The Truncated Euler-Maruyama Method for Stochastic Differential Delay Equations
aa r X i v : . [ m a t h . NA ] M a r The Truncated Euler–Maruyama Method forStochastic Differential Delay Equations
Qian Guo , Xuerong Mao ∗ , Rongxian Yue Department of Mathematics,Shanghai Normal University, Shanghai, China. Department of Mathematics and Statistics,University of Strathclyde, Glasgow G1 1XH, U.K.
Abstract
The numerical solutions of stochastic differential delay equations (SDDEs) under thegeneralized Khasminskii-type condition were discussed by Mao [15], and the theory thereshowed that the Euler–Maruyama (EM) numerical solutions converge to the true solutions in probability . However, there is so far no result on the strong convergence (namely in L p )of the numerical solutions for the SDDEs under this generalized condition. In this paper, wewill use the truncated EM method developed by Mao [16] to study the strong convergenceof the numerical solutions for the SDDEs under the generalized Khasminskii-type condition. Key words:
Brownian motion, stochastic differential delay equation, Itˆo’s formula, trun-cated Euler–Maruyama, Khasminskii-type condition.
Mathematical Subject Classifications (2000) : 60H10, 60J65.
In the study of stochastic differential delay equations (SDDEs), the classical existence-and-uniquenesstheorem requires the coefficients of the SDDEs satisfy the local Lipschitz condition and the lineargrowth condition (see, e.g., [4, 8, 11, 12, 21]). However, there are many SDDEs which do notsatisfy the linear growth condition. In 2002, Mao [14] generalized the the well-known Khasminskiitest [6] from stochastic differential equations (SDEs) to SDDEs. The Khasminskii-type theoremestablished in [14] for SDDEs gives the conditions, in terms of Lyapunov functions, under whichthe solutions to SDDEs will not explode to infinity at a finite time. The Khasminskii-type theoremenables us to verify if a given nonlinear SDDE has a unique global solution under the local Lipschitzcondition but without the linear growth condition. In 2005, Mao and Rassias [17] demonstratedthat there are many important SDDEs which are not covered by the Khasminskii-type theoremgiven in [14], and established a generalized Khasminskii-type theorem which covers a very wideclass of nonlinear SDDEs.On the other hand, there are in general no explicit solutions to nonlinear SDDEs, whencenumerical solutions are required in practice. The numerical solutions under the linear growthcondition plus the local Lipschitz condition have been discussed intensively by many authors (see, ∗ Corresponding author. E-mail: [email protected]. in probability . However, there is so far noresult on the strong convergence (namely in L p ) of the numerical solutions for the SDDEs underthe generalized Khasminskii-type condition.Recently, Mao [16] develops a new explicit numerical method, called the truncated EMmethod, for SDEs under the Khasminskii-type condition plus the local Lipschitz condition and es-tablishes the strong convergence theory. In this paper, we will use this new truncated EM methodto study the strong convergence of the numerical solutions for the SDDEs under the generalizedKhasminskii-type condition.This paper is organized as follows: We will introduce necessary notion, state the generalizedKhasminskii-type condition and define the