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We find upper bounds for the rate of convergence when the EulerMaruyama approximation is used in order to compute the expectation of nonsmooth functionals of some stochastic differential equations whose diffusion coefficient is constant, whereas the drift coefficient may be very irregular. As a byproduct of our method, we establish the weak order of the EulerMaruyama approximation for a diffusion processes killed when it leaves an open set. We also apply our method to the study of the weak approximation of reflected stochastic differential equations whose drift is H¨older continuous.
Approximation for non-smooth functionals of stochastic differential equations with irregular drift Hoang-Long Ngo∗ and Dai Taguchi† Abstract We find upper bounds for the rate of convergence when the Euler-Maruyama approximation is used in order to compute the expectation of non-smooth functionals of some stochastic differential equations whose diffusion coefficient is constant, whereas the drift coefficient may be very irregular. As a byproduct of our method, we establish the weak order of the EulerMaruyama approximation for a diffusion processes killed when it leaves an open set. We also apply our method to the study of the weak approximation of reflected stochastic differential equations whose drift is H¨ older continuous. 2010 Mathematics Subject Classification: 60H35, 65C05, 65C30 Keywords: Euler-Maruyama approximation, Irregular drift, Monte Carlo method, Reflected stochastic differential equation, Weak approximation 1 Introduction Let (Xt )0≤t≤T be the solution to dXt = b(Xt )dt + σ(Xt )dWt , X0 = x0 ∈ Rd , 0 ≤ t ≤ T, where W is a d-dimensional Brownian motion. The diffusion (Xt )0≤t≤T is used to model many random dynamical phenomena in many fields of applications. In practice, one often encounters the problem of evaluating functionals of the type E[f (X)] for some given function f : C[0, T ] → R. For example, in mathematical finance the function f is commonly referred as a payoff function. Since they are rarely analytically tractable, these expectations are usually approximated using numerical schemes. One of the most popular approximation methods is the Monte Carlo Euler-Maruyama method which consists of two steps: 1. The diffusion process (Xt )0≤t≤T is approximated using the Euler-Maruyama scheme (Xth )0≤t≤T with a small time step h > 0: dXth = b(Xηhh (t) )dt + σ(Xηhh (t) )dWt , X0h = x0 , ηh (t) = kh, ∗ Hanoi National University of Education, 136 Xuan Thuy - Cau Giay - Hanoi - Vietnam, email: ngolong@hnue.edu.vn † Ritsumeikan University, 1-1-1 Nojihigashi, Kusatsu, Shiga, 525-8577, Japan, email: dai.taguchi.dai@gmail.com 1 for t ∈ [kh, (k + 1)h), k ∈ N. 2. The expectation E[f (X)] is approximated using independent copies of X h . 1 N N i=1 f (X h,i ) where (X h,i )1≤i≤N are N This approximation procedure is influenced by two sources of errors: a discretization error and a statistical error Err(f, h) := Err(h) := E[f (X)] − E[f (X h )], and E[f (X h )] − 1 N N f (X h,i ). i=1 We say that the Euler-Maruyama approximation (X h ) is of weak order κ > 0 for a class H of functions f if there exists a constant K(T ) such that for any f ∈ H, |Err(f, h)| ≤ K(T )hκ . The effect of the statistical error can be handled by the classical central limit theorem or large deviation theory. Roughly speaking, if f (XTh ) has a bounded variance, the L2 -norm of the statistical error is bounded by N −1/2 V ar(XTh )1/2 . Hence, if the Euler-Maruyama approximation is of weak order κ, the optimal choice of the number of Monte Carlo iterations should be N = O(h−2κ ) in order to minimize the computational cost. Therefore, it is of both theoretical and practical importance to understand the weak order of the Euler-Maruyama approximation. It has been shown that under sufficient regularity on the coefficients b and σ as well as f , the weak order of the Euler-Maruyama approximation is 1. This fact is proven by writing the discretization error Err(f, h) as a sum of terms involving the solution of a parabolic partial differential equation (see [1, 28, 24, 14, 7]). It should be noted here that besides the Monte Carlo Euler-Maruyama method, there are many other related approximation schemes for E[f (XT )] which have either higher weak order or lower computational cost. For example, one