j s) dws), V j t) := v i s) dws), v j s) dws).

10 274 Y. HU, S.-E. A. MOHAMMED AND F. YAN Then E li U i lj U j li V i lj V j = E li U i V i ) lj U j li V i lj U j V j ) E li U i V i ) lj U j E li V i lj U j V j ) ) 1/2 ) 1/2 E li U i V i ) E 2 lj U j ) 2 ) 1/2 ) 1/2 E li V i ) E 2 lj U j V j ) 2. By an L p estimate of the Skorohod integral [23], Proposition 3.5, and [22], page 158), we have E lj U j 2 l s j 2 = E u j s) dws) t l 1 s j a 2 = E I tl 1 s j,t l s j ]s)u j s) dws) a I tl 1 s j,t l s j ]s)eu 2 j s) ds a a I tl 1 s j,t l s j ]s)e D t u j s) ) 2 dsdt a a a = Eu 2 j s) ds E D t u j s) ) 2 dsdt = u j 2 1,2. Hence we obtain the following inequality: E li U i lj U j li V i lj V j 2.14) u i v i 1,2 u j 1,2 v i 1,2 u j v j 1,2.

11 DISCRETE-TIME APPROXIMATIONS OF SDDEs 275 Since L 1,2 L 4 [,a]) is dense in L 1,2, it suffices to prove 2.12) for the case u L 1,2 L 4 [,a]). Set { ut), t a si, 2.15) u i t) :=, t <ort>a s i. Define u n i t) := I tl 1 s i,t l s i ]t) tl s i 2.16) us) ds t l t l 1 t l 1 s i and u n j similarly. Let U i t) := u i s) dws), Ui n t) := u n i s) dws), 2.17) U j t) := u j s) dws), Vj n t) := u n j s) dws). Using 2.14) it is easy to check that lim E 2.18) n li Ui n lj Uj n li U i lj U j =. By the formula for the Skorohod integral of a process multiplied by a random variable [23], Theorem 3.2), we get l li Ui n s i = t l 1 s i 1 = t l t l 1 k=1 1 t l t l 1 I tk 1 s i,t k s i ]t) t k t k 1 l s i k s i t k 1 s i u i s) ds dwt) t l 1 s i u i s) ds[wt l s i ) Wt l 1 s i )] l s i t l 1 s i = P li li W Q li, l s i t l 1 s i D t u i s) ds dt where 1 P li := t l t l 1 l s i t l 1 s i u i s) ds, 1 Q li := t l t l 1 l s i t l 1 s i l s i t l 1 s i D t u i s) ds dt.

12 276 Y. HU, S.-E. A. MOHAMMED AND F. YAN Therefore, li Ui n lj Uj n = P li li W Q li )P lj lj W Q lj ) = P li P lj ) li W lj W) P li Q lj ) li W P lj Q li ) lj W Q li Q lj. By Hölder s inequality, Q 2 tl li s i l s i 2.19) D t u i s) 2 dsdt. t l 1 s i t l 1 s i Thus lim n E n Q 2 li =. Now P li li W) 2 li W) 2 l s i ) 2 = t l t l 1 ) 2 u i s) ds t l 1 s i li W) 2 l s i = u n i t l t s))2 ds. l 1 t l 1 s i It is easy to check that E u n i )2 L 2 [,as i ]) E u2 i L 2 [,as i ]) and 2.2) lim n E un i )2 u 2 i L 2 [,as i ]) =. By an argument similar to the one used in the proof of Lemma A.2, we can show that { n P li li W) 2,n 1} is uniformly integrable. Applying Lemma A.2, we have lim E asi 2.21) P n li li W) 2 u 2 i s) ds =. The Cauchy Schwarz-type inequality E P li li W)Q li 2.22) E P li li W) 2 E Q 2 li together with 2.19) and 2.21) implies that lim n E n P li li W)Q li =. Now consider the case i j. The Cauchy Schwarz inequality implies E Q lj Q li 2.23) E Q 2 lj E Q 2 li.

13 DISCRETE-TIME APPROXIMATIONS OF SDDEs 277 We may write 2.24) where 2.25) P li P lj ) li W lj W) li W lj W tl s i l s j = t l t l 1 ) 2 u i s) ds u j s) ds t l 1 s i t l 1 s j li W lj W tl s i = u n i t l t s)ûn j s) ds, l 1 t l 1 s i û n j s) = m I tl 1 s i,t l s i ]s) t l t l 1 l s i t l 1 s i u j s s j s i )ds. Similar to the case i = j, wehave lim E 2.26) P n li P lj ) li W lj W) =. This completes the proof of the proposition. Suppose that X tl = X tl 1 α l Xtl X tl 1 ) for some random variables α l 1, l = 1,...,n. Denote 2.27) )) )) ) ) X tl = Xtl = Xtl Xtl 1, 2.28) ) ) X tl = Xtl 1 αl ) X tl, 2.29) lix = Xt l s i ) Xt l 1 s i ), for 1 i k and 1 l n. PROPOSITION 2.3. Suppose that φ C 1,2 T R k, R), and let 1 i, j k. Under the hypotheses of Proposition 2.2, we have 2.3) 2 φ tl 1, )) X tl li X lj X x i si 2 φ s, Xs x i 2 ) ) u 2 s) ds, i = j,, i j as n, in probability.

14 278 Y. HU, S.-E. A. MOHAMMED AND F. YAN PROOF. For1 i, j k, 2.31) li X lj X = li U li V ) lj U lj V) = li U lj U li U lj V li V lj U li V lj V, where U,V are defined by 2.2). Since U,V are continuous and V [,a] is of bounded variation, it follows that 2.32) lim n lim n lim n li U lj V =, li V lj U =, li V lj V =, in probability, for all i, j n. To handle the term n li U lj U,we adapt an approach by Nualart and Pardoux cf. [23], Theorem 3.4, or [22], Theorem 3.2.1). Set Ys):= 2 φ s, Xs xi 2 ) ) I [,t] s) and Y n 2 φ s) := Y)I {} s) tl 1, )) 2.33) X tl Itl 1,t l ]s). xi 2 Then Y n s) Ys) as n, uniformly in s [,t]. Applying Proposition 2.2 and Lemma A.3, we get 2.34) in probability as n. 2 φ tl 1, X tl )) li X lj X si 2 φ δ ij s, Xs ) ) u 2 s) ds x 2 i PROPOSITION 2.4. Suppose that φ C 1,2 T R k ) and let Xt) be a continuous stochastic process defined by 2.1), where u L 2,4 loc, v L1,4 loc and η C[ τ,], R m ) is of bounded variation. Assume that π n : τ = s < <s n =

15 DISCRETE-TIME APPROXIMATIONS OF SDDEs 279 are partitions of [ τ,] such that π n as n. Then, for each 1 i k and each t T, we have φ lim tl 1, )) X tl 1 li X n 2.35) in probability. = φ s, Xs ) ) dxs s i ) k j=i1 k 2 φ s, Xs ) ) u 2 s s i )ds 2 φ s, Xs ) ) [ ssj D ssi ur) dwr) ssj ] D ssi vr)dr us s i )ds PROOF. By a localization argument, we may assume that φ C 1,2 b T R k, R).Let π n < min {1 i k} s i s i 1.Fix1 i k,1 l n,andset F l := φ tl 1, )) 2.36) X tl 1. By property of the Skorohod integral [23], Theorem 3.2), it follows that l s i l s i 2.37) F l li U = us)f l dws) D r F l )ur) dr, t l 1 s i t l 1 s i where U is defined by 2.2). The chain rule for weak derivatives) yields 2.38) D r F l ) = k 2 φ tl 1, X tl 1 )) Dr Xt l 1 s j ). Now, taking the Malliavin derivative D r in 2.1) gives 2.39) Consequently, D r Xt) = ur)i {r t} D r us) dws) D r vs)ds. φ tl 1, X tl 1 )) li U = c 1 c 2 c 3 c 4,

16 28 Y. HU, S.-E. A. MOHAMMED AND F. YAN where c 1 := c 2 := l s i t l 1 s i l s i t l 1 s i φ tl 1, X tl 1 )) us) dws), k 2 φ tl 1, X tl 1 )) I{r tl 1 s j }u 2 r) dr, 2.4) c 3 := l s i k t l 1 s i 2 φ tl 1, X tl 1 )) l 1 s j D r us) dws) ur) dr, c 4 := l s i k t l 1 s i 2 φ tl 1, X tl 1 )) l 1 s j D r vs)dsur)dr. We will study the limits of the above expressions as n. Step ) First we show that the limit of c 2 is given by k si 2 φ c 2 r si, )) X r si u 2 r) dr j=i1 a.s. If j i, then t l 1 s i t l 1 s j. So when t l 1 s i <r<t l s i, I {r tl 1 s j } =. We have c 2 = k j=i1 l s i t l 1 s i 2 φ tl 1, X tl 1 )) I{r tl 1 s j }u 2 r) dr a.s. as n. k j=i1 si 2 φ r si, X r si )) u 2 r) dr Step ) Next we study the limit of c 3 as n. We claim that k si 2 φ c 3 r si, )) X r si r si s j D r us) dws) ur) dr

17 as k in probability. In fact, T n j := l s i t l 1 s i l s i t l 1 s i l s i t l 1 s i 2 φ DISCRETE-TIME APPROXIMATIONS OF SDDEs 281 [ 2 φ tl 1, X tl 1 )) si l 1 s j 2 φ r si, X r si )) 2 φ tl 1, )) rsj s i X tl 1 t l 1 s j [ 2 φ tl 1, X tl 1 )) l s i rsj s i t l 1 s i t l 1 s j sup 2 φ 1 l n r [t l 1 s i,t l s i ] sup D r us) dws) r si s j ] D r us) dws) ur) dr D r us) dws) ur) dr r si s j 2 φ r si, ] )) X r si D r us) dws) ur) dr tl 1, X tl 1 )) D r us) dws) ur) dr 2 φ r si, )) X r si x j l s i r si s j D r us) dws)ur) dr = Tj1 n T j2 n, where Tj1 n and T j2 n denote the first and second term on the right-hand side of the last inequality. Using the Cauchy Schwarz inequality and the L p inequality for the Skorohod integral [23], Proposition 3.5, and [22], page 158), we have ETj1 n 2 φ asi ) 1/2 x E u 2 r) dr j { l s i rsj s i E D r us) 2 dsdr E t l 1 s i l s i t l 1 s i t l 1 s j rsj s i a t l 1 s j D θ Dr us) ) 2 dθ dsdr} 1/2

18 282 Y. HU, S.-E. A. MOHAMMED AND F. YAN as n. The uniform continuity of in probability. T n j 2 φ implies Tj2 n a.s.soasn, Step 3. Nowwewillshowthat k si 2 φ c 4 r si, )) X r si 2.43) rsj s i D r vs)dsur)dr a.s. As in Step 2, we have l s i [ 2 φ tl 1, )) tl 1 s j X tl 1 D r vs)ds t l 1 s i 2 φ r si, )) r si s j ] X r si D r vs)ds ur) dr 2 φ l s i rsj s i x D r vs)ds ur) dr j t l 1 s i t l 1 s j sup sup 2 φ tl 1, )) X tl 1 1 l n r [t l 1 s i,t l s i ] 2 φ r si, )) X r si x j l s i r si s j D r vs)ds ur) dr a.s.asn. Step 4. Finally, we study the limit of c 1 as n. We shall show that si φ c 1 s si, )) 2.44) X s si us) dws) in L 2, R) as n. To see this, define 2.45) u n s) := us) It suffices to show that 2.46) φ tl 1, X tl 1 )) Itl 1 s i,t l s i ]s). u n s) φ s si, X ts si )) us)i,tsi ]s)

19 DISCRETE-TIME APPROXIMATIONS OF SDDEs 283 in L 1,2 as n. It is clear that the sequence {u n s)} converges to φ s s i, X s si ))us)i,tsi ]s) in L 2 T,R). It remains to show that the sequence {D r u n s)} n=1,r,s T, converges in L2 T 2, R) to D r [ φ s s i, X s si ))us)i,tsi ]s)]. Now D r u n s) = D r us) us) φ tl 1, X tl 1 )) Itl 1 s i,t l s i ]s) [ k 2 φ tl 1, ] )) tl 1 s j X tl 1 D r us )dws ) I tl 1 s i,t l s i ]s) [ k 2 φ us) tl 1, ] )) tl 1 s j X tl 1 D r vs )ds I tl 1 s i,t l s i ]s) [ k 2 φ us) tl 1, ] )) X tl 1 ur)i[,tl 1 s ]r) j I tl 1 s i,t l s i ]s) = d 1 d 2 d 3 d 4, where d 1,d 2,d 3 and d 4 stand for the first, second, third and fourth terms on the right-hand side of the above equality, respectively. It is easy to see that d 1 D r us) φ )) s si,x s si I,tsi ]s) in L 2, R). Since for all 1 j k, us) ss j s i D r vθ)dθ belongs to L 2 T 2, R), then by Lebesgue s dominated convergence theorem, the L 2 T 2, R) limit of the function q 3 s, r) defined by k [ 2 φ q 3 := us) tl 1, )) ssj s i X tl 1 x i ] D r vθ)dθ I tl 1 s i,t l s i ]s) is k [ 2 φ us) s si, )) ssj s i ] X s si D r vθ)dθ I,tsi ]s). Since v L 1,4 and u L 4 T,R), the following argument shows that the