truncated EM numerical solutions for SDDEs in Section2. We will establish the strong convergence theory for the truncated EM numerical solutions inSections 3 and 4 and discuss the convergence rates in Section 5. In each of these three sections wewill illustrate our theory by examples. We will see from these examples that the truncated EMnumerical method can be applied to approximate the solutions of many highly nonlinear SDDEs.We will finally conclude our paper in Section 6. Throughout this paper, unless otherwise specified, we use the following notation. Let | · | bethe Euclidean norm in R n . If A is a vector or matrix, its transpose is denoted by A T . If A is a matrix, its trace norm is denoted by | A | = p trace( A T A ). Let R + = [0 , ∞ ) and τ > C ([ − τ, R n ) the family of continuous functions from [ − τ,
0] to R n with the norm k ϕ k = sup − τ ≤ θ ≤ | ϕ ( θ ) | . Let (Ω , F , {F t } t ≥ , P ) be a complete probability space with a filtration {F t } t ≥ satisfying the usual conditions (i.e., it is increasing and right continuous while F containsall P -null sets). Let B ( t ) = ( B ( t ) , · · · , B m ( t )) T be an m -dimensional Brownian motion definedon the probability space. Moreover, for two real numbers a and b , we use a ∨ b = max( a, b ) and a ∧ b = min( a, b ). If G is a set, its indicator function is denoted by I G , namely I G ( x ) = 1 if x ∈ G and 0 otherwise. If a is a real number, we denote by ⌊ a ⌋ the largest integer which is less or equalto a , e.g., ⌊− . ⌋ = − ⌊ . ⌋ = 2.Consider a nonlinear SDDE dx ( t ) = f ( x ( t ) , x ( t − τ )) dt + g ( x ( t ) , x ( t − τ )) dB ( t ) , t ≥ , (2.1)with the initial data given by { x ( θ ) : − τ ≤ θ ≤ } = ξ ∈ C ([ − τ, R n ) . (2.2)Here f : R n × R n → R n and g : R n × R n → R n × m . We assume that the coefficients f and g obey the Local Lipschitz condition: Assumption 2.1
For every positive number R there is a positive constant K R such that | f ( x, y ) − f (¯ x, ¯ y ) | ∨ | g ( x, y ) − g (¯ x, ¯ y ) | ≤ K R ( | x − ¯ x | + | y − ¯ y | ) for those x, y, ¯ x, ¯ y ∈ R n with | x | ∨ | y | ∨ | ¯ x | ∨ | ¯ y | ≤ R . Assumption 2.2
There are constants K > , K ≥ and β > such that x T f ( x, y ) + 12 | g ( x, y ) | ≤ K (1 + | x | + | y | ) − K | x | β + K | y | β (2.3) for all ( x, y ) ∈ R n × R n . To have a feeling about what type of nonlinear SDDEs to which our theory may apply, pleaseconsider, for example, the scalar SDDE dx ( t ) = [ a + a x ( t − τ ) − a x ( t )] dt + [ a | x ( t ) | / + a | x ( t − τ ) | / ] dB ( t ) , t ≥ , where a > a , a , a , a ∈ R (see Example 3.7 for the details). The following result,established in [17], is a generalized Khasminskii-type theorem on the existence and uniqueness ofthe solution to the SDDE. Lemma 2.3
Let Assumptions 2.1 and 2.2 hold. Then for any given initial data (2.2), there isa unique global solution x ( t ) to equation (2.1) on t ∈ [ − τ, ∞ ) . Moreover, the solution has theproperty that sup − τ ≤ t ≤ T E | x ( t ) | < ∞ , ∀ T > . (2.4)It has been shown (see, e.g., [15]) that under Assumptions 2.1 and 2.2, the EM numericalsolutions converge to the true solution in probability. But, to our best knowledge, there is sofar no result on the strong convergence under these assumptions . In this paper, we will use thetruncated EM method developed in [16] and show that the truncated EM solutions will convergeto the true solution in L q for some q ≥ µ : R + → R + such that µ ( r ) → ∞ as r → ∞ andsup | x |∨| y |≤ r (cid:0) | f ( x, y ) | ∨ | g ( x, y ) | (cid:1) ≤ µ ( r ) , ∀ r ≥ . (2.5)Denote by µ − the inverse function of µ and we see that µ − is a strictly increasing continuousfunction from [ µ (0) , ∞ ) to R + . We also choose a constant ∆ ∗ ∈ (0 ,
