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We consider a class of stochastic functional differential equations with distributed delays whose coefficients are superlinear growth and H¨older continuous with respect to the delay components. We introduce an EulerMaruyama approximation scheme for these equations and study their strong rate of convergence
Strong approximation for non-Lipschitz stochastic functional differential equations with distributed delays Ngo Hoang Long∗ Pham Thi Tuyen Hanoi National University of Education Abstract We consider a class of stochastic functional differential equations with distributed delays whose coefficients are super-linear growth and H¨older continuous with respect to the delay components. We introduce an Euler-Maruyama approximation scheme for these equations and study their strong rate of convergence. 1 Introduction We are concerned with strong approximation for a class of stochastic functional differential equations (SFDEs) with distributed delays and non-Lipschitz coefficients. More precisely, we consider the following multidimensional equation on a filtered probability space (Ω, F, (Ft )t≥0 , P), t H(t, s, X(s − τ ))ds dt dX(t) =b X(t), 0 t G(t, s, X(s − τ ))ds dW (t), t ∈ [0, T ], + σ X(t), (1) 0 with initial data X(s) = ξ(s), s ∈ [−τ, 0], where τ is a fixed positive constant, W = (Wt , Ft )t≥0 is a standard m-dimensional Brownian motion. SFDEs with distributed delays have been extensively studied since they can be used to model the dynamical behavior of processes having long period memory, which appear in many application domains, such as population dynamics, economy and engineering (see [4] [11], [14], [9], [12] and the references therein). Since these equations are not usually analytically solvable, we need to approximate their solutions by some numerical schemes. One of the most popular approximation scheme is the Euler-Maruyama (EM) scheme, which is stated as follows: For each integers n ≥ 1, the continuous EM approximation scheme with step size h = hn = T /n = τ /l associated with (1) is defined by t dY n (t) = b Y n (κn (t)), H(t, s, Y n (κn (s) − τ ))ds dt 0 t + σ Y n (κn (t)), G(t, s, Y n (κn (s) − τ ))ds dW (t), 0 ∗ Correspoding author. Email: ngolong@hnue.edu.vn 1 t ∈ [0, T ], (2) where Y n (s) = ξ(s) for s ∈ [−τ, 0], and κn (t) = ih for t ∈ [ih, (i + 1)h). The strong error of this approximation is E[ sup |X(t) − Y n (t)|p ]. 0≤t≤T for some p > 0. Recently, there have been many studies on the rates of convergence of strong error of EM type approximation schemes for stochastic functional differential equations, e.g. in [10], [13], [12], [1], [7], [8] for stochastic differential delay equations, and in [2], [3] for general SFDEs. We note that all these papers mentioned above as well as most of other papers in literature consider approximation for SFDEs with