20 284 Y. HU, S.-E. A. MOHAMMED AND F. YAN difference between d 3 and q 3 converges to as n in L 1, R): l s i a [ u 2 2 φ s) tl 1, ] )) 2 [ ssj s ] i 2 X tl 1 D r vθ)dθ dr ds t l 1 s i t l 1 s j π n 2 φ x j. 2 a u 2 s) ds a a Dr vθ) ) 2 dr dθ Hence, the L 2 T 2, R) limit of d 3 isthesameasthatofq 3, namely, k [ 2 φ us) s si, )) ssj s i ] X s si D r vθ)dθ I,tsi ]s) in L 2 T 2, R). To find the limit of d 2, we need to check that for all j, the two-parameter process us) ss j s i D r uθ) dwθ), s,r a) belongs to L 2 T 2, R). This follows from the following estimates: a a [ E u 2 ssj s i 2 s) D r uθ) dwθ)] dsdr { E a { C E a u 4 s) ds E [ u 4 s) ds E a a [ ssj s i a a E a a a 2 2 } 1/2 D r uθ) dwθ)] dr) ds D r uθ) 2 dθ dr ) 2 D α Dr uθ) ) dθ dr dα) 2 ]} 1/2. Here we have used a slight modification of the L p estimate of the Skorohod integral for p = 4 cf. [23], Exercise 3.2.7). Using similar L p estimates to the above, we obtain l s i a [ u 2 2 φ s) tl 1, ] )) 2 X tl 1 t l 1 s i 2.47) [ ssj s ] i 2 D r uθ) dwθ) dr ds t l 1 s j 2 φ 2 a ) 1/2 x Eu 4 s) ds j { n l s i [ a ssj s i E t l 1 s i t l 1 s j ) 2 ] } 2 1/2 D r vθ)dθ dr ds.

21 DISCRETE-TIME APPROXIMATIONS OF SDDEs 285 Note that the right-hand side of the above inequality tends to zero as n. Thus 2.48) d 2 k [ 2 φ us) s si, )) X s si ssj s i ] D r uθ) dwθ) I,tsi ]s) in L 2 T 2, R) as n. It is easy to check that k 2 φ d 4 us) s si, )) X s si ur)i[,ssj s x i ]r)i,tsi ]s) j as n in L 2, R). Therefore, D r u n s) D r [us) 2 φ s si, ] )) 2.49) X s si I,tsi ]s) in L 2 T 2, R). Finally it is easy to see that si c 1 in L 2, R) as n. φ s si, X s si )) us) dws) Step 5. The convergence φ tl 1, )) X tl 1 li V x i 2.5) si φ s si, )) X s si dvs) s i as n, is easy to verify. a.s. We complete the section by giving a Stratonovich version of the Itô formula 2.7). Suppose that k 1andp 2. The set L k,p d,c cf. [23], Definition 7.2, and [22], page 167) is the class of processes u L k,p d such that the mappings s D s t us t) and s D s t us t) are continuous in L p ), uniformly in t T, and sup s,t T E D s ut) p )<.

22 286 Y. HU, S.-E. A. MOHAMMED AND F. YAN The space L 1,2 d,c,loc is the class of processes that are locally in L1,2 d,c.forany u L 1,2 d,c, the following limits, d D t ut) = lim Dt i ui t ε), ε 2.51) d Dt ut) = lim Dt i ui t ε), ε exist in L 2 ) uniformly in t,weset =D D,thatis, u)t) = D t ut) Dt ut). Consider the process η) us) dws) vs)ds, t >, 2.52) Xt) = ηt), τ t, where η belongs to C and is of bounded variation, u = u 1,...,u m ) T, u i L 2,4 d,c,loc, u) L1,4 loc, v = v1,...,v m ) T, v i L 1,4 loc, and the stochastic integral is a Stratonovich one. Assume also that the process X is continuous. Using the relationship between the Skorohod and Stratonovich integrals [23], Theorem 7.3, and [22], Theorem 3.11) and Theorem 2.3, we can easily obtain the following Stratonovich version of Itô s formula for the segment process X t cf. [28]). COROLLARY 2.5. Suppose that the process Xt) is defined by 2.52), and let φ C 1,2 T R mk, R). Then φ t, X t ) ) φ, X ) ) φ = s, Xs ) ) ds s k 2.53) t φ s, Xs x i ) ) us s i ) dws s i ) for all t T a.s. k φ x i s, Xs ) ) vs s i )ds 3. Weak differentiability of solutions of SDDEs. In this section, we will study the weak differentiability of the solution of the Itô SDDE 1.6). Bell and Mohammed [6] have applied the Malliavin calculus to study regularity of solutions of SDDEs with a single delay in the noise term. Their analysis relies on weak