1] and a strictly decreasingfunction h : (0 , ∆ ∗ ] → (0 , ∞ ) such that h (∆ ∗ ) ≥ µ (1) , lim ∆ → h (∆) = ∞ and ∆ / h (∆) ≤ , ∀ ∆ ∈ (0 , ∆ ∗ ] . (2.6)For example, we may choose ∆ ∗ ∈ (0 ,
1) sufficiently small such that 1 / ∆ ∗ ≥ ( µ (1)) and define h (∆) = ∆ − / for ∆ ∈ (0 , ∆ ∗ ]. For a given step size ∆ ∈ (0 , ∆ ∗ ], let us define a mapping π ∆ from R n to the closed ball { x ∈ R n : | x | ≤ µ − ( h (∆)) } by π ∆ ( x ) = ( | x | ∧ µ − ( h (∆))) x | x | , x/ | x | = 0 when x = 0. That is, π ∆ will map x to itself when | x | ≤ µ − ( h (∆)) and to µ − ( h (∆)) x/ | x | when | x | > µ − ( h (∆)). We then define the truncated functions f ∆ ( x, y ) = f ( π ∆ ( x ) , π ∆ ( y )) and g ∆ ( x, y ) = g ( π ∆ ( x ) , π ∆ ( y )) (2.7)for x, y ∈ R n . It is easy to see that | f ∆ ( x, y ) | ∨ | g ∆ ( x, y ) | ≤ µ ( µ − ( h (∆))) = h (∆) , ∀ x, y ∈ R n . (2.8)That is, both truncated functions f ∆ and g ∆ are bounded although f and g may not. Moreusefully, these truncated functions preserve the generalized Khasminskii-type condition to a verynice degree as described in the following lemma. Lemma 2.4
Let Assumption 2.2 hold. Then, for every ∆ ∈ (0 , ∆ ∗ ] , we have x T f ∆ ( x, y ) + 12 | g ∆ ( x, y ) | ≤ K (1 + | x | + | y | ) − K | π ∆ ( x ) | β + K | π ∆ ( y ) | β (2.9) for all x, y ∈ R n .Proof . Fix any ∆ ∈ (0 , ∆ ∗ ]. Recalling that h (∆ ∗ ) ≥ µ (1), we see that µ − ( h (∆ ∗ )) ≥
1. But h isdecreasing while µ − is increasing, so µ − ( h (∆)) ≥ x ∈ R n with | x | ≤ µ − ( h (∆)) and any y ∈ R n , we have, by (2.3), x T f ∆ ( x, y ) + 12 | g ∆ ( x, y ) | = π ∆ ( x ) T f ( π ∆ ( x ) , π ∆ ( y )) + 12 | g ( π ∆ ( x ) , π ∆ ( y )) | ≤ K (1 + | π ∆ ( x ) | + | π ∆ ( y ) | ) − K | π ∆ ( x ) | β + K | π ∆ ( y ) | β ≤ K (1 + | x | + | y | ) − K | π ∆ ( x ) | β + K | π ∆ ( y ) | β , (2.10)which implies the desired assertion (2.9). On the other hand, for x ∈ R n with | x | > µ − ( h (∆))and any y ∈ R n , we have x T f ∆ ( x, y ) + 12 | g ∆ ( x, y ) | = π ∆ ( x ) T f ( π ∆ ( x ) , π ∆ ( y )) + 12 | g ( π ∆ ( x ) , π ∆ ( y )) | + ( x − π ∆ ( x )) T f ( π ∆ ( x ) , π ∆ ( y )) ≤ K (1 + | π ∆ ( x ) | + | π ∆ ( y ) | ) − K | π ∆ ( x ) | β + K | π ∆ ( y ) | β + (cid:16) | x | µ − ( h (∆)) − (cid:17) π ∆ ( x ) T f ( π ∆ ( x ) , π ∆ ( y )) , (2.11)where (2.3) has been used. But once again we see from (2.3) that π ∆ ( x ) T f ( π ∆ ( x ) , π ∆ ( y )) ≤ K (1 + | π ∆ ( x ) | + | π ∆ ( y ) | ) − K [ µ − ( h (∆))] β + K | π ∆ ( y ) | β ≤ K (1 + | π ∆ ( x ) | + | π ∆ ( y ) | ) . x T f ∆ ( x, y ) + 12 | g ∆ ( x, y ) | ≤ K | x | µ − ( h (∆)) (1 + | π ∆ ( x ) | + | π ∆ ( y ) | ) − K | π ∆ ( x ) | β + K | π ∆ ( y ) | β ≤ K | x | (1 + | x | + | y | ) − K | π ∆ ( x ) | β + K | π ∆ ( y ) | β ≤ K (1 + | x | + | y | ) − K | π ∆ ( x ) | β + K | π ∆ ( y ) | β . (2.12)Namely, we have showed that the required assertion (2.9) also holds for x ∈ R n with | x | >µ − ( h (∆)) and any y ∈ R n . The proof is hence complete. ✷ From now on, we will let the step size ∆ be a fraction of τ . That is, we will use ∆ = τ /M for some positive integer M . When we use the terms of a sufficiently small ∆, we mean that wechoose M sufficiently large.Let us now form the discrete-time truncated EM solutions. Define t k = k ∆ for k = − M, − ( M − , · · · , , , , · · · . Set X ∆ ( t k ) = ξ ( t k ) for k = − M, − ( M − , · · · , X ∆ ( t k +1 ) = X ∆ ( t k ) + f ∆ ( X ∆ ( t k ) , X ∆ ( t k − M ))∆ + g ∆ ( X ∆ ( t k ) , X ∆ ( t k − M ))∆ B k (2.13)for k = 0 , , , · · · , where ∆ B k = B ( t k +1 ) − B ( t k ). In our analysis, it is more convenient towork on the continuous-time approximations. There are two continuous-time versions. One is thecontinuous-time step process ¯ x ∆ ( t ) on t ∈ [ − τ, ∞ ) defined by¯ x ∆ ( t ) = ∞ X k = − M X ∆ ( t k ) I [ k ∆ , ( k +1)∆) ( t ) . (2.14)The other one is the continuous-time continuous process x ∆ ( t ) on t ∈ [ − τ, ∞ ) defined by x ∆ ( t ) = ξ ( t ) for t ∈ [ − τ,
0] while for t ≥ x ∆ ( t ) = ξ (0) + Z t f ∆ (¯ x ∆ ( s ) , ¯ x ∆ ( s − τ )) ds + Z t g ∆ (¯ x ∆ ( s ) , ¯ x ∆ ( s − τ )) dB ( s ) . (2.15)We see that x ∆ ( t ) is an Itˆo process on t ≥ dx ∆ ( t ) = f ∆ (¯ x ∆ ( t ) , ¯ x ∆ ( t − τ )) dt + g ∆ (¯ x ∆ ( t ) , ¯ x ∆ ( t − τ )) dB ( t ) . (2.16)It is useful to know that X ∆ ( t k ) = ¯ x ∆ ( t k ) = x ∆ ( t k ) for every k ≥ − M , namely they coincide at t k .Of course, ¯ x ∆ ( t ) is computable but x ∆ ( t ) is not in general. However, the following lemma showsthat x ∆ ( t ) and ¯ x ∆ ( t ) are close to each other in the sense of L p . This indicates that it is sufficientto use ¯ x ∆ ( t ) in practice. On the other hand, in our analysis, it is more convenient to work on bothof them. Lemma 2.5
For any ∆ ∈ (0 , ∆ ∗ ] and any p ≥ , we have E | x ∆ ( t ) − ¯ x ∆ ( t ) | p ≤ c p ∆ p/ ( h (∆)) p , ∀ t ≥ , (2.17) where c p is a positive constant dependent only on p . Consequently lim ∆ → E | x ∆ ( t ) − ¯ x ∆ ( t ) | p = 0 , ∀ t ≥ . (2.18)5 roof . In what follows, we will use c p to stand for generic positive real constants dependent onlyon p and its values may change between occurrences. Fix ∆ ∈ (0 , ∆ ∗ ] arbitrarily. For any t ≥ k ≥ t k ≤ t < t k +1 . By (2.8) and the properties of the Itˆointegral (see, e.g., [13]), we then derive from (2.16) that E | x ∆ ( t ) − ¯ x ∆ ( t ) | p = E | x ∆ ( t ) − x ∆ ( t k ) | p ≤ c p (cid:16) E (cid:12)(cid:12)(cid:12) Z tt k f ∆ (¯ x ∆ ( s ) , ¯ x ∆ ( s − τ )) ds (cid:12)(cid:12)(cid:12) p + E (cid:12)(cid:12)(cid:12) Z tt k g ∆ (¯ x ∆ ( s ) , ¯ x ∆ ( s − τ )) dB ( s ) (cid:12)(cid:12)(cid:12) p (cid:17) ≤ c p (cid:16) ∆ p − E Z tt k | f ∆ (¯ x ∆ ( s ) , ¯ x ∆ ( s − τ )) | p ds + ∆ ( p − / E Z tt k | g ∆ (¯ x ∆ ( s ) , ¯ x ∆ ( s − τ )) | p ds (cid:17) ≤ c p ∆ p/ ( h (∆)) p , which is (2.17). Noting from (2.6) that ∆ p/ ( h (∆)) p ≤ ∆ p/ , we obtain (2.18) from (2.17) imme-diately. ✷ L q for q ∈ [1 , From now on we will fix
T > arbitrarily . In this section we will show thatlim ∆ → E | x ∆ ( T ) − x ( T ) | q = 0 and lim ∆ → E | ¯ x ∆ ( T ) − x ( T ) | q = 0for every 1 ≤ q <
2. By (2.8), it is obvious that for every p ≥ E | x ∆ ( t ) | p < ∞ , ∀ t ≥ . The following lemma gives an upper bound, independent of ∆, for the second moment.
Lemma 3.1