either local Lipschitz continuous or one-sided Lipschitz continuous coefficients. In this paper, we study the strong rates of convergence of Y n under the assumption that the coefficients b and σ are H¨ older continuous and super-linear growth with respect to the second component. This work extend the result of recent papers [6] and [1], which consider the strong rate of approximation for ordinary stochastic differential equations with H¨older continuous diffusion coefficients and for stochastic delay differential equation with super-linear growth coefficients, respectively. It is worth mentioning that the strong rates is a key factor to establish a Multi-level Monte Carlo scheme which is very effective method to approximate expectations of functional of X (see [5]). The paper is organized as follows. In Section 2 we introduce some conditions on the coefficients of equation (1) and prove the existence and uniqueness of its solution under these conditions. In Section 3 we investigate the strong rate of convergence of the EM approximation scheme for equation (1). 2 Framework For a positive integer d, let (Rd , ·, · , |.|) be the d-dimensional Euclidean space and ||A|| := the Hilbert-Schmidt norm for a matrix A, where A∗ is its transpose. We introduce the following assumptions. √ traceA∗ A (A1) b : Rd × Rd1 → Rd and σ : Rd × Rd2 → Rd×m are measurable function and there exist L > 0 and α, β ∈ [ 21 , 1] such that |b(x1 , y1 ) − b(x2 , y2 )| ≤ L|x1 − x2 | + V (|y1 |, |y2 |)|y1 − y2 |α , and ||σ(x1 , z1 ) − σ(x2 , z2 )|| ≤ L|x1 − x2 | + V (|z1 |, |z2 |)|z1 − z2 |β , for all xi ∈ Rd , yi ∈ Rd1 , zi ∈ Rd2 , i = 1, 2, where V : R+ × R+ → R+ is polynomial bounded, i.e., there exist K > 0 and q ≥ 1 such that V (u1 , u2 ) ≤ K(1 + uq1 + uq2 ), for all u1 , u2 ∈ R+ . (3) (A2) H : [0, T ] × [0, T ] × Rd −→ Rd1 and G : [0, T ] × [0, T ] × Rd −→ Rd2 are measurable function and there exist L > 0 and γ, µ ∈ [ 21 , 1] such that |H(t, s, x) − H(t, s, y)| ≤ L|x − y|γ , |G(t, s, x) − G(t, s, y)| ≤ L|x − y|µ and sup |H(t, s, 0)| ∨ 0≤s≤t≤T sup |G(t, s, 0)| ≤ L, 0≤s≤t≤T for all x, y ∈ Rd . 2 We denote CFb 0 ([−τ, 0]; Rd ) the class of all continuous process ξ defined on [−τ, 0] such that supt∈[−τ,0] |ξ(t)| < C < ∞ and ξ(t) are F0 -measurable for all t ∈ [−τ, 0]. The following theorem establishes the existence of a unique strong solution to equation (1). Theorem 2.1. Assume that (A1) holds. Then, for any initial data ξ ∈ CFb 0 ([−τ, 0]; Rd ) there exists a unique strong solution X(t) to equation (1). This theorem can be proven by following the argument of the proof of Theorem 3.1 in [12] and will be skipped. Throughout the paper, C > 0 denotes a generic constant whose values may change from line to line. 3 Strong rates of Euler-Maruyama approximation Let Y n be defined by (2). We first prepare a lemma. Lemma 3.1. Assume that (A1), (A2) holds