23 DISCRETE-TIME APPROXIMATIONS OF SDDEs 287 differentiability of the solution of the SDDE. In Section 5 of this article, the weak differentiability of the solution to the SDDE 1.6) together with the Itô formula 2.1) are used to develop higher-order numerical schemes for solving the SDDE. The next three results Proposition 3.1, Lemma 3.2 and Proposition 3.3) are analogous to those in [22], Theorem 2.2.1, Lemma and Theorem Denote D k, m := p 2 D k,p m,fork N. Recall that Dr l, 1 l d, standforweak differentiation with respect to the lth component of W. PROPOSITION 3.1 cf. [22], Proposition 1.2.3). In the Itô SDDE 1.6), assume that g C,1 b T Rk1m,LR d, R m )) and h C,1 b T Rk2m, R m ). Let X be the solution of 1.6). Then Xt) D 1, m for all t T, and ) 3.1) sup E sup D r Xs) p < r a r s a for all p 2. Furthermore, the partial weak derivatives Dr l Xj t) with respect to the lth coordinate of W satisfy the following linear SDDEs a.s.: g jl r, 1 Xr j )) k 1 g jl s, 1 X s ) ) Dr l x Xj s s 1,i )dw l s) i 3.2) D l r Xj t) = r k 2 h j s, 2 X s ) ) Dr l x Xj s s 2,i )ds, t r, i, t <r, for l = 1,...,d, j = 1,...,m. In 3.2), g jl is the j, l) entry of the m d matrix g, and h j is the jth coordinate of h. PROOF. For simplicity, we will only consider the one-dimensional case d = m = 1 { η), t, X t) = ηt), τ t<, X n1 t) = η) g 3.3) s, 1 Xs n )) dws) h s, 2 Xs n )) ds. It is easy to see that D r g ) s, 1 Xs n )) dws) 3.4) = g r, 1 Xr n )) D r g s, 1 Xs n ))) dws) r s 1,k1

24 288 Y. HU, S.-E. A. MOHAMMED AND F. YAN and D r h ) s, 2 Xs n )) 3.5) ds = D r h s, 2 Xs n ))) ds. r s 2,k2 Since g and h have bounded space derivatives, it is easy to see that there is a positive constant K such that D r g s, 1 Xs n ))) K sup D r X n u), r u s 3.6) D r h s, 2 Xs n ))) K sup D r X n u), r u s almost surely. From the Burkholder Davis Gundy inequality and 3.3) 3.6), it follows that X n t) D 1, for all t [,a], and there are positive constants C 1,C 2 such that 3.7) ) E sup D r X n1 u) p r u t C 1 1 E X n r p ) t C C2 E r sup r u s D r X n u) p ) ds. By induction on n, the above inequality implies that Esup r s a D r X n s) p ) are uniformly bounded in n for all p 2. By [22], Proposition 1.5.5, it follows that Xt) D 1, for all t. Applying the operator D to 1.6) and using [22], Proposition 1.2.3), we obtain the linear SDDE 3.2) for the weak derivative of Xt). The estimate 3.1) follows from 3.2), Burkholder Davis Gundy s inequality and Gronwall s lemma. The following lemma may be proved using similar ideas. Its proof is left to the reader. LEMMA 3.2. Suppose that the real-valued process α ={αr,t): t [r, a]} is adapted and continuous. Assume that the processes at) = a 1 t),...,a k1 t)) R k 1 and bt) = b 1 t),...,b k2 t)) R k 2 are adapted, continuous and uniformly bounded. Furthermore, suppose that the random variables αr,t), at) and bt) belong to D 1, and satisfy the conditions ) ) sup E sup αr,t) p sup E sup D s αr,t) p <, r a r t a r,s a s t a { ) )} 3.8) sup E sup at) p E sup D s at) p <, s a { sup s a E s t a sup s t a bt) p ) E s t a sup s t a D s bt) p )} <

25 DISCRETE-TIME APPROXIMATIONS OF SDDEs 289 for all p 2. Let Y ={Yt): t [,a]} be the solution of the linear SDDE αr,t) as), 1 Y s ) R k 1 dws) bs), 2 Y s ) R k 2 ds, 3.9) Yt)= r r t r,, t r. Then Yt)belongs to D 1,, and for all integers p 2, we have ) sup E sup D s Yt) p <, s a s t a 3.1) ) sup E sup Yt) p <. s a s t a Furthermore, the weak derivative D s Yt)of Yt)satisfies the linear SDDE D s Yt)= D s αr,t) as), 1 Y s ) R k 1 I {r s t} [ ] 3.11) Ds av), 1 Y v ) R k 1 av), 1 D s Y v ) R k 1 dwv) r r [ Ds bv), 2 Y v ) R k 2 bv), 2 D s Y v ) R k 2 ] dv, s < t. The next proposition follows from Proposition 3.1 and Lemma 3.2. PROPOSITION 3.3. Let X ={Xt): t T =[,a]} be the solution of the SDDE 1.6), where g C,2 b T Rk1m,LR d, R m )), h C,2 b T Rk2m, R m ) have bounded first and second partial derivatives in the space variables. Then Xt) D 2, m for all t T, and sup E sup D l 1 r1 D l 2 r2 Xs) ) 3.12) p < r 1,r 2 a r 1 r 2 s a for l 1,l 2 = 1,...,d, and all p Strong approximation of multiple Stratonovich integrals. The following iterated Stratonovich integrals are used in the Milstein scheme for the SDDE 1.6): 4.1) J i,j t,t 1 ; b) := 1 b s b t b t dw i v) dw j s), where <t <t 1,b. We will adopt the discretization scheme in [17], Section 5.8, in order to handle the above double stochastic integral. For alternative discretization approaches to iterated stochastic integrals, see [11] and [26].

26 29 Y. HU, S.-E. A. MOHAMMED AND F. YAN Set 4.2) Jt,t 1 ; b) := J 1,1 t,t 1 ; b), t := t 1 t and r := 2π/t. We choose a complete orthonormal basis of L 2 [,t] as { } 1 t, ) sin nrs, cosnrs : n = 1, 2,..., s t. t t Set W i s) := W i s t ) W i t ) and B j s) := W j s b) W j b), s, 1 i,j d. Using the Kahunen Loève expansion technique, we have W i s) s W i t) = ai t) [ a i 4.4) t 2 n t) cos nrs bn i t) sin nrs] n=1 and B j s) s t B j t) = aj,b t) [ a j,b 4.5) n t) cos nrs bj,b n t) sin nrs] 2 n=1 where an i t) = 2 W i s) s ) W i t) cosnrs ds, t t 4.6) bn i t) = 2 W i s) s ) W i t) sin nrs ds t t and 4.7) an j,b t) = 2 t bn j,b t) = 2 t B j s) s ) t B j t) cosnrs ds, B j s) s ) t B j t) sin nrs ds for n 1. The convergences in 4.4) and 4.5) are in L 2 [,t]). It is easy to see that if n 1, an i t), bi n t), aj,b n t) and bn j,b t) are normally distributed with mean and variance t/2π 2 n 2 [17], page 198). Furthermore, {an i t), bi n t)} and {an j,b t), bn j,b t)} are pairwise independent [17], page 198). One can use wellknown random number generators to simulate these random coefficients cf. [12], Section 3.1.2, [17], Section 1.3, and [18], Section 1.2). 4.8) Let t, t. Then J i,j t,t t; b) = 1 2 W i t) B j t) ) 1 2 W i t)a j,b t ) B j t)a i t ) ) LEMMA 4.1. π n=1 n [ an i t )bn j,b t ) bn i t )an j,b t ) ], 1 i, j d, where the infinite series converges in L 2, R).