Let Assumptions 2.1 and 2.2 hold. Then sup < ∆ ≤ ∆ ∗ sup ≤ t ≤ T E | x ∆ ( t ) | ≤ C, (3.1) where, and from now on, C stands for generic positive real constants dependent on T, K , K , ξ (and ¯ p , K etc. as well in the next sections) but independent of ∆ and its values may changebetween occurrences.Proof . Fix ∆ ∈ (0 , ∆ ∗ ] and the initial data ξ arbitrarily. By the Itˆo formula, we derive from (2.16)that for 0 ≤ t ≤ T , E | x ∆ ( t ) | = | ξ (0) | + E Z t (cid:16) x T ∆ ( s ) f ∆ (¯ x ∆ ( s ) , ¯ x ∆ ( s − τ )) + | g ∆ (¯ x ∆ ( s ) , ¯ x ∆ ( s − τ )) | (cid:17) ds = | ξ (0) | + E Z t (cid:16) x T ∆ ( s ) f ∆ (¯ x ∆ ( s ) , ¯ x ∆ ( s − τ )) + | g ∆ (¯ x ∆ ( s ) , ¯ x ∆ ( s − τ )) | (cid:17) ds + E Z t x ∆ ( s ) − ¯ x ∆ ( s )) T f ∆ (¯ x ∆ ( s ) , ¯ x ∆ ( s − τ )) ds. By Lemma 2.4, we get E | x ∆ ( t ) | ≤ | ξ (0) | + 4 K E Z t (1 + | ¯ x ∆ ( s ) | + | ¯ x ∆ ( s − τ ) | ) ds − K E Z t | π ∆ (¯ x ∆ ( s )) | β ds + 2 K E Z t | π ∆ (¯ x ∆ ( s − τ )) | β ds + 2 E Z t | x ∆ ( s ) − ¯ x ∆ ( s ) || f ∆ (¯ x ∆ ( s ) , ¯ x ∆ ( s − τ )) | ds. (3.2)6owever, it is easy to show that | ξ (0) | + 4 K E Z t (1 + | ¯ x ∆ ( s ) | + | ¯ x ∆ ( s − τ ) | ) ds ≤ C + 8 K Z t (cid:16) sup ≤ u ≤ s E | x ∆ ( u ) | (cid:17) ds. (3.3)Moreover, − K E Z t | π ∆ (¯ x ∆ ( s )) | β ds + 2 K E Z t | π ∆ (¯ x ∆ ( s − τ )) | β ds = − K E Z t | π ∆ (¯ x ∆ ( s )) | β ds + 2 K E Z t − τ − τ | π ∆ (¯ x ∆ ( s )) | β ds ≤ K Z − τ | π ∆ (¯ x ∆ ( s )) | β ds ≤ τ K k ξ k β . (3.4)Furthermore, by Lemma 2.5 with p = 2 and inequalities (2.8) and (2.6), we derive that E Z t | x ∆ ( s ) − ¯ x ∆ ( s ) || f ∆ (¯ x ∆ ( s ) , ¯ x ∆ ( s − τ )) | ds ≤ h (∆) Z T E | x ∆ ( s ) − ¯ x ∆ ( s ) | ds ≤ h (∆) Z T ( E | x ∆ ( s ) − ¯ x ∆ ( s ) | ) / ds ≤ C ( h (∆)) ∆ / ≤ C. (3.5)Substituting (3.3)-(3.5) into (3.2) yields E | x ∆ ( t ) | ≤ C + 8 K Z t (cid:16) sup ≤ u ≤ s E | x ∆ ( u ) | (cid:17) ds. As this holds for any t ∈ [0 , T ] while the sum of the right-hand-side (RHS) terms is non-decreasingin t , we then see sup ≤ u ≤ t E | x ∆ ( u ) | ≤ C + 8 K Z t (cid:16) sup ≤ u ≤ s E | x ∆ ( u ) | (cid:17) ds. The well-known Gronwall inequality yields thatsup ≤ u ≤ T E | x ∆ ( u ) | ≤ C. As this holds for any ∆ ∈ (0 , ∆ ∗ ] while C is independent of ∆, we obtain the required assertion(3.1). ✷ Let us present two more lemmas before we state one of our main results in this paper.
Lemma 3.2
Let Assumptions 2.1 and 2.2 hold. For any real number
R > k ξ k , define the stoppingtime τ R = inf { t ≥ | x ( t ) | ≥ R } , here throughout this paper we set inf ∅ = ∞ (and as usual ∅ denotes the empty set). Then P ( τ R ≤ T ) ≤ CR . (3.6) (Recall that C stands for generic positive real constants dependent on T, K , K , ξ so C here isindependent of R .)Proof . By the Itˆo formula and Assumption 2.2, we derive that for 0 ≤ t ≤ T , E | x ( t ∧ τ R ) | ≤ | ξ (0) | + 2 K E Z t ∧ τ R (1 + | x ( s ) | + | x ( s − τ ) | ) ds − K E Z t ∧ τ R | x ( s ) | β ds + 2 K E Z t ∧ τ R | x ( s − τ ) | β ds ≤ | ξ (0) | + 2 K T + 2 K E Z t (cid:0) | x ( s ∧ τ R ) | + | x (( s − τ ) ∧ τ R ) | (cid:1) ds + 2 K Z − τ | ξ ( s ) | β ds ≤ C + 2 K Z t (cid:0) E | x ( s ∧ τ R ) | + E | x (( s − τ ) ∧ τ R ) | (cid:1) ds ≤ C + 4 K Z t (cid:16) sup ≤ u ≤ s E | x ( u ∧ τ R ) | (cid:17) ds But the sum of the RHS terms is non-decreasing in t , we hence havesup ≤ u ≤ t E | x ( u ∧ τ R ) | ≤ C + 4 K Z t (cid:16) sup ≤ u ≤ s E | x ( u ∧ τ R ) | (cid:17) ds. The Gronwall inequality shows sup ≤ u ≤ T E | x ( u ∧ τ R ) | ≤ C. In particular, we have E | x ( T ∧ τ R ) | ≤ C. This implies, by the Chebyshev inequality, R P ( τ R ≤ T ) ≤ C and the assertion follows. ✷ Lemma 3.3