and the initial data ξ ∈ CFb 0 ([−τ, 0]; Rn ). Then, for any p ≥ 2 there exists a constant C > 0 independent of h such that E[ sup |X(t)|p ] ∨ E[ sup |Y n (t)|p ] ≤ C 0≤t≤T (4) 0≤t≤T and p E |Y n (t) − Y n (κn (t))|p ≤ Ch 2 . (5) E sup |X(t)|p < ∞. (6) Proof. We first show that 0≤t≤T For each integer N > 1, we define the stopping time τN = T ∧ inf {t ∈ [−τ, T ] : |X(t)| ≥ N } . Clearly, |X(s)| ≤ N for all s ∈ [−τ, τN ] and τN ↑ T. Set X N (t) = X(t ∧ τN ) for t ∈ [0, T ]. Then X N (t) satisfies the equation t N s b X N (s), X (t) = ξ(0) + 0 0 t s σ X N (s), + H(s, r, X N (r − τ ))dr I[0≤s≤τN ] ds 0 0 G(s, r, X N (r − τ ))dr I[0≤s≤τN ] dW (s), t ∈ [0, T ]. Using the elementary inequality |a + b + c|p ≤ 3p−1 (|a|p + |b|p + |c|p ), we can show that for t ∈ [0, T ] UtN,p := E sup |X N (s)|p 0≤s≤t s ≤ C + CE 0≤s≤t s + CE 0 0 0 H(s, r, X N (r − τ ))du I[0≤r≤τN ] dr s σ X N (s), sup 0≤s≤t s b X N (s), sup 0 G(s, r, X N (r − τ ))dr I[0≤r≤τN ] dW (r) 3 p p . By the H¨ older and Burkholder-Davis-Gundy inequalities that UtN,p t ≤C 1+E N H(s, r, X (r − τ ))dr b X (s), 0 ds 0 t p s σ X N (s), + p s N G(s, r, X N (r − τ ))dr 0 ds , 0 for all t ∈ [0, T ]. It follows from assumption (A1) that t UtN,p ≤ C E t t s |X N (s)|p ds + E |X N (r − τ )|pαγ + |X N (r − τ )|pβµ drds 0 0 0 s E |X N (r − τ )|p(q+α)γ + |X N (r − τ )|p(q+β)µ drds + 1 . + 0 0 Using the elementary inequality ap + p − 1 ≥ pa for a > 0 and p ≥ 1, we have UtN,p ≤ C 1 + E t t s |X N (s)|p ds + 0 E |X N (r − τ )|p(q+α)γ + |X N (r − τ )|p(q+β)µ drds . 0 0 This, together with the Gronwall inequality, yields that for p ≥ 2 and t ∈ [0, T ], E sup |X N (s)|p 0≤s≤t t s E |X N (r − τ )|p(q+α)γ + |X N (r − τ )|p(q+β)µ drds . ≤C 1+ 0 (7) 0 Set ρ := [(q + α)γ] ∨ [(q + β)µ] and pi := ([T /τ ] + 2 − i)pρ[T /τ ]+1−i , for i = 1, 2, ..., [T /τ ] + 1, where [a] denotes the integer part of the real number a. Thus, due to ρ ≥ 1 and p ≥ 2, it is easy to see that pi ≥ 2, pi+1 ρ ≤ pi and p[T /τ ]+1 = p. Since X(s) = ξ(s) for s ∈ [−τ, 0] and ξ ∈ CFb 0 ([−τ, 0]; Rn ), we obtain that E sup |X N (s)|p1 ≤ C. (8) 0≤s≤τ Furthermore, it follows from (7), the fact p2 ρ < p1 and the Liapunov inequality that τ E s sup |X N (s)|p2 ≤ C 1 + 0≤s≤2τ 0 τ [E|X N (r − τ )|p1 ] 0 E p2 (q+α)γ p1 drds 0 s + Thanks to (8), we get E [E|X N (r − τ )|p1 ] p2 (q+β)µ p1 drds . 0 sup |X N (s)|p2 ≤ C. Repeating the previous procedures gives 0≤s≤2τ sup |X N (s)|p = E 0≤s≤T sup |X N (s)|p[T /τ ]+1 ≤ C. 0≤s≤T a.s Since X N (t) −→ X(t), applying the Fatou Lemma we get (6). By a similar argument we also have E sup |Y n (s)|p ≤ C. This concludes (4). 