27 DISCRETE-TIME APPROXIMATIONS OF SDDEs 291 PROOF. It suffices to show 4.8) for t =. Fix t>. For simplicity of notation, we write 4.9) and 4.1) a j n := aj n ), bj n := bj n ), a j,b n It is easy to check that := an j,b ), bj,b n := bn j,b ) WN i s) := s t W i t) ai N 2 an i cos nrs bi n sin nrs). n=1 4.11) b s b b dw i N v) dwj s) b s b b dw i v) dw j s) in L 2 ) as N. Then we may write b J i,j,t; b) = W i s b) dw j s) For any n 1, we have b b = b b s b) b t [ n=1 cos nrs b)dw j s) = cos nrs d B j s) = = cosnrs nr a i n b b b i n W i t) dw j s) ai 2 B j t) b cos nrs d B j s) s ) t B j t) = t 2 nrbj,b n. B j s) s ) t B j t t) B j s) s t B j t) b ) cos nrs b)dw j s) ] sin nrs b)dw j s). sin nrs ds B j t) t ) s cos nrs d t B j t) cosnrs ds

28 292 Y. HU, S.-E. A. MOHAMMED AND F. YAN Similarly, we have 4.12) So 4.13) Now, Therefore, 4.14) b b J i,j,t; b) = W i t) t sin nrs b)dw j s) = t 2 nraj,b n. sd B j s) ai 2 B j t) π sd B j s) = t B j t) = t 2 B j t) = t 2 n=1 B j t) a j,b ). B j s) ds na i n bj,b n b i n aj,b n ). B j s) s ) t B j t) ds J i,j,t; b) = 1 2 W i t) B j t) 1 2 W i t)a j,b B j t)a i π n=1 na i n bj,b n b i n aj,b n ). The expansion of J i,j,t; b) is a generalization of the expansion of 4.15) s see [11, 17] and [18]). Set J p i,j t,t t; b) 4.16) dw i v) dw j s) = 1 2 W i t)w j t) ) 1 [ 2 W i t)a j,b W j t)a i ] π nan i bj n bi n aj n ) n=1 := 1 2 W i t) B j t) ) 1 2 p π n=1 [ W i t)a j,b t ) B j t)a i t ) ] n [ an i t )bn j,b t ) bn i t )an j,b t ) ]. Then J p i,j t,t t; b) can be used to approximate J i,j t,t t; b) in the mean )

29 DISCRETE-TIME APPROXIMATIONS OF SDDEs 293 square. The rate of convergence is given in Lemma 4.2. LEMMA ) For any integer p 1 and t>, we have E J p i,j,t; b) J i,j,t; b) 2 t2 2π 2 p. PROOF. 4.18) Letp 1 be any integer. Then n=p1 1 n 2 1 p u 2 du = 1 p. Since a i n and bi n are independent, Eai n bi n ) = andeaj,b n E J p i,j,t; b) J i,j,t; b) 2 = π 2 n=p1 = π 2 = t2 2π 2 n=p1 n=p1 n 2 Ea i n bj,b n b i n aj,b n )2 n 2[ Ean i bj,b n )2 Ebn i aj,b n )2] 1 n 2 bn j,b ) =, we have t2 2π 2 p. 5. The strong Milstein scheme. In this section we construct a strong Milstein scheme of order 1 for the SDDE 1.6). Our construction relies heavily on the Itô formula for tame functions Theorem 2.1). Throughout this section, we assume that in 1.6) the coefficients g C 1,2 T R k1m,lr d, R m )) and h C 1,2 T R k2m, R m ). For convenience, set Ws) = W) =, for all s. We also define { g t, 1 X ut) := t ) ), t a,, t <, 5.1) { h t, 2 X vt) := t ) ), t a, ηt), t <. We first derive the Milstein scheme for the case d = m = 1.

30 294 Y. HU, S.-E. A. MOHAMMED AND F. YAN 5.1. Itô Taylor expansion. Assume that <t <t,and x = x 1,...,x k1 ) R k 1. Applying the Itô formula 2.1), we have g t, 1 X t ) ) g )) t, 1 Xt g = s, 1 X s ) ) ds s t 5.2) k 1 s1,i t s 1,i g s s1,i, 1 Xs s1,i )) us) dws) k 1 t [ g s, 1 X s ) ) vs s 1,i ) g x 2 i s, 1 X s ) ) ] s 1,i Xs), s 1,i Xs) ds, where ± s 1,i Xs) are defined by 2.9). Applying the Itô formula 2.1) again and using similar notations for h, we obtain h t, 2 X t ) ) h )) t, 2 Xt h = s, 2 X s ) ) ds s t 5.3) k 2 s2,i t s 2,i h s s2,i, 2 Xs s2,i )) us) dws) 1 2 k 2 t [ h s, 2 X s ) ) vs s 2,i ) g x 2 i s, 2 X s ) ) ] s 2,i Xs), s 2,i Xs) ds. Substituting 5.2) and 5.3) into 1.6), we get the following approximate Itô Taylor) expansion of 1.6): Xt) = Xt ) g t, 1 Xt )) [Wt) Wt )]h t, 2 Xt )) t t ) 5.4) k 1 g t, 1 Xt )) ut s 1,i ) 1 s 1,i t t s 1,i dwt 2 )dwt 1 ) Rt,t),