can use Romberg extrapolation technique to obtain very high weak order as long as Err(h) can be expanded in terms of powers of h (see [28]). When f is a Lipschitz function and the strong rate of approximation is known, one can implement a Multi-level Monte Carlo simulation which can significantly reduce the computation cost of approximating E[f (X)] in many cases (see [5]). It is also worth looking at some algebraic schemes introduced in [18]. However, all the accelerated schemes mentioned above require sufficient regularity condition on the coefficients b, σ and the test function f . The stochastic differential equations with non-smooth drift appear in many applications, especially when one wants to model sudden changes in the trend of a certain random dynamical phenomenon (see e.g., [14]). There are many papers studying the Euler-Maruyama approximations in this context. In [9] (see also [2]), it is shown that when the drift is only measurable, the diffusion coefficient is non-degenerate and Lipschitz continuous then the Euler-Maruyama approximations converges to the solution of stochastic differential equation. The weak order of the Euler-Maruyama scheme when both coefficients b and σ as well as payoff functions f are H¨older continuous has been studied in [14, 23]. In the papers [15] and [25], the authors studied the weak and strong convergent rates of the Euler-Maruyama scheme for specific classes of stochastic differential equations with discontinuous drift. The aim of the present paper is to investigate the weak order of the Euler-Maruyama approximation for stochastic differential equations whose diffusion coefficient σ is constant, whereas the 2 drift coefficient b may have a very low regularity, or could even be discontinuous. More precisely, we consider a class of function A which contains not only smooth functions but also some discontinuous one such as indicator function. b will then be assumed to be either in A or α-H¨older continuous. It should be noted that no smoothness assumption on the payoff function f is needed in our framework. As a by product of our method, we establish the weak order of the EulerMaruyama approximation for a diffusion processes killed when it leaves an open set. We also apply our method to study the weak approximation of reflected stochastic differential equation whose drift is H¨ older continuous. The remainder of this paper is organized as follows. In the next section we introduce some notations and assumptions for our framework together with the main results. All proofs are deferred to Section 3. 2 2.1 Main Results Notations A function ζ : Rd → R is called exponentially bounded or polynomially bounded if there exist positive constants K, p such that |ζ(x)| ≤ KeK|x| or |ζ(x)| ≤ K(1 + |x|p ), respectively. Let A be a class of exponentially bounded functions ζ : Rd → R such that there exists a sequence of functions (ζN ) ⊂ C 1 (Rd ) satisfying: ζN → ζ in L1loc (Rd ), A(i) : A(ii) : supN |ζN (x)| + |ζ(x)| ≤ KeK|x| , −|x|2 /u A(iii) : supN,u>0; a∈Rd e−K|a|−Ku Rd |∇ζN (x + a)| ue (d−1)/2 dx < K, for some positive constant K. We call (ζN ) an approximation sequence of ζ in A. The following propositions shows that this class is quite large. Proposition 2.1. i) If ξ, ζ ∈ A then ξζ ∈ A and a1 ξ + a2 ζ ∈ A for any a1 , a2 ∈ R. ii) Suppose that A is a non-singular d × d-matrix, B ∈ Rd . Then ζ ∈ A iff ξ(x) := ζ(Ax + B) ∈ A. It is easy to verify that the class A contains all C 1 (Rd ) functions which has all first order derivatives polynomially bounded. Furthermore, the class A contains also some non-smooth functions of the type ζ(x) = (x1 − a)+ or ζ(x) = Ia 0, denote the Euler-Maruyama approximation of X, t Xth = x0 + 0 b(Xηhh (s) )ds + σWt , t ∈ [0, T ], (2) where ηh (s) = kh if kh ≤ s < (k + 1)h for some nonnegative integer k. In this paper, we study the convergent rates of the error Err(h) = E[f (X)] − E[f (X h )] as h → 0 for some payoff function f : C[0, T ] → R. A Borel measurable function ζ : Rd → Rd is called sub-linear growth if ζ is bounded on compact sets and ζ(y) = o(|y|) as y → ∞. ζ is called linear growth if |ζ(y)| < c1 |y| + c2 for some positive constants c1 , c2 . It has been shown recently in [11] that when b is of super-linear growth, i.e., there exist constants C > 0 and θ > 1 such that |ζ(y)| ≥ |y|θ for all |y| > C, then the Euler-Maruyama