Let Assumptions 2.1 and 2.2 hold. For any real number
R > k ξ k and ∆ ∈ (0 , ∆ ∗ ] ,define the stopping time ρ ∆ ,R = inf { t ≥ | x ∆ ( t ) | ≥ R } . Then P ( ρ ∆ ,R ≤ T ) ≤ CR . (3.7) (Please recall that C is independent of ∆ and R .) roof . We simply write ρ ∆ ,R = ρ . In the same way as (3.2) was obtained, we can show that for0 ≤ t ≤ T , E | x ∆ ( t ∧ ρ ) | ≤ | ξ (0) | + 4 K E Z t ∧ ρ (1 + | ¯ x ∆ ( s ) | + | ¯ x ∆ ( s − τ ) | ) ds − K E Z t ∧ ρ | π ∆ (¯ x ∆ ( s )) | β ds + 2 K E Z t ∧ ρ | π ∆ (¯ x ∆ ( s − τ )) | β ds + 2 E Z t ∧ ρ | x ∆ ( s ) − ¯ x ∆ ( s ) || f ∆ (¯ x ∆ ( s ) , ¯ x ∆ ( s − τ )) | ds. (3.8)In the same way as we performed in the proofs of Lemmas 3.1 and 3.2, we can then show that E | x ∆ ( t ∧ ρ ) | ≤ C + 8 K Z t (cid:16) sup ≤ u ≤ s E | ¯ x ∆ ( u ∧ ρ ) | (cid:17) ds + 2 E Z t | x ∆ ( s ) − ¯ x ∆ ( s ) || f ∆ (¯ x ∆ ( s ) , ¯ x ∆ ( s − τ )) | ds. (3.9)This, together with (3.5), implies E | x ∆ ( t ∧ ρ ) | ≤ C + 8 K Z t (cid:16) sup ≤ u ≤ s E | ¯ x ∆ ( u ∧ ρ ) | (cid:17) ds. Noting that the sum of the RHS terms is increasing in t whilesup ≤ u ≤ s E | ¯ x ∆ ( u ∧ ρ ) | ≤ sup ≤ u ≤ s E | x ∆ ( u ∧ ρ ) | , we get sup ≤ u ≤ t E | x ∆ ( u ∧ ρ ) | ≤ C + 8 K Z t (cid:16) sup ≤ u ≤ s E | x ∆ ( u ∧ ρ ) | (cid:17) ds. The Gronwall inequality shows sup ≤ u ≤ T E | x ∆ ( u ∧ ρ ) | ≤ C. This implies the required assertion (3.7) easily. ✷ For the numerical solutions to converge to the true solution in L q , we need to assume that theinitial data are H¨older continuous with exponent γ (or γ -H¨older continuous). This is a standardcondition which is also needed for the classical EM method under the global Lipschitz condition(see, e.g., [18, 19, 22]). Assumption 3.4
There is a pair of constants K > and γ ∈ (0 , such that the initial data ξ satisfies | ξ ( u ) − ξ ( v ) | ≤ K | u − v | γ , − τ ≤ v < u ≤ . We can now show one of our main results in this paper.
Theorem 3.5
Let Assumptions 2.1, 2.2 and 3.4 hold. Then, for any q ∈ [1 , , lim ∆ → E | x ∆ ( T ) − x ( T ) | q = 0 and lim ∆ → E | ¯ x ∆ ( T ) − x ( T ) | q = 0 . (3.10)9 roof . Let τ R and ρ ∆ ,R be the same as before. Set θ ∆ ,R = τ R ∧ ρ ∆ ,R and e ∆ ( T ) = x ∆ ( T ) − x ( T ) . Obviously E | e ∆ ( T ) | q = E (cid:16) | e ∆ ( T ) | q I { θ ∆ ,R >T } (cid:17) + E (cid:16) | e ∆ ( T ) | q I { θ ∆ ,R ≤ T } (cid:17) . (3.11)Let δ > a q b = ( δa ) q/ (cid:16) b / (2 − q ) δ q/ (2 − q ) (cid:17) (2 − q ) / ≤ qδ a + 2 − q δ q/ (2 − q ) b / (2 − q ) , ∀ a, b > , we have E (cid:16) | e ∆ ( T ) | q I { θ ∆ ,R ≤ T } (cid:17) ≤ qδ E | e ∆ ( T ) | + 2 − q δ q/ (2 − q ) P ( θ ∆ ,R ≤ T ) . By Lemmas 2.3 and 3.1, we have E | e ∆ ( T ) | ≤ C, while by Lemmas 3.2 and 3.3, P ( θ ∆ ,R ≤ T ) ≤ P ( τ R ≤ T ) + P ( ρ ∆ ,R ≤ T ) ≤ CR . We hence have E (cid:16) | e ∆ ( T ) | q I { θ ∆ ,R ≤ T } (cid:17) ≤ Cqδ C (2 − q )2 R δ q/ (2 − q ) . Substituting this into (3.11) yields E | e ∆ ( T ) | q ≤ E (cid:16) | e ∆ ( T ) | q I { θ ∆ ,R >T } (cid:17) + Cqδ C (2 − q )2 R δ q/ (2 − q ) . (3.12)Now, let ε > δ sufficiently small for Cqδ/ ≤ ε/ R sufficiently large for C (2 − q )2 R δ q/ (2 − q ) ≤ ε . We then see from (3.12) that for this particularly chosen R , E | e ∆ ( T ) | q ≤ E (cid:16) | e ∆ ( T ) | q I { θ ∆ ,R >T } (cid:17) + 2 ε . (3.13)If we can show that for all