31 DISCRETE-TIME APPROXIMATIONS OF SDDEs 295 where Rt,t):= k 1 { 1 s 1,i t t s 1,i [ g t2 s 1,i, 1 Xt2 s 1,i )) ut2 ) g ] } )) t, 1 Xt ut s 1,i ) dwt 2 )dwt 1 ) 1 k 1 [ g )) t2, 1 Xt2 vt2 s 1,i ) t t x i 1 2 ] g )) t2 2 xi 2, 1 Xt2 s1,i X t2, s 1,i X t2 dt 2 dwt 1 ) 5.5) k 2 1 s 2,i h )) t2 s 2,i, 2 Xt2 s t t s 2,i x 2,i ut2 )dwt 2 )dt 1 i 1 k 2 [ h )) t2, 2 Xt2 vt2 s 2,i ) t t x i 1 2 ] h )) t2 2 xi 2, 2 Xt2 s2,i X t2, s 2,i X t2 dt 2 dt 1 1 [ ] g )) h )) t2, 1 Xt2 t2, 2 Xt2 dt 2 dt 1. t t t 2 t 2 In the above expression, the stochastic integrals 1 s 1,i g )) t2 s 1,i, 1 Xt2 s t s 1,i x 1,i ut2 )dwt 2 ) i and 1 s 2,i t s 2,i h t2 s 2,i, 2 Xt2 s 2,i )) ut2 )dwt 2 ) are Skorohod integrals. Define 5.6) It s i,j,t s i,j ; s i,j ) := 1 s i,j t t s i,j dwt 2 )dwt 1 ), for i = 1, 2andj = 1,...,k i. Recall the definition of Jt s i,j,t s i,j ; s i,j ) in 4.1). Note that if s i,j <, then 5.7) It s i,j,t s i,j ; s i,j ) = 1 s i,j t t s i,j dwt 2 ) dwt 1 );

32 296 Y. HU, S.-E. A. MOHAMMED AND F. YAN if s i,j =, then 5.8) It s i,j,t s i,j ; s i,j ) = [Wt 1 ) Wt )] dwt 1 ) t = Wt) Wt )) 2 2 t t The one-dimensional Milstein scheme d = m = 1). Assume d = m = 1. Let π : τ = t l < <t = < <t n = a be a partition of [ τ,a]. We introduce the Milstein scheme for the SDDE 1.6) as follows: X π t) = X π t k ) h t k, 2 X π )) tk t tk ) g t k, 1 X π )) tk Wt) Wtk ) ) 5.9) k 1 g tk, 1 X π )) x tk u π t k s 1,i )I t k s 1,i,t s 1,i ; s 1,i ) i for t k <t t k1,where and Recall the notation u π t) = { g t, 1 Xt π)), t,, τ t<, 1 s 1,i It k s 1,i,t s 1,i ; s 1,i ) = dwt 2 ) dwt 1 ). t k t k s 1,i { tk, t k s<t k1, s := t nt, t nt s t, and introduce the following notation: { tk1, t k <s t k1, s = t, t nt <s t. In view of 5.7) and Lemma 4.2, we will use J p t i,t; s 1,i ) to approximate It i,t; s 1,i ). Denote by Z π t) := X π t) Xt), t [ τ,a] the global truncation error for the Milstein scheme, with X the unique solution of the SDDE 1.6). LEMMA 5.1. In the SDDE 1.6) with d = m = 1), suppose that g C 2 b Rk 1, R), h C 2 b Rk 2, R), have bounded first and second derivatives. Then for

33 DISCRETE-TIME APPROXIMATIONS OF SDDEs 297 each integer p 1, there exists a constant Kp) > such that E 2 g s, 1 x 2 X s ) s 1,i X s, p s 1,i X s Kp), 5.1) E 2 h s, 2 x 2 X s ) ) s 2,i X s, p s 2,i X s Kp), for all t [,a]. PROOF. By the definition of ± s 2,i Xs) [see 2.9)], we have 5.11) and 5.12) Therefore, 5.13) If r>, then 5.14) and 5.15) s 1,i,s 1,j Xs) = 2us s 1,i )I {s1,i <s 1,j } us s 1,i )δ ij 2 ss1,j D ss1,i ur) dwr) 2 s 1,i,s 1,j Xs) = us s 1,i )δ ij. ss1,j 2 g s, 1 x 2 X s ) ) s 1,i Xs), s 1,i Xs) k 1 { 2 g = 2 s, 1 X s ) ) us s 1,i ) x i [ us s 1,i )I {s1,i <s 1,j } 1 2 us s 1,i)δ ij ss1,j D ss1,i ur) dwr) ss1,j ]} D ss1,i vr)dr. D t D s ur) = D s ur) = D s g 1 X r ) ) k 1 = i, k 1 k 1 g r, 1 X r ) ) D s Xr s 1,i ) D ss1,i vr)dr 2 g r, 1 X r ) ) D s Xr s 1,i )D t Xr s 1,j ) g r, 1 X r ) ) D t D s Xr s 1,i ).

34 298 Y. HU, S.-E. A. MOHAMMED AND F. YAN By Proposition 3.1 and Proposition 3.3, there exists a constant C 1 > such that ) sup E D s Xr) 2 C 1, s a sup E s,t a sup s r a sup s t r a D t D s Xr) 2 ) C 1. Since g has bounded first and second derivatives, then there is a positive constant C 2 such that ) sup E sup D s ur) 2 s a s r a and sup E s,t a If r<s s 1,i,then Therefore, E s1,j C 2 k 1 ts 1,i s1,j s1,j sup s a sup s t r a ) 2 D ts1,i ur) dwr) ts 1,i C 2 k 2 1 C2 1 2C 2k 1 C 1 =: K 1. E sup s r a D s Xr) 2 ) C 1 C 2 k 1 D t D s ur) 2 ) C 2 1 C2 2 k 1 C 1 C 2 k 1. D ss1,i ur) =, D ss1,i vr) =. ts 1,i E D s D ts1,i ur) ) 2 dr ds s1,j ts 1,i Similarly, there exists a constant K 2 > such that s1,j ) 2 E D ts1,i vr)dr K 2. ts 1,i E D ts1,i ur) ) 2 dr So the first inequality of 5.1) follows from the above two inequalities and the Lipschitz and bounded conditions on h, g [1.4) and 1.5)]. The second estimate of 5.1) is proved by a similar argument. THEOREM 5.2. Consider the Milstein scheme 5.9) for the SDDE 1.6). Recall that Z π := X π X is the global truncation error for any partition π of [ τ,a].