approximation (2) converges neither in the strong mean square sense nor in weak sense to the exact solution at a finite time point. It means that if E[|XT |p ] < ∞ for some p ∈ [1, ∞) then lim E |XT − XTh |p = ∞ h→0 lim E |XT |p − |XTh |p and h→0 = ∞. Thus, in this paper we will consider the case that b is of at most linear growth. Remark 2.3. In the one-dimensional case, d = 1, it is well-known that if σ = 0 and b is of linear growth, then the strong existence and path-wise uniqueness hold for the equation (1) (see [3]). In the multidimensional case, d > 1, it has been shown in [30] that if b is bounded then the equation (1) has a strong solution and the solution of (1) is strongly unique. Moreover, if σ is the identity matrix, then the equation (1) has a unique strong solution in the class of continuous T processes such that P 0 |b(Xs )|2 ds < ∞ = 1 provided that Rd |b(y)|p dy < ∞ for some p > d ∨ 2 (see [17]). Throughout this paper, we suppose that equation (1) has a weak solution which is unique in the sense of probability law (see Chapter 5 [13])). Our main results requires no assumption on the smoothness of f . Theorem 2.4. Suppose that b ∈ B(α) and b is of linear growth. Moreover, assume that f : C[0, T ] → R is bounded. Then lim E[f (X h )] = E[f (X)]. h→0 4 If b is of sub-linear growth, we can obtain the rate of convergence as follows. Theorem 2.5. Suppose that b ∈ B(α) and b is of sub-linear growth. Moreover, assume that f : C[0, T ] → R satisfies E[|f (x0 + σW )|r ] < ∞ for some r > 2. Then there exists a constant C which does not depend of h such that α 1 |E[f (X)] − E[f (X h )]| ≤ Ch 2 ∧ 4 . For an integral type functional, we obtain the following corollary. Corollary 2.6. Let h = T /n for some n ∈ N. If the drift coefficient b ∈ B(α) is bounded, then for any Lipschitz continuous function f and g ∈ B(β) with β ∈ (0, 1], there exists a constant C which does not depend of h such that T E f T g(Xs )ds −E f 0 0 g(Xηhh (s) )ds α β 1 ≤ Ch 2 ∧ 2 ∧ 4 . Remark 2.7. In the paper [22], the author considered the weak rate of convergence of the EulerMaruyama scheme for equation (1) in the case of a one-dimensional diffusion. It was claimed that if b was Lipschitz continuous, the weak rate of approximation is of order 1. However, we would like to point out that the given proof contains several gaps (see for instance Lemma 2 of [22] and Remark 3.3 below) which leave us unsure about the claim. Remark 2.8. It has been shown in [14, 23] that for a stochastic differential equation with α-H¨older continuous drift and diffusion coefficients with α ∈ (0, 1), one has |E[f (XT )] − E[f (XTh )]| ≤ Chα/2 , where f ∈ Cb2 and the second derivative of f is α-H¨older continuous. On the other hand, in [10], Gy¨ ongy and R´ asonyi have obtained the strong rate of convergence for a one-dimensional stochastic differential equation whose drift is the sum of a Lipschitz continuous and a monotone decreasing H¨ older continuous function, and its diffusion coefficient is H¨older continuous. In [25], the authors improve the results in [10]. More precisely, we assume that the drift coefficient b is a bounded and one-sided Lipschitz function, i.e., there exists a positive constant L such that for any x, y ∈ Rd , x − y, b(x) − b(y) Rd ≤ L|x − y|2 , bj ∈ A for any j = 1, ..., d and the diffusion coefficient σ is bounded, uniformly elliptic and 1/2 + α-H¨older continuous with α ∈ [0, 1/2]. Then for h = T /n, it holds that −1 if α = 0 and d = 1, C(log 1/h) α h Ch if α ∈ (0, 1/2] and d = 1, E[|XT − XT |] ≤ Ch1/2 if α = 1/2 and d ≥ 2. Therefore, if the payoff function f is Lipschitz continuous, it is straightforward to verify that −1 if α = 0 and d = 1, C(log 1/h) α h Ch if α ∈ (0, 1/2] and d = 1, |E[f (XT ) − f (XT )]| ≤ Ch1/2 if α = 1/2 and d ≥ 2. 5 In the following we consider a special case of the functional f . More precise, we are interested in the law at time T of the diffusion X killed when it leaves an open set. Let D be an open subset of Rd and denote τD = inf{t > 0 : Xt ∈ D}. Quantities of the type E[g(XT )1(τD >T ) ] appear in many domains, e.g. in financial mathematics when one computes the price of a barrier option on a d-dimensional asset price random variable Xt with characteristics f, T and D (see [6, 8] and h h the references therein for more detail). We approximate τD by τD = inf{kh > 0 : Xkh ∈ D, k = 0, 1, . . .