sufficiently small ∆, E (cid:16) | e ∆ ( T ) | q I { θ ∆ ,R >T } (cid:17) ≤ ε , (3.14)we have lim ∆ → E | e ∆ ( T ) | q = 0 , and then by Lemma 2.5, we also havelim ∆ → E | x ( T ) − ¯ x ∆ ( T ) | q = 0 . In other words, to complete our proof, all we need is to show (3.14). For this purpose, we definethe truncated functions F R ( x, y ) = f (cid:16) ( | x | ∧ R ) x | x | , ( | y | ∧ R ) y | y | (cid:17) and G R ( x, y ) = g (cid:16) ( | x | ∧ R ) x | x | , ( | y | ∧ R ) y | y | (cid:17) x, y ∈ R n . Without loss of any generality, we may assume that ∆ ∗ is already sufficiently smallfor µ − ( h (∆ ∗ )) ≥ R . Hence, for all ∆ ∈ (0 , ∆ ∗ ], we have that f ∆ ( x, y ) = F R ( x, y ) and g ∆ ( x, y ) = G R ( x, y )for those x, y ∈ R n with | x | ∨ | y | ≤ R . Consider the SDDE dz ( t ) = F R ( z ( t ) , z ( t − τ )) dt + G R ( z ( t ) , z ( t − τ )) dB ( t ) (3.15)on t ≥ z ( u ) = ξ ( u ) on u ∈ [ − τ, F R ( x, y ) and G R ( x, y ) are globally Lipschitz continuous with the Lipschitz constant K R . So theSDDE (3.15) has a unique global solution z ( t ) on t ≥ − τ . It is straightforward to see that P { x ( t ∧ τ R ) = z ( t ∧ τ R ) for all 0 ≤ t ≤ T } = 1 . (3.16)On the other hand, for each step size ∆ ∈ (0 , ∆ ∗ ], we can apply the (classical) EM method tothe SDDE (3.15) and we denote by z ∆ ( t ) the continuous-time continuous EM solution. It is againstraightforward to see that P { x ∆ ( t ∧ ρ ∆ ,R ) = z ∆ ( t ∧ ρ ∆ ,R ) for all 0 ≤ t ≤ T } = 1 . (3.17)However, it is well known (see, e.g., [18, 19]) that E (cid:16) sup ≤ t ≤ T | z ( t ) − z ∆ ( t ) | q (cid:17) ≤ H ∆ q (0 . ∧ γ ) , (3.18)where H is a positive constant dependent on K R , T, ξ, q but independent of ∆. Consequently, E (cid:16) sup ≤ t ≤ T | z ( t ∧ θ ∆ ,R ) − z ∆ ( t ∧ θ ∆ ,R ) | q (cid:17) ≤ H ∆ q (0 . ∧ γ ) . Using (3.16) and (3.17), we then have E (cid:16) sup ≤ t ≤ T | x ( t ∧ θ ∆ ,R ) − x ∆ ( t ∧ θ ∆ ,R ) | q (cid:17) ≤ H ∆ q (0 . ∧ γ ) , (3.19)which implies E (cid:16) | x ( T ∧ θ ∆ ,R ) − x ∆ ( T ∧ θ ∆ ,R ) | q (cid:17) ≤ H ∆ q (0 . ∧ γ ) . Finally E (cid:16) | e ∆ ( T ) | q I { θ ∆ ,R >T } (cid:17) = E (cid:16) | e ∆ ( T ∧ θ ∆ ,R ) | q I { θ ∆ ,R >T } (cid:17) ≤ E (cid:16) | x ( T ∧ θ ∆ ,R ) − x ∆ ( T ∧ θ ∆ ,R ) | q (cid:17) ≤ H ∆ q (0 . ∧ γ ) . (3.20)This implies (3.14) as desired. The proof is therefore complete. ✷ Let make a useful remark which will be used in next sections before we discuss an example toillustrate our theory.
Remark 3.6
It is known (see, e.g., [18, 19]) that (3.18) holds for any q ≥ . We hence see fromthe proof above that both (3.19) and (3.20) hold for any q ≥ too. xample 3.7 Consider the scalar SDDE dx ( t ) = x ( t ) (cid:16) [ a + a x ( t − τ ) − a x ( t )] dt + [ a x ( t ) + a x ( t − τ )] dB ( t ) (cid:17) , t ≥ , (3.21)with the initial data { x ( θ ) : − τ ≤ θ ≤ } = ξ ∈ C ([ − τ, , ∞ )), where B ( t ) is a scalar Brownianmotion and a i (1 ≤ i ≤