35 DISCRETE-TIME APPROXIMATIONS OF SDDEs 299 Let <γ 1. Suppose that η : [ τ,] R m is of bounded variation and is γ 2 )-Hölder continuous. Let g C1,2 T R k 1, R), h C 1,2 T R k 2, R) have bounded first and second spatial derivatives. Assume that sup Z π s) C π γ τ s for some positive constant C. Then there exists a constant C>depending on a and independent of π) such that sup E Z π s) 2 C π 2γ. τ s a PROOF. We express the global error in the form Z π t) = Z π ) I π t) R π t), where I π n t [ t) = h ti 1, 2 X π )) ))] ti 1 h ti 1, 2 Xti 1 ti t i 1 ) and n t [ g ti 1, 1 X π )) ))] ) ti 1 g ti 1, 1 Xti 1 Wti W ti 1 [ h t nt, 2 X π )) )) ] ) tn t h tnt, 2 Xtn t t tnt [ g t nt, 1 X π )) )) ] )) tn t g tnt, 1 Xtn t Wt) W tnt n t k { 1 k { 1 [ g It i 1,t i ; s 1,j ) ti 1, 1 X π )) x ti 1 u π t i 1 s 1,j ) j I t nt,t; s 1,j ) [ g tnt, 1 X π tn t g ] )) } ti 1, 1 Xti 1 uti 1 s 1,j ) )) u π t nt s 1,j ) g )) ) ]} tnt, 1 Xtn t u tnt s 1,j n t R π t) = Rt i 1,t i ) R t nt,t ). We shall decompose R π t) into five parts: R π t) = R1 π t) Rπ 2 t) Rπ 3 t) Rπ 4 t) Rπ 5 t),

36 3 Y. HU, S.-E. A. MOHAMMED AND F. YAN where n t R1 π t) := = k 1 {i ss1,j t i 1 t i 1 s 1,j k 1 { ss1,j t n t k 1 { ss1,j R π 2 t) := k 1 R π 3 t) := k 2 R π 4 t) := k 2 and s t n t s 1,j s s 1,j s [ g r s1,j, 1 Xr s1,j )) ur) g ti 1, 1 Xti 1 )) [ g r s1,j, 1 Xr s1,j )) ur) ] } ut i 1 s 1,j ) dwr)dws) g )) ) ] } tnt, 1 Xtn t u tnt s 1,j dwr)dws) [ g r s1,j, 1 Xr s1,j )) ur) g ] } )) s, 1 X s u s s1,j ) dwr)dws), [ g r, 1 X r ) ) vr s 1,j ) 1 2 ss2,j s s s 2,j s 2 g r, 1 x 2 X r ) ) ] s 1,j X r, s 1,j X r dr dws), h r s2,j, 2 Xr s2,j )) ur) dwr) ds, [ h r, 2 X r ) ) vr s 2,j ) 1 2 R π 5 t) := s 2 h r, 2 x 2 X r ) ) ] s 2,j X r, s 2,j X r dr ds s { h r r, 2 X r ) ) g r, 1 X r ) )} dr ds. r

37 DISCRETE-TIME APPROXIMATIONS OF SDDEs 31 By the Itô isometry and the formula for the covariance between two Skorohod integrals [22], Section 1.3.1), we have sup E R1 π s) 2 s t k 1 k 1 E k 1 k 1 k 1 k 1 { ss1,j s s 1,j [ g r s1,j, 1 Xr s1,j )) ur) g ] )) 2 s, 1 X s u s s1,j ) dwr)} ds ss1,j [ g )) E r s1,j, 1 Xr s1,j ur) s s 1,j ss1,j s s 1,j g ] )) 2 s, 1 X s u s s1,j ) dr ds ss1,j { [ g E D r1 r s1,j, s s 1,j )) 1 Xr s1,j ur) g )) s, 1 X s = k 1 R π 11 t) k 1R π 12 t). By assumption, the function u s s 1,j )]} 2 dr 1 dr 2 ds G j s,x,z)= g s, x)gs s 1,j,z), x R k 1 and z R k 1 is Lipschitz; that is, there exists a constant L 1 > such that Using and G j s, z) G j s, w) L 1 z w z, w) R 2k 1 and 1 j k 1. { g r, 1 X ur) = r ) ), r,, r <, sup E Xβ) Xα) 2 C 2 r 2 r 1 γ, τ r 1 α<β r 2 a

38 32 Y. HU, S.-E. A. MOHAMMED AND F. YAN it follows that k 1 R11 π t) 2k 1 L 2 1 ss1,j ) E [G j r s1,j, 1 Xr s1,j, 1 X r ) ) s s 1,j k 1 ss1,j G j s, 1 X s ), 1 X s s1,j )) ] 2 I { s s1,j } dr ds s s 1,j 2a 1)k 2 1 L2 1 C 2 π 2γ. Now for all r and1 j k 1, ) g )) D s r s1,j, 1 Xr s1,j ur) = g r, 1 X r ) ) k 1 g r s1,j, 1 Xr s1,j )) sup E Xr 2 ) Xr 1 ) 2 dr ds 1 r 1 <r 2 a, r 2 r 1 π 2 g r s1,j, 1 Xr s1,j )) Ds Xr s 1,j s 1,i ) k 1 g r, 1 X r ) ) D s Xr s 1,i ). By Proposition 3.1, there exists a constant C 3 > such that ) sup E sup D r Xs) 2 C 3. r a s a By 1.8), 1.1) and boundedness of the spatial derivatives of g, thereexistsa constant C 4 > such that ) g )) 2) sup sup E D s r s1,j, 1 Xr s1,j ur) 2C 4 k1 2 r a s a x. j Therefore R π 12 t) k 1 k 1 ss1,j ss1,j s s 1,j 4a 1)C 4 k 4 1 π 2. Hence there is a constant C 5 > such that 5.16) s s 1,j 4C 4 k 2 1 dr 1 dr 2 ds sup E R1 π s) 2 C 5 π 2γ. s t