}. Theorem 2.9. Assume the hypotheses of Theorem 2.5. Furthermore, we assume (i) D is of class C ∞ and ∂D is a compact set (see [4] and [6]); (ii) g : Rd → R is a measurable function, satisfying d(Supp(g), ∂D) ≥ 2 for some g ∞ = supx∈Rd |g(x)| < ∞. > 0 and Then for any p > 1, there exist constants C and Cp independent of h such that α 1 E[g(XT )1(τD >T ) ] − E[g(XTh )1(τDh >T ) ] ≤ Ch 2 ∧ 4 + 2.3 1 Cp g p ∞ 2p h . 1 ∧ 4/p (3) Weak approximation of reflected stochastic differential equations We first recall the Skorohod problem. Lemma 2.10 ([13], Lemma III.6.14). Let z ≥ 0 be a given number and y : [0, ∞) → R be a continuous function with y0 = 0. Then there exists unique continuous function = ( t )t≥0 satisfying the following conditions: (i) xt := z + yt + (ii) t ≥ 0, 0 ≤ t < ∞; is a non-decreasing function with Moreover, 0 = 0 and t = t 0 1(xs = 0)d s . = ( t )t≥0 is given by t = max{0, max (−z − ys )} = max max(0, 0≤s≤t 0≤s≤t s − xs ). Let us consider the following one-dimensional reflected stochastic Xt valued in [0, ∞) such that t b(Xs )ds + σWt + L0t (X), x0 ∈ [0, ∞), t ∈ [0, T ], Xt = x0 + (4) 0 t L0t (X) = 1(Xs =0) dL0s (X), 0 where (L0t (X))0≤t≤T is a non-decreasing continuous process stating at the origin and it is called local time of X at the origin. In this paper, we assume that the SDE (4) has a weak solution and the uniqueness in the sense of probability law holds (see [27, 29]). Using Lemma 2.10, we have L0t (X) = sup max 0, L0s (X) − Xs . 0≤s≤t 6 Now we define the Euler-Maruyama scheme X h = (Xth )0≤t≤T for the reflected stochastic differential equation (4). Let X0h := x0 and define t Xth = x0 + 0 b(Xηhh (s) )ds + σWt + L0t (X h ). The existence of the pair (Xth , L0t (X h ))0≤t≤T is deduced from Lemma 2.10. Moreover t L0t (X h ) = 0 1(Xsh =0) dL0s (X h ). By the definition of the Euler-Maruyama scheme, we have the following representation. For each k = 0, 1, ..., h h h h X(k+1)h = Xkh + b(Xkh )h + σ(W(k+1)h − Wkh ) + max(0, Ak − Xkh ), where Ak := sup h −b(Xkh )(s − kh) − σ(Ws − Wkh ) . kh≤sT ) ] − E[g(XTh )1(τDh >T ) ] = E[g(x0 + σWT )(ZT 1(τDW >T ) − ZTh 1(τ W,h >T ) )], (7) D for all measurable functions f : C[0, T ] → R and g : Rd → R provided that all the above expectations are integrable. Proof. Let σ −1 be the inverse matrix of σ. Since b is of linear growth, so is σ −1 b. Thus, there exist constants c1 , c2 > 0 such that |b(x)| < c1 |x| + c2 for any x ∈ Rd . For any 0 ≤ t ≤ t0 ≤ T , t |Xt | ≤ |x0 | + |σWt | + |b(Xs )|ds 0 t ≤ |x0 | + |σ| sup |Ws | + c2 t0 + c1 0≤s≤t0 |Xs |ds. 0 8 Applying Gronwall’s inequality for t ∈ [0, t0 ], one obtains |Xt0 | ≤ (|x0 | + |σ| sup |Ws | + c2 t0 )ec1 t0 0≤s≤t0 ≤ (|x0 | + c2 T )ec1 T + |σ|ec1 T sup |Ws |. (8) 0≤s≤t0 On the other hand, for each integer k ≥ 1, one has h h h | + h|b(X(k−1)h )| + 2|σ| sup |Wt | |Xkh | ≤ |X(k−1)h 0≤t≤kh ≤ (1 + h hc1 )|X(k−1)h | + hc2 + 2|σ| sup |Wt |. 0≤t≤kh Hence, a simple iteration yields that h |Xkh | ≤ (1 + hc1 )k |x0 | + (hc2 + 2|σ| sup |Wt |) 0≤t≤kh (1 + hc1 )k−1 − 1 . hc1 Thus, for any t ∈ (0, T ], |Xηhh (t) | ≤ (1 + hc1 )T /h |x0 | + (1 + hc1 )T /h c2 (1 + hc1 )T /h + 2|σ| c1 hc1 sup |Ws |. 0≤s≤ηh (t) Moreover, |Xth − Xηhh (t) | ≤ c1 h|Xηhh (t) | + c2 h + 2|σ| sup |Wt |. 0≤s≤t Therefore, for any t ∈ (0, T ], we have |Xth | ≤ (1 + c1 h)1+T /h 2|σ|(1 + hc1 )1+T /h + 2hc1 c1 |x0 | + c2 + c2 h + sup |Ws |. c1 hc1 0≤s≤t (9) We define a new measure P and Ph as dP = exp − dQ dPh = exp − dQ T (σ −1 b)j (Xs )dWsj − 0 1 2 T |σ −1 b(Xs )|2 ds , 0 T 0 (σ −1 b)j (Xηhh (s) )dWsj − 1 2 T 0 |σ −1 b(Xηhh (s) )|2 ds . It follows from Corollary 3.5.16 [13] together with estimates (8) and (9) that P and Ph are probability measures. Furthermore, it follows from Girsanov theorem that processes B = {(Bt1 , . . . , Btd ), 0 ≤ t ≤ T } and B h = {(Bth,1 , . . . , Bth,d ), 0 ≤ t ≤ T } defined by t t (σ −1 b)j (Xs )ds, Bth,j = Wtj + Btj = Wtj + 0 (σ −1 b)j (Xηh (s) )ds, 1 ≤ j ≤ d, 0 ≤ t ≤ T, 0 are d-dimensional Brownian motions with respect to P and Ph , respectively. Note that Xs = x0 + σBs and Xsh = x0 + σBsh . Therefore, E[f (X)] = EP f (X) dQ dP 9 T (σ −1 b)j (x0 + σBs )dBsj − = EP f (x0 + σB) exp 0 1 2 T |σ −1 b(x0 + σBs )|2 ds 0 = E[f (x0 + σW )ZT ]. Repeating the previous argument leads to E[f (X h )] = E[f (x0 + σW )ZTh ], which implies (6). The proof of (7) is similar and will be omitted. From now on, we will use the representation formulas in Lemma 3.1 to analyze the weak rate of convergence. We need the following estimates on the moments of Z and Z h . Lemma 3.2. Suppose that b is of sub-linear growth. Then for any p > 0, E[|ZT |p + |ZTh |p ] ≤ C < ∞, for some constant C which is not depend on h. Proof. For each p > 0, T (σ −1 b)j (x0 + σWs )dWsj − E[epYT ] = E exp p 0 T p 2 |σ −1 b(x + σWs )|2 ds 0 T tn (σ −1 b)j (x0 + σWs )dWsj − p2 = E exp p 0 p + (p2 − ) 2 |σ −1 b(x0 + σWs )|2 ds+ tn−1 T |σ −1 b(x0 + σWs )|2 ds 0 T T (σ −1 b)j (x0 + σWs )dWsj − 2p2 ≤ E exp 2p 0 |σ −1 b(x0 + σWs )|2 ds 1/2 0 T |σ −1 b(x0 + σWs )|2 ds × E exp (2p2 − p) 1/2 . 