The Convergence and MS Stability of Exponential Euler Method for Semilinear Stochastic Differential Equations

and Applied Analysis 3 2.1. Exponential Euler Method We consider the n-dimensional semi-linear SDEs dX t FX t f t, X t dt g t, X t dW t , X 0 X0, 2.1 where initial dataX0 ∈ LpF0 Ω;R , F ∈ Rn×n is the generator of a strongly continuous analytic semigroup S S t t≥0 on a Banach space 17 , f : 0, T × R → R, g : 0, T × R → R and W t is a scalar Wiener process. In our analysis, it will be more natural to work with the equivalent expression X t eX0 ∫ t 0 e t−s f s,X s ds ∫ t 0 e t−s g s,X s dW s . 2.2 Now, we introduce the exponential Euler method for 2.1 . Given a stepsize h > 0, the exponential Euler approximate solution is defined by yk 1 eyk ef ( tk, yk ) h eg ( tk, yk ) ΔWk, 2.3 where yk is an approximation to X tk with tk kh, y0 X0 and ΔWk W tk 1 −W tk is the Wiener increment. It is convenient to use the continuous exponential Euler approximate solution and hence y t is defined by y t : ey0 ∫ t 0 e t−s f s, Y s ds ∫ t 0 e t−s g s, Y s dW s , 2.4 where s s/h h and x denote the largest integer, which is smaller than x and Y t is the step function which defined by Y t : ∞ ∑ k 0 I tk ,tk 1 t yk, 2.5 where I A is the indicator function of set A. Obviously, y tk Y tk yk for any integer k ≥ 0; that is the continuous exponential Euler solution y t and the step function Y t coincide with the discrete solution at the grid point. 2.2. Strong Convergence In this subsection, we are taking aim at the convergence of exponential Euler method applying to 2.1 . To show this, some conditions are imposed to the functions f and g in 2.1 . 4 Abstract and Applied Analysis Assumption 2.1. Assume that f and g satisfy the globally Lipschitz condition and the linear growth condition, that is, there exist two constants L1, L2 such that ∣ ∣f t, X − f t, Y ∣2 ∨ ∣g t, X − g t, Y ∣2 ≤ L1|X − Y |, 2.6 ∣ ∣f t, X ∣ ∣2 ∨ ∣g t, Y ∣2 ≤ L2 ( 1 |X| ) , 2.7 for allX,Y ∈ LpF0 Ω;R . Furthermore, f and g are supposed to satisfy the following property: ∣ ∣f t, X − f s,X ∣2 ∨ ∣g t, X − g s,X ∣2 ≤ L3 ( 1 |X| ) |t − s|, 2.8 where L3 is a constant and t, s ∈ 0, T with t > s. The following lemma illustrates that the continuous exponential Euler approximate solution 2.4 is bounded in MS sense and the relationship between continuous approximate solution 2.4 and the step function Y t . Lemma 2.2. Under Assumption 2.1, there exist two constants C1, C2 independent of h such that E ( sup 0≤t≤T ∣y t ∣2 ) ≤ C1, 2.9 E ∣y t − Y t ∣2 ) ≤ C2h, 2.10 for any t ∈ 0, T . Proof. From 2.4 and the elementary inequality a b c 2 ≤ 3 a2 b2 c2 , we have ∣y t ∣2 ≤ 3 ⎡ ⎣ ∣∣∣eFty0 ∣∣ 2 ∣∣∣∣ ∫ t 0 e t−s f s, Y s ds ∣∣∣∣ 2 ∣∣∣∣ ∫ t 0 e t−s g s, Y s dW s ∣∣∣∣ 2 ⎤ ⎦. 2.11 Taking the expectation on both sides and using the Hölder inequality and Doom’s martingale inequality yields E ( sup 0≤s≤t ∣y s ∣2 ) ≤ 3 [∣∣eFt ∣∣ 2 E ∣y0 ∣∣2 TE ∫ t 0 ∣∣eF t−s ∣∣ ∣f s, Y s ∣2ds 4E (∫ t 0 ∣∣eF t−s ∣∣ ∣g s, Y s ∣2ds )] . 2.12 Abstract and Applied Analysis 5 Letting M max{|eFT |, 1}, by the linear growth condition 2.7 , thenand Applied Analysis 5 Letting M max{|eFT |, 1}, by the linear growth condition 2.7 , then E ( sup 0≤s≤t ∣ ∣y s ∣ ∣2 ) ≤ 3M [ E ∣ ∣y0 ∣ ∣2 T ∫ t 0 E ∣ ∣f s, Y s ∣ ∣ds 4E ∫ t 0 E ∣ ∣g s, Y s ∣ ∣ds ] ≤ 3M [ E ∣y0 ∣∣2 L2T ∫ t 0 ( 1 E|Y s | ) ds 4L2 ∫ t 0 ( 1 E|Y s |ds )] ≤ 3M [ E ∣ ∣y0 ∣ ∣2 L2T T 4 L2 T 4 ∫ t 0 ( E|Y s |ds )] ≤ 3M ( E ∣y0 ∣2 L2T T 4 ) 3ML2 T 4 ∫ t 0 E ( sup 0≤r≤s ∣y r ∣2 ) ds. 2.13 Now using the Gronwall inequality yields that E ( sup 0≤s≤t ∣y s ∣∣2 ) ≤ C1, 2.14 where C1 3M E|y0| L2T T 4 e3ML2 T 4 T . From the definition of Y t and 2.4 , for t ∈ tk, tk 1 , we can obtain y t − Y t e t−tk yk ∫ t tk e t−s f ( tk, yk ) ds ∫ t tk e t−s g ( tk, yk ) dW s − yk. 2.15 Using Hölder inequality gives |y t − Y t | ≤ 3 ⎡ ⎣ ∣∣eF t−tk − In ∣∣ ∣yk ∣2 h ∫ t tk ∣∣eF t−s ∣∣ ∣f ( tk, yk ∣2ds ∣∣∣∣ ∫ t tk e t−s g ( tk, yk ) dW s ∣∣∣∣ 2 ⎤ ⎦, 2.16 where In is the n dimension identity matrix. Taking the expectation of both sides, we have E ∣y t − Y t ∣∣2 ≤ 3 [∣∣eF t−tk − In ∣∣ 2 E ∣yk ∣∣2 hE ∫ t tk ∣∣eF t−s ∣∣ ∣f ( tk, yk ∣2ds E ∫ t tk ∣∣eF t−s ∣∣ ∣g ( tk, yk ∣2ds ] . 2.17 6 Abstract and Applied Analysis In view of 2.7 , 2.9 , and |eF t−tk − In| ∼ O h2 , we can obtain E ∣ ∣y t − Y t ∣2 ≤ 3 [∣ ∣ ∣e t−tk − In ∣ ∣ ∣ 2 C1 hML2 1 C1 h ML2 1 C1 h ] ≤ 3M 1 C1 L2h O ( h2 ) . 2.18