39 DISCRETE-TIME APPROXIMATIONS OF SDDEs 33 Applying Fubini s theorem, we can rewrite R3 π t) as n t R3 π t) = So we have n t R3 π t) = n t = k 2 i t i 1 t n t ss2,j t i 1 s 2,j k 2 ss2,j k 2 i s 2,j t i 1 s 2,j k 2 s2,j k 2 = t n t s 2,j k 2 i s 2,j t n t s 2,1 i r s 2,j r s 2,j h r s2,j, 2 Xr s2,j )) ur) dwr) ds h r s2,j, 2 Xr s2,j )) ur) dwr) ds. h r s2,j, 2 Xr s2,j )) ur)dsdwr) h r s2,j, 2 Xr s2,j )) ur)dsdwr) t i 1 s 2,j t i s 2,j r) h r s2,j, 2 Xr s2,j )) ur) dwr) k 2 s2,j s2,j s 2,j t n t s 2,j t s 2,j r) h r s2,j, 2 Xr s2,j )) ur) dwr) r s 2,j s 2,j r) h r s2,j, 2 Xr s2,j )) ur) dwr). Applying the formula for covariance between two Skorohod integrals [22], Section 1.3.1) and Proposition 3.1, we can show that there exists a constant C 6 > such that 5.17) sup s t E R3 π s) 2 C 6 π 2. Similarly, by Lemma 5.1, we can easily show that there exist C 7 > such that sup E R2 π s) 2 C 7 π 2, s t 5.18) sup E R4 π s) 2 C 7 π 2, s t sup s t E R5 π s) 2 C 7 π 2. By arguments similar to the ones used in the proof of Theorem 3.4 in [15], we obtain the following inequality: 5.19) sup E I π u) 2 C 1 sup E Z π u) 2) ds u t τ u s

40 34 Y. HU, S.-E. A. MOHAMMED AND F. YAN for some constant C 1 >. From 5.16) 5.19), there exist C 8 > andc 9 > such that 5.2) sup E Z π u) 2 E Z π ) 2 C 8 π 2γ C 9 sup E Z π u) 2 ds. u t τ u s So 5.21) sup E Z π u) 2 2C C 8 ) π 2γ C 9 sup E Z π u) 2 ds. τ u t τ u s By Gronwall s lemma, there exists a constant C>such that E sup τ s t Z π s) 2 C π 2γ. REMARKS. 1. Let us consider a particular case when g and h are of the form 5.22) g s, 1 X s ) ) = h s, 2 X s ) ) = k 1 k 2 a i s,xs s 1,i ) ), b j s,xs s 2,j ) ), where a i,b j C 1,2 b T R) for 1 i k 1 and 1 j k 2. In this case, one may also apply the nonanticipating Itô formula to a i t s1,1,xt s 1,1 ) ) a i t s 1,1,Xt s 1,1 ) ), a i t s1,1,xt s 1,1 ) ) a i t s 1,1,Xt s 1,1 ) ) to prove Theorem 5.2 cf. [28]). 2. One can allow the initial process η to be a sample continuous semimartingale in the following way. Replace W by an extended Brownian motion Wt), t τ, with the associated Brownian filtration F t ) τ t a. Assume that ηt) D 1, m for all t [ τ,],andη is an F t) τ t -continuous semimartingale satisfying 5.23) sup E ηβ) ηα) 2 C 2 β α γ, τ α<β sup E Z π s) 2) C π 2γ τ s for some positive constants C 2 and C. The arguments in Section 2 and the proof of Theorem 5.2 may be adapted to include this generalization.

41 DISCRETE-TIME APPROXIMATIONS OF SDDEs 35 We can rewrite the SDDE 1.6) in Stratonovich form, namely, if t, Xt) = η) g s, 1 X s ) ) dws) 5.24) [ h s, 2 X s ) ) 1 g s, 1 X s ) ) g s, 1 X s ) ) ] ds, 2 x k1 if s k1 =. If s k1 <, then the SDDE is of the same form as 1.6) except the Itô integral is replaced by a Stratonovich integral, that is, Xt) = η) g s, 1 X s ) ) dws) h s, 2 X s ) ) ds. Bell and Mohammed [5, 6]) derived a similar result in the case of a single delay. From Corollary 2.5, we can obtain the following Stratonovich Taylor expansion of Xt) cf. [28]): Xt) = Xt ) g )) t, 1 Xt [Wt) Wt )] h )) t, 2 Xt t t ) 5.25) where Rt,t) k 1 g t, 1 Xt )) ut s 1,i ) 1 s 1,i t t s 1,i dwt 2 ) dwt 1 ) Rt,t), 5.26) = k 1 { 1 s 1,i t 1 t t t s 1,i k 1 k 2 1 s 2,i t t s 2,i [ g t2 s 1,i, 1 Xt2 s 1,i )) ut2 ) g ] } )) t, 1 Xt ut s 1,i ) dwt 2 ) dwt 1 ) g t2, 1 Xt2 )) vt2 s 1,i )dt 2 dwt 1 ) h x i t2 s 2,i, 2 Xt2 s 2,i )) ut2 ) dwt 2 )dt 1 and 5.27) 1 t t k 2 h x i t2, 2 Xt2 )) vt2 s 2,i )dt 2 dt 1 h := h 1 2 g k 1 g, vt) := { h t, 2 X t ) ), t a, ηt), t <.

42 36 Y. HU, S.-E. A. MOHAMMED AND F. YAN One can also derive the Milstein scheme for 5.24) using the Stratonovich Taylor expansion 5.25) of Xt) as follows. Let t k <t t k1.then 5.28) X π t) = X π t k ) h t k, 2 X π tk )) t tk ) g t k, 1 X π tk )) Wt) Wtk ) ) k 1 g tk, 1 X π )) x tk u π t k s 1,i )J t k s 1,i,t s 1,i ; s 1,i ), i where { u π g t, 1 X t) = t π)), t,, τ t< The multidimensional Milstein scheme. Write hs, x) = h 1 s, x),..., h m s, x)) T, x R mk 1, x 11,..., x 1k1 x =... x m1,..., x mk1 Denote by g jl s, x) the j, l) element of the m d matrix gs, x). Tosimplify notation, we use below the summation convention on repeated indices. Recall the notations for the partition τ = t y < <t = < <t n = t introduced in Section 2. We formulate the Milstein scheme for the SDDE 1.6) as follows: if t k <t t k1,theith coordinate X i t) of Xt) = X 1 t),...,x m t)) T is approximated by 5.29) X i,π t) = X i,π t k ) h i t k, 2 X π tk )) t tk ) g ij t k, 1 X π tk )) W j t) W j t k ) ) gij 1 j 1 tk, 1 X π tk )) u i 1 j 1,π t k s 1,j1 ) I j,j1 tk s 1,j1,t s 1,j1 ; s 1,j1 ), where { u i 1j 1,π g i 1 j 1 t, 1 X t) = t π)), t,, τ t<. As in the SODE case [17, 18] and in view of Lemma 4.2, it is possible to further discretize the double stochastic integral I j,j1 t k s 1,j1,t s 1,j1 ; s 1,j1 ) in 5.29)