0 Since b is of linear growth, so is σ −1 b and it follows from Corollary 3.5.16 [13] that T T (σ −1 b)j (x0 + σWs )dWsj − 2p2 E exp 2p 0 |σ −1 b(x0 + σWs )|2 ds = 1. (10) 0 On the other hand, since b is bounded on compact sets and b(y) = o(|y|), for any δ > 0 sufficiently small, there exists a constant M > 0 such that |σ −1 b(x0 + σy)|2 ≤ δ|y|2 + M for any y ∈ Rd . Thus, T T |σ −1 b(x0 + σWs )|2 ds ≤ 0 (δ|Ws |2 + M )ds ≤ T M + T δ sup |Ws |2 s≤T 0 d ((sup Wsj )2 + ( inf Wsj )2 ). ≤ TM + Tδ j=1 s≤T Hence, T |σ −1 b(x0 + σWs )|2 ds E exp (2p2 − p) 0 10 s≤T d ≤ e(2p 2 −p)M T E exp T δ(2p2 − p) ((sup Wsj )2 + ( inf Wsj )2 ) j=1 ≤ e(2p 2 −p)M T s≤T s≤T E exp 2T δ(2p2 − p)|WT1 |2 d/2 , where the last inequality follows from H¨older inequality and the fact that law law sup Wsj = − inf Wsj = |WT1 |. s≤T s≤T < ∞ if one chooses δ < (4T 2 (2p2 −p))−1 , we obtain E[|ZT |p ] < Because E exp 2T δ(2p2 −p)WT2 ∞. Furthermore, since equation (10) still holds if one replaces b(x0 + σWs ) with b(x0 + σWηh (s) ), a similar argument yields E[|ZTh |p ] < ∞. Remark 3.3. In general, the conclusion of Lemma 3.2 is no longer correct if we only suppose that b is of linear growth or even Lipschitz. Indeed, consider the particular case that d = 1, σ = 1 and b(x) = x, which is a Lipschitz function. It follows from H¨ older inequality that T p 2 E exp T Ws2 ds Ws dWs − E exp p 0 0 = e−pT /2 E exp p 2 T Ws2 ds E exp 0 p 2 p 2 p W − 2 T 2 T Ws2 ds 0 T Ws2 ds 0 2 2 ≥ e−pT /2 E epWT /4 . 2 Furthermore, for any p, T > 0 such that pT ≥ 2 and pT 2 < 1/2, we have E epWT /4 = ∞, whereas E exp p 2 T Ws2 ds 0 pT sup |Ws |2 2 s≤T pT pT ≤ E exp (sup Ws )2 + ( inf Ws )2 ) 2 s≤T 2 s≤T ≤ E exp 2 ≤ E epT |WT | 2 < ∞. Therefore, T Ws dWs − E exp p 0 3.2 p 2 T Ws2 ds = ∞, 0 Some auxiliary estimates The following result plays a crucial role in our argument. 11 if pT ≥ 2, pT 2 < 1 . 2 Lemma 3.4. For any ζ ∈ A, any p ≥ 1, t > s > 0, √ t−s E[|ζ(Wt ) − ζ(Ws )|p ] ≤ Cp √ , s (11) for some constant Cp not depending on neither t nor s. On the other hand, if ζ is α-H¨older continuous then E[|ζ(Wt ) − ζ(Ws )|p ] ≤ Cp (t − s)p/2 . (12) Proof. If ζ ∈ A, let (ζN ) be an approximate sequence of ζ in A. Since ζN → ζ in L1loc (Rd ) and ζ and ζN are uniformly exponential bounded, we have E[|ζ(Wt ) − ζ(Ws )|p )] = lim E[|ζN (Wt ) − ζN (Ws )|p ]. (13) N →∞ Next, we will show that √ t−s sup E[|ζN (Wt ) − ζN (Ws )| ] ≤ C √ . s N p (14) Indeed, we write E[|ζN (Wt ) − ζN (Ws )|p ] 2 2 |ζN (x + y) − ζN (x)|p = Rd Rd e−|x| /2s e−|y| /2(t−s) dxdy (2πs)d/2 (2π(t − s))d/2 2 |ζN (x + y) − ζN (x)|(eK(|x+y| + eK|y| )p−1 ≤C Rd d Rd i=1 Rd 1 ≤C √ Rd 0 1 ≤C t − s Rd 2 2 ∂ζN (y + θx) K(p−1)(|x|+|y|) e−|x| /2s e−|y| /2(t−s) dθdxdy yi e ∂xi (2πs)d/2 (2π(t − s))d/2 d i=1 2 e−|x| /2s e−|y| /2(t−s) dxdy (2πs)d/2 (2π(t − s))d/2 Rd 0 2 2 ∂ζN (y + θx) e−|x| /4s e−|y| /4(t−s) dθdxdy. ∂xi (2πs)d/2 (2π(t − s))d/2 It follows from A(iii) that d 2 sup N i=1 Rd ∂ζN (y + θx) e−|y| /4(t−s) dy ≤ CeK|θx| , ∂xi (2π(t − s))(d−1)/2 thus √ t−s E[|ζN (Wt ) − ζN (Ws )| ] ≤ C √ s √ t−s ≤C √ . s 2 1 CeK|θx| p Rd 0 e−|x| /4s dθdx (2πs)d/2 From (13) and (14) we get (11). The proof of (12) is straightforward. 12 Lemma 3.5. Suppose that b ∈ B(α) and b is of linear growth. Then there exists a constant C > 0 such that YT − YTh 2 ≤ Ch1/4 + Chα/2 . Proof. Using Minkowski’s inequality, we obtain E[|YT − YTh |2 ] ≤ C(S1 + S2 ) where T [(σ −1 b)j (x0 + σWs )dWsj − (σ −1 b)j (x0 + σWηh (s) )]dWsj S1 = E 2 , 0 T [|σ −1 b(x0 + σWs )|2 − |σ −1 b(x0 + σWηh (s) )|2 ]ds S2 = E 2 . 