0≤s≤T 4 Next we will show (5). Since t s Y n (t) − Y n (κn (t)) = b Y n (κn (s)), H(s, r, Y n (κn (r) − τ ))dr ds κn (t) t 0 s σ Y n (κn (s)), + G(s, r, Y n (κn (r) − τ ))dr dW (s), κn (t) (9) 0 by the elementary inequality |a + b|p ≤ 2p−1 (|a|p + |b|p ), the H¨older inequality and the BurkholderDavis-Gundy inequality, we have E |Y n (t) − Y n (κn (t))|p t + 2p−1 C(t − κn (t)) p s ≤ 2p−1 (t − κn (t))p−1 b Y n (κn (s)), E κn (t) t p −1 2 H(s, r, Y n (κn (r) − τ ))dr ds 0 p s E σ Y n (κn (s)), κn (t) G(s, r, Y n (κn (r) − τ ))dr ds. 0 Thanks to conditions A1, A2, estimate (4) and the fact that t − κn (t) ≤ h, we obtain (5). We are now in a position to state the main theorem of this paper. Theorem 3.2. Suppose (A1), (A2) hold and ξ ∈ CFb 0 ([−τ, 0]; Rn ). For any p ≥ 2 there exists C > 0 independent of h and n such that p [T /τ ]+1 E sup |X(s) − Y n (s)|p ≤ Ch 2 ν , 0≤s≤T where ν = (αγ) ∧ (βµ). Proof. We use the method of Yamada and Watanabe to approximate the function φ(x) = |x|. Let δ > 1 and ∈ (0, 1). Since /δ x1 dx = lnδ, there is a continuous nonnegative function ψδ (x), x ≥ 0, 2 which is zero outside [ /δ, ] satisfies /δ ψδ (x) = 1 and ψδ (x) ≤ xlnδ . Define x y φδ (x) := ψδ (z)dzdy, x > 0. 0 0 Then φδ ∈ C 2 (R+ ; R+ ) possesses the following properties: x − ≤ φδ (x) ≤ x, x > 0 and 0 ≤ φδ (x) ≤ 1, φδ (x) ≤ 2 1 xlnδ [ /δ, ] (x), x > 0. 2 δ (x) δ (x) Define Vδ (x) := φδ (|x|), x ∈ Rd , (Vδ )x (x) := ( ∂V∂x , ..., ∂V∂x ), and (Vδ )xx (x) := ( ∂∂xViδ∂x(x) )d×d , x ∈ 1 j d Rd . By a straightforward computation, we deduce that 0 ≤ |(Vδ )x (x)| ≤ 1, ||(Vδ )xx (x)|| ≤ 2d(1 + 5 1 1 ) 1 ln σ |x| [ /δ, ] (|x|). (10) For any t ∈ [−τ, T ], set Z(t) := X(t) − Y n (t), Z(t) := Y n (t) − Y n (κn (t)). Applying the Itˆ o formula, we get Vδ (Z(t)) = I1 (t) + I2 (t) + I3 (t) where t s I1 (t) = H(s, r, X(r − τ ))dr (Vδ )x (Z(s)), b X(s), 0 0 s H(s, r, Y n (κn (r) − τ ))dr − b Y n (κn (s)), ds 0 I2 (t) = t 1 2 s trace G(s, r, X(r − τ ))dr σ X(s), 0 0 ∗ s G(s, r, Y n (κn (r) − τ ))dr −σ Y n (κn (s)), 0 s G(s, r, X(r − τ ))dr (Vδ )xx (Z(s)) σ X(s), 0 s −σ Y n (κn (s)), G(s, r, Y n (κn (r) − τ ))dr ds 0 t I3 (t) = s G(s, r, X(r − τ ))dr (Vδ )x (Z(s)), σ X(s), 0 0 s −σ Y n (κn (s)), G(s, r, Y n (κn (r) − τ ))dr dW (s). 0 E sup |I1 (s)|p 0≤s≤t t ≤C s 0 s 0 s s H(s, r, Y n (κn (r) − τ ))dr H(s, r, X(r − τ ))dr, 0 0 pα s n H(s, r, X(r − τ ))dr − 0 H(s, r, Y (r − τ ))dr) 0 s +V p s H(s, r, Y n (κn (r) − τ ))dr H(s, r, X(r − τ ))dr, 0 0 s s n × ds 0 E |Z(s)|p + |Z(s)|p + V p × H(s, r, Y n (κn (r) − τ ))dr 0 t ≤C p s H(s, r, X(r − τ ))dr − b Y n (κn (s)), E b X(s), pα H(s, r, Y n (κn (r) − τ ))dr) H(s, r, Y (r − τ ))dr − 0 ds. 0 By (A1), (A2) and (4), we have s E V 2p s H(s, r, Y n (κn (r) − τ ))dr H(s, r, X(r − τ ))dr, 0 ≤ C. 0 This fact together with assumption A2 and H¨older’s inequality implies that s t E|Z(s)|p + E|Z(s)|p + E sup |I1 (s)|p ≤ C 0≤s≤t E|Z(r − τ )|2pαγ dr 1 2 ds. 0 Similarly, thanks to (10), condition A1 and the H¨older inequality, we have t E sup |I2 (s)|p ≤ CE 0≤s≤t 0 1 2 ds 0 0 s + E|Z(r − τ )|2pαγ dr 1 1 |Z(s)|p [ /δ, ] (|Z(s)|) 6 |Z(s)|2p + |Z(s)|2p (11) s +V 2p s G(s, r, Y n (κn (r) − τ ))dr