5) are all positive numbers with a > a + a . (3.22)This is a stochastic delay population system (see, e.g., [1, 2, 20]). It can be shown that given theinitial data { x ( θ ) : − τ ≤ θ ≤ } = ξ ∈ C ([ − τ, , ∞ )), the solution will remain positive forall t ≥ R with thecoefficients f ( x, y ) = x ( a + a y − a x ) and g ( x, y ) = x ( a x + a y ) , x, y ∈ R . It is obvious that these coefficients are locally Lipschitz continuous, namely, they satisfy As-sumption 2.1. We also assume that the initial data satisfy Assumption 3.4. Moreover, we set δ = a − a − a , which is positive by (3.22), and derive xf ( x, y ) + 12 | g ( x, y ) | ≤ a x + a x | y | − a x + a x + a x y ≤ a x + ( a / δ ) y − ( a − δ − a − . a ) x + 0 . a y ≤ ( a ∨ ( a / δ ))(1 + x + y ) − . a x + 0 . a y . That is, Assumption 2.2 is satisfied as well. We can therefore apply the truncated EM method toobtain the numerical solutions of the SDDE (3.21). For this purpose, we observe that, for r ≥ | x |∨| y |≤ r ( | f ( x, y ) | ∨ | g ( x, y ) | ) ≤ ( a r + a r + a r ) ∨ (( a + a ) r ) ≤ ar , where a = ( a + a + a ) ∨ ( a + a ). We can therefore define µ : R + → R + by µ ( r ) = ar , r ≥ . Its inverse function µ − : R + → R + has the form µ − ( r ) = (cid:16) ra (cid:17) / , r ≥ . Let ρ ∈ (0 , /
4] and ∆ ∗ = (1 ∨ (8 a )) − /ρ ∈ (0 , h (∆) = ∆ − ρ for ∆ ∈ (0 , ∆ ∗ ]. We then seethat h (∆ ∗ ) ≥ a = µ (2), lim ∆ → h (∆) = ∞ and∆ / h (∆) = ∆ / − ρ ≤ , ∀ ∆ ∈ (0 , ∆ ∗ ]as required by (2.6). With these chosen functions µ and h , we can then apply the truncatedEM method to obtain the numerical solutions x ∆ ( t ) and ¯ x ∆ ( t ) of the SDDE (3.21). Moreover,Theorem 3.5 shows that these numerical solutions will converge to the true solution x ( t ) in thesense that lim ∆ → E | x ∆ ( t ) − x ( t ) | q = 0 and lim ∆ → E | ¯ x ∆ ( t ) − x ( t ) | q = 0for any q ∈ [1 , Convergence in L q for q ≥ In the previous section, we showed that the truncated EM solutions x ∆ ( T ) and ¯ x ∆ ( T ) will convergeto the true solution x ( T ) in L q for any q ∈ [1 , L q for q ≥
2. Forthis purpose, we impose a stronger Khasminskii-type condition.
Assumption 4.1
There is a pair of constants ¯ p > and K > such that x T f ( x, y ) + ¯ p − | g ( x, y ) | ≤ K (1 + | x | + | y | ) (4.1) for all ( x, y ) ∈ R n × R n . Once again, the truncated functions f ∆ and g ∆ preserve this condition nicely. Lemma 4.2
Let Assumption 4.1 hold. Then, for every ∆ ∈ (0 , ∆ ∗ ] , we have x T f ∆ ( x, y ) + ¯ p − | g ∆ ( x, y ) | ≤ K (1 + | x | + | y | ) (4.2) for all x, y ∈ R n . This lemma can be proved in the same way as Lemma 2.4 was proved. We also cite a strongerresult than Lemma 2.3 from [17].
Lemma 4.3
Let Assumptions 2.1 and 4.1 hold. Then for any given initial data (2.2), there isa unique global solution x ( t ) to equation (2.1) on t ∈ [ − τ, ∞ ) . Moreover, the solution has theproperty that sup − τ ≤ t ≤ T E | x ( t ) | ¯ p < ∞ . (4.3)Let us now establish a stronger result than Lemma 3.1. Lemma 4.4
Let Assumptions 2.1 and 4.1 hold. Then sup < ∆ ≤ ∆ ∗ sup ≤ t ≤ T E | x ∆ ( t ) | ¯ p ≤ C. (4.4) Proof . Fix any ∆ ∈ (0 , ∆ ∗ ]. By the Itˆo formula, we derive from (2.16) that, for 0 ≤ t ≤ T , E | x ∆ ( t ) | ¯ p ≤ | ξ (0) | ¯ p + E Z t ¯ p | x ∆ ( s ) | ¯ p − × (cid:16) x T ∆ ( s ) f ∆ (¯ x ∆ ( s ) , ¯ x ∆ ( s − τ )) + ¯ p − | g ∆ (¯ x ∆ ( s ) , ¯ x ∆ ( s − τ )) | (cid:17) ds = | ξ (0) | ¯ p + E Z t ¯ p | x ∆ ( s ) | ¯ p − × (cid:16) ¯ x T ∆ ( s ) f ∆ (¯ x ∆ ( s ) , ¯ x ∆ ( s − τ )) + ¯ p − | g ∆ (¯ x ∆ ( s ) , ¯ x ∆ ( s − τ )) | (cid:17) ds + E Z t ¯ p | x ∆ ( s ) | ¯ p − ( x ∆ ( s ) − ¯ x ∆ ( s )) T f ∆ (¯ x ∆ ( s ) , ¯ x ∆ ( s − τ )) ds.
13y Lemma 4.2 and the Young inequality a ¯ p − b ≤ ¯ p − p a ¯ p + 2¯ p b ¯ p/ , ∀ a, b ≥ , we then have E | x ∆ ( t ) | ¯ p ≤ | ξ (0) | ¯ p + E Z t pK | x ∆ ( s ) | ¯ p − (1 + | ¯ x ∆ ( s ) | + | ¯ x ∆