0 It follows from Doob’s inequality (see [13]) that, d T S1 ≤ C [(σ −1 b)j (x0 + σWs ) − (σ −1 b)j (x0 + σWηh (s) )]2 ds E 0 j=1 d T E[(σ −1 b)j (x0 + σWs ) − (σ −1 b)j (x0 + σWηh (s) )]2 ds. ≤C 0 j=1 Since b is of linear growth, d h E[(σ −1 b)j (x0 + σWs ) − (σ −1 b)j (x0 + σWηh (s) )]2 ds ≤ Ch. 0 j=1 Furthermore, it follows from Proposition 2.1 ii) and Lemma 3.4 that d T E[(σ −1 b)j (x0 + σWs ) − (σ −1 b)j (x0 + σWηh (s) )]2 ds h j=1 T s − ηh (s) |s − ηn (s)|α + ≤C ηh (s) h ds ≤ C(hα + h1/2 ). Therefore, S1 ≤ C(hα + h1/2 ). Now we estimate S2 . Using H¨ older’s inequality, we obtain T 2 |σ −1 b(x0 + σWs )|2 − |σ −1 b(x0 + σWηh (s) )|2 ds S2 ≤ E 0 d T E |(σ −1 b)j (x0 + σWs )|2 − |(σ −1 b)j (x0 + σWηh (s) )|2 ≤C 2 ds. 0 j=1 2 Note that by Proposition 2.1, (σ −1 bA j ) ∈ A. Dividing the integral into two parts: from 0 to h and from h to T , and applying a similar argument as above, we obtain T S2 ≤ C h Thus, YT − YTh 2 1/4 ≤ Ch s − ηh (s) ηh (s) 1/2 ds + Chα ≤ C(h1/2 + hα ). + Chα/2 . 13 3.3 Proof of Theorem 2.4 It follows from Lemma 3.5 that YTh converges in probability to YT as h → 0. Thus ZTh also converges in probability to ZT as h → 0. Moreover E[ZTh ] = E[ZT ] = 1 for all h > 0. Therefore, it follows from Proposition 4.12 [12] that lim E[|ZTh − ZT |] = 0. (15) h→0 On the other hand, since f is bounded, it follows from (6) that |E[f (X) − f (X h )]| ≤ CE[|ZTh − ZT |]. This estimate together with (15) implies the desired result. 3.4 Proof of Theorem 2.5 It is clear that |ex − ey | ≤ (ex + ey )|x − y|. This estimate and H¨older inequality imply that |E[f (X) − f (X h )]| is bounded by E f (x0 + σW )(ZT + ZTh )(YT − YTh ) ≤ f (x0 + σW )(ZT + ZTh ) ≤ E |f (x0 + σW )|r 2/r 2 YT − YTh 2 E |ZT + ZTh |2r/(r−2) (r−2)/r YT − YTh 2 . Thanks to the integrability condition of f and Lemma 3.2, E |f (x0 + σW )|r 2/r E |ZT + ZTh |2r/(r−2) (r−2)/r ≤ C < ∞. This together with Lemma 3.5 implies the desired result. 3.5 Proof of Corollary 2.6 We first note that if b is bounded, then it holds from Theorem 2.1 in [19] (see also Corollary 3.2 in [25]) that there exists a density function pht of Xth for t ∈ (0, T ] and it satisfies the following Gaussian upper bound, i.e., pht (x) ≤C e− |x−x0 |2 2ct td/2 . for some positive constants C and c. r T Now we prove that E f 0 g(x0 + Ws )ds is finite for any r > 2. Since |g(x)| ≤ KeK|x| , it follows from Jensen’s inequality that for any r > 2, r T E f g(x0 + Ws )ds 0 T T d i E[|g(x + Ws )|r ]ds ≤ C + C ≤C +C 0 E[erKWs ]ds. 0 14 i=1 Since x2 /(4s) + K 2 r2 s ≥ Krx, we have r T T edKrs ds < ∞. ≤C +C g(x0 + Ws )ds f E 0 0 Thanks to Theorem 2.5, it remains to prove that T T g(Xsh )ds E f 0 ≤ Chβ/2 . g(Xηhh (s) )ds −E f 0 Since f is a Lipschitz continuous function, we have T T T g(Xsh )ds E f −E f 0 0 g(Xηhh (s) )ds E g(Xsh ) − g(Xηhh (s) ) ds ≤C 0 If s ∈ (0, h], then by using the Gaussian upper bound for phs (x), we have h 0 h E g(Xsh ) − g(Xηhh (s) ) ds ≤ E[|g(Xsh )|]ds + |g(x0 )|h 0 h ≤C |g(x)| ds 0 |x−x0 |2 2cs e− sd/2 Rd + |g(x0 )|h ≤ Ch. On the other hand, for s ∈ [h, T ], using the Gaussian upper bound for phηh (s) and following the proof of Lemma 3.4 (see also Lemma 3.5 of [25]), we have E[|g(Xsh ) − g(Xηhh (s) )|] ≤ C s − ηh (s) ηh (s) + Chβ/2 . Therefore, we conclude the proof of the statement. 3.6 Proof of Theorem 2.9 It suffices to proof the statement for the case that g is positive. It follows from (7) that E[g(XT )1(τD >T ) ]− E[g(XTh )1(τDh >T ) ] = E1 + E2 where E1 = E[g(x0 + σWT )(ZT − ZTh )1(τ W,h >T ) ], D E2 = E[g(x0 + σWT )ZT (1(τDW >T ) − 1(τ W,h >T ) )]. D It follows from the proof of Theorem 2.5 that α 1 |E1 | ≤ E[|g(x0 + σWT )(ZT − ZTh )|] ≤ Ch 2 ∧ 4 . Applying H¨ older’s inequality, we have |E2 | ≤ ZT q g(x0 + σWT )(1(τDW >T ) − 1(τ W,h >T ) ) p , D 15 (16) W,h W where q is the conjugate of p. Thanks to Lemma 3.2 and the fact τD ≥ τD , we have |E2 | ≤ Cp E g p (x0 + σWT )1(τ W,h ≥T ) − E g p (x0 + σWT )1(τDW ≥T ) 1/p . D It follows from Theorem 2.4 in [6] that there exists a constant K(T ) such that K(T ) g p 1∧ 4 |E2 | ≤ Cp 1/p ∞ 1 h 2p . Combining this estimate with (16) completes the proof. 