G(s, r, X(r − τ ))dr, 0 0 s s G(s, r, Y n (κn (r) − τ ))dr G(s, r, X(r − τ ))dr − × 2pβ ds. 0 0 This implies t E sup |I2 (s)|p ≤ C 0≤s≤t + E|Z(s)|p + 0 s 1 E|Z(r − τ )| p 4pβµ 1 2 dr + 0 1 p E|Z(s)|2p s 1 E|Z(s − τ )| p 4pβµ dr 1 2 ds. (12) 0 By (10), the Burkholder-Davis-Gundy inequality, the inequality H¨older, we obtain that E sup |I3 (s)|p 0≤s≤t t ≤C s G(s, r, Y (κn (r) − τ ))dr 0 t ≤C 0 ds 0 s p s G(s, r, X(r − τ ))dr −σ Y n (κn (s), E σ X(s), p/2 n G(s, r, X(r − τ ))dr −σ Y (κn (s)), E σ X(s), 0 2 s n G(s, r, Y n (κn (r) − τ ))dr 0 ds. 0 Thanks to condition (A1), E sup |I3 (s)|p 0≤s≤t s t E |Z(s)|p + |Z(s)|p + V p ≤C 0 0 s × s 0 s G(s, r, Y n (κn (r) − τ ))dr G(s, r, X(r − τ ))dr − 0 G(s, r, Y n (κn (r) − τ ))dr G(s, r, X(r − τ ))dr, pβ ds. 0 Therefore, we also have t p E sup |I3 (s)| ≤C p E|Z(r − τ )| E|Z(s)| + E|Z(s)| + 0≤s≤t 0 2pβµ dr 0 1 2 s 2pβµ E|Z(r − τ ))| + 1 2 s p dr ds. (13) 0 Since p E sup |Z(s)|p ≤ + sup Vδ (Z(s)) ≤ 2p−1 0≤s≤t p 0≤s≤t + E sup Vδp (Z(s)) 0≤s≤t combining (11), (12), (13), (5), we obtain that for any t ∈ [0, T ] and any p ≥ 2, E sup |Z(s)|p ≤ C p p + h2 + h pαγ 2 + 0≤s≤t t 0 p hp + 1 p hpβµ + h s E|Z(s)|p + + 1 E|Z(r − τ )|2pαγ dr 0 7 1 2 pβµ 2 , + s 1 1 2 E|Z(r − τ )|4pβµ dr p s E|Z(r − τ )|2pβµ dr + 0 1 2 ds . 0 This, together with the Gronwall inequality, further implies that E sup |Z(s)|p ≤ C p p pαγ 2 + h2 + h 1 + 0≤s≤t t s 2pαγ E|Z(r − τ )| + 0 + dr p hpβµ + h pβµ 2 1 2 0 s 1 E|Z(r − τ )|4pβµ dr p 1 hp + p 1 2 s E|Z(r − τ )|2pβµ dr + 1 2 ds . (14) 0 0 We fix p ≥ 2, and denote pi := ([T /τ ] + 2 − i)p4[T /τ ]+1−i , i = 1, 2, ..., [T /τ ] + 1. We have 4pi+1 < pi , ν and p[T /τ ]+1 = p, i = 1, 2, ..., [T /τ ]. For s ∈ [0, τ ], we have Z(s − τ ) = 0 and taking = h 2 , ν = (αγ) ∧ (βµ). It follows from (14) that sup |Z(s)|p1 ≤ Ch E Next, taking E =∆ ν2 2 p1 ν 2 . (15) 0≤s≤τ , by using (14) and the Liapunov inequality, we obtain that sup |Z(s)|p2 ≤ C h p2 2 ν 2 +h p2 2 +h p2 αγ 2 + hp2 βµ− p2 ν 2 2 +h p2 βµ 2 0≤s≤2τ 2τ s 0 + 0 s 1 p2 E|Z(r − τ )|4p2 βµ dr p2 2 ν 2 τ 1 E|Z(r)|2p2 αγ dr 1 2 E|Z(r)|4p2 βµ dr 1 2 E|Z(r)|2p2 βµ dr ds 0 p2 2 ν 2 τ 2p2 αγ p1 s p1 + E|Z(r)| dr 1 2 0 s p1 E|Z(r)| 4p2 βµ p1 dr 1 2 2p2 βµ p1 s p1 + E|Z(r)| dr 1 2 ds 0 0 ≤C h ds 1 2 s + 0 1 1 2 0 0 p2 E|Z(r − τ )|2p2 βµ dr s + s ≤C h + s + 0 0 p2 1 2 0 ≤C h + 1 2 E|Z(r − τ )|2p2 αγ dr + p2 2 ν 2 +h p2 ναγ 2 +h 2p2 p ν2 νβµ− 22 2 +h p2 νβµ 2 . Therefore, it follows that E sup |Z(s)|p2 ≤ Ch p2 2 ν 2 , 0≤s≤2τ where ν = (αγ)∧(βµ). The desired assertion then follows by repeating the previous procedures. 8 Acknowledgment This research was supported by the Vietnam National Foundation for Science and Technology Development (NAFOSTED) under Grant Number 101.03-2014.14. This work was done during a stay of the first author at Vietnam Institute for Advance Study in Mathematics. 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