3.7 Proof of Theorem 2.12 In the same way as in subsection 3.1, we have the following Lemma. Lemma 3.6. If b is a measurable function with sub-linear growth then E[f (X)] − E[f (X h )] = E[f (U )(ZˆT − ZˆTh )] for all measurable functions f : C[0, T ] → R provided that the above expectations are integrable. Here the process U = (Ut )0≤t≤T is the unique solution of the equation Ut = x0 + σWt + L0t (U ) and ˆ Zˆt := eYt , t Yˆt := b(Us )dWs − 0 t 1 2 b2 (Us )ds 0 t ˆh Zˆth := eYt , Yˆth := b(Uηh (s) )dWs − 0 1 2 t b2 (Uηh (s) )ds, 0 ˆ and P ˆ h as Proof. We define new measures P ˆ dP = exp − dQ ˆh dP = exp − dQ T σ −1 b(Xs )dWs − 0 1 2 T |σ −1 b(Xs )|2 ds , 0 T 0 σ −1 b(Xηhh (s) )dWs − 1 2 T 0 |σ −1 b(Xηhh (s) )|2 ds . L0t (X) Since b is of sub-linear growth and the fact that 0 ≤ ≤ |σ| sup0≤s≤t |Ws |, by following the h ˆ ˆ proof of Lemma 3.1 we can show that P and P are probability measures. Furthermore, it follows ˆ = (B ˆt )0≤t≤T and B ˆ h = (B ˆth )0≤t≤T defined by from the Girsanov theorem that the processes B t ˆ t = Wt + B t ˆth = Wt + σ −1 b(Xs )ds, B 0 0 σ −1 b(Xηhh (s) )ds, 0 ≤ t ≤ T, ˆ and P ˆ h respectively. Note that Xs = x0 + σ B ˆs + L0s (X) are Brownian motions with respect to P h h 0 h ˆ and Xs = x0 + σ Bs + Ls (X ). Therefore, E[f (X)] = EPˆ f (X) dQ ˆ dP 16 T = EPˆ f (X) exp 0 T ˆs − 1 σ −1 b(Xs )dB 2 T ˆ + L0 (X)) exp = EPˆ f (x0 + σ B 0 |σ −1 b(Xs )|2 ds 0 ˆs + L0s (X))dB ˆs − 1 σ −1 b(x0 + σ B 2 T ˆs + L0s (X))|2 ds |σ −1 b(x0 + σ B 0 d ˆ ˆ = (U, W )|Q , the above term equals to Since (X, B)| P T σ −1 b(x0 + σWs + L0s (U ))dWs − E f (x0 + σW + L0 (U )) exp 0 T σ −1 b(Us )dWs − = E f (U ) exp 0 1 2 1 2 T |σ −1 b(x0 + σWs + L0s (U ))|2 ds 0 T |σ −1 b(Us )|2 ds 0 = E[f (U )ZˆT ]. Repeating the previous argument leads to E[f (X h )] = E[f (U )ZˆTh ], which concludes the statement. In the same way as Lemma 3.2, we have the following estimate of the moments of Zˆ and Zˆ h . Lemma 3.7. Suppose that b is of sub-linear growth. Then for any p > 0, E[|ZˆT |p + |ZˆTh |p ] ≤ C < ∞, for some constant C which is not depend on h. Finally, we introduce the following auxiliary estimate. Lemma 3.8. Let U as in Lemma 3.6. Suppose that ζ is α-H¨older continuous with α ∈ (0, 1], then for any t > s > 0, E[|ζ(Ut ) − ζ(Us )|p ] ≤ Cp (t − s)pα/2 . Proof. By H¨ older continuity of ζ, we have E[|ζ(Ut ) − ζ(Us )|p )] ≤ E[|Ut − Us |αp ] ≤ CE[|Wt − Ws |pα ] + CE[|L0t (U ) − L0s (U )|pα ]. Hence it is sufficient to prove that E[|L0t (U ) − L0s (U )|p α] ≤ Cp (t − s)pα/2 . Using Lemma 2.10, we have L0s (U ) ≤ L0t (U ) ≤ L0s (U ) + sup max (0, −σ(Wu − Ws )) . s≤u≤t Therefore, |L0t (U )−L0s (U )| ≤ |σ| sups≤u≤t |Wu −Ws |. Hence applying the Burkholder-Davis-Gundy inequality, we have E[|L0t (U ) − L0s (U )|pα ] ≤ CE[ sup |Wu − Ws |pα ] ≤ C(t − s)pα/2 . s≤u≤t This concludes the proof. 17 . Proof of Theorem 2.12 Using Lemmas 3.6 and the elementary estimate |ex − ey | ≤ (ex + ey )|x − y|, we have |E[f (X)] − E[f (X h )]| ≤ E f (U )(ZˆT + ZˆTh )(YˆT − YˆTh ) . Thanks to Lemma 3.7 and the H¨ older inequality, for some r > 2, we have |E[f (X)] − E[f (X h )]| ≤ CE[|f (U )|r ]2/r ||YˆT − YˆTh ||2 . By a similar argument as the proof of Lemma 3.5, we can show that ||YˆT − YˆTh ||2 ≤ Chα/2 , which concludes the proof. Remark 3.9. The conclusion of Theorem 2.12 still holds if we relax the condition f bounded to E[|f (U )|r ] < ∞ for some r > 2. Acknowledgment. This work was completed while the H.N. was staying at Vietnam Institute for Advanced Study in Mathematics (VIASM). He would like to thank the institute for support. This work was also partly supported by the Vietnamese National Foundation for Science and Technology Development (NAFOSTED) under Grant Number 101.03-2014.14. The authors thank Prof. Arturo Kohatsu-Higa for his helpful comments. 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[27] Rozkosz, A., and Slominski, L. (1997) On stability and existence of solutions of SDEs with reflection at the boundary. Stochastic Process. Appl., 68(2), 285-302 . [28] Talay, D. and Tubaro, L. (1990) Expansion of the global error for numerical schemes solving stochastic differential equations. Stochastic Anal. Appl., 8, 94–120. [29] Tanaka, H. (1979) Stochastic differential equations with reflecting boundary condition in convex regions. Hiroshima Math. J., 9(1), 163-177. [30] Veretennikov, A.Yu. (1981) On strong solution and explicit formulas for solutions of stochastic integral equations. Math. USSR Sb., 39, 387– 403. 20 [...]... (1991) Rate of convergence of the Euler approximation for diffusion processes Math Nachr., 151 , 233–239 [24] Milstein, G.N (1985) Weak approximatoin of solutions of systems of stochastic differential equations Theory Probab Appl., 30, 750–766 19 [25] Ngo, H-L., and Taguchi, D (2013) Strong rate of convergence for the Euler-Maruyama approximation of stochastic differential equations with irregular coefficients... Slominski, L (1997) On stability and existence of solutions of SDEs with reflection at the boundary Stochastic Process Appl., 68(2), 285-302 [28] Talay, D and Tubaro, L (1990) Expansion of the global error for numerical schemes solving stochastic differential equations Stochastic Anal Appl., 8, 94–120 [29] Tanaka, H (1979) Stochastic differential equations with reflecting boundary condition in convex... Makhlouf, A and Ngo, H-L (2014) Approximations of non-smooth integral type functionals of one dimensional diffusion processes Stochastic Process Appl., Vol 124(5) 1881-1909 [17] Krylov, N.V and R¨ ockner, M (2005) Strong solutions of stochastic equations with singular time dependent drift Probab Theory Relat Fields, 131, 154–196 [18] Kusuoka, S (2001) Approximation of expectation of diffusion process and... authors thank Prof Arturo Kohatsu-Higa for his helpful comments References [1] Bally, V and Talay, D (1996) The law of the Euler scheme for stochastic differential equations: I Convergence rate of the distribution function Probab Theory Relat Fields., 104, 43–60 [2] Chan, K.S and Stramer, O (1998) Weak Consistency of the Euler Method for Numerically Solving Stochastic Dierential Equations with Discontinuous... Foundations of Modern Probability Second edition, Springer [13] Karatzas, I and Shreve, S E (1991) Brownian motion and stochastic calculus Second edition, Springer [14] Kloeden, P and Platen, E (1995) Numerical Solution of Stochastic Differential Equations Springer [15] Kohatsu-Higa, A., Lejay, A and Yasuda, K (2012) On Weak Approximation of Stochastic Differential Equations with Discontinuous Drift Coefficient... Sharp estimates for the convergence of the density of the Euler scheme in small time Elect Comm in Probab., 13, 352-363 [8] Gobet, E and Menozzi, S (2004) Exact approximation rate of killed hypoelliptic diffusions using the discrete Euler scheme Stochastic Process Appl., 112 , 201-223 18 [9] Gy¨ ongy, I and Krylov, N.V (1996) Existence of strong solutions for Itˆo’s stochastic equations via approximations... bound for phs (x), we have h 0 h E g(Xsh ) − g(Xηhh (s) ) ds ≤ E[|g(Xsh )|]ds + |g(x0 )|h 0 h ≤C |g(x)| ds 0 |x−x0 |2 2cs e− sd/2 Rd + |g(x0 )|h ≤ Ch On the other hand, for s ∈ [h, T ], using the Gaussian upper bound for phηh (s) and following the proof of Lemma 3.4 (see also Lemma 3.5 of [25]), we have E[|g(Xsh ) − g(Xηhh (s) )|] ≤ C s − ηh (s) ηh (s) + Chβ/2 Therefore, we conclude the proof of the... Bounds for the Euler Scheme Electron J Probab., 15, 1645-1681 [20] L´epingle, D (1993) Un sch´ema d’Euler pour ´eqations diff´erentielles stochastiques re´efl´echies C.R.A.S Paris, 316, 601-605 [21] L´epingle, D (1995) Euler scheme for reflected stochastic differential equations Math Comput Simulation, 38, 119-126 [22] Mackevicius, V (2003) On the Convergence Rate of Euler Scheme for SDE with Lipschitz Drift. .. 105, 143 – 158 [10] Gy¨ ongy, I and R´ asonyi, M (2011) A note on Euler approximations for SDEs with H¨older continuous diffusion coefficients Stoch Proc Appl., 121, 2189–2200 [11] Hutzenthaler, M., Jentzen, A and Kloeden, P E (2011) Strong and weak divergence in finite time of Euler’s method for stochastic differential equations with non-globally Lipschitz continuous coefficients Proc R Soc Lond Ser... H-J (2005) Singular Stochastic Differential Equations Lecture Notes in Math Vol 1858 Springer [4] Gilbarg, D and Trudinger, N.S (2002) Elliptic partial differential equations of second order Springer, Berlin, Second edition [5] Giles, M.B (2008) Multilevel Monte Carlo path simulation Oper Res., 56, 607–617 [6] Gobet, E (2000) Weak approximation of killed diffusion using Euler schemes Stochastic Process ... Solution of Stochastic Differential Equations Springer [15] Kohatsu-Higa, A., Lejay, A and Yasuda, K (2012) On Weak Approximation of Stochastic Differential Equations with Discontinuous Drift Coefficient... convergent rates of the Euler-Maruyama scheme for specific classes of stochastic differential equations with discontinuous drift The aim of the present paper is to investigate the weak order of the Euler-Maruyama... Rate of convergence of the Euler approximation for diffusion processes Math Nachr., 151 , 233–239 [24] Milstein, G.N (1985) Weak approximatoin of solutions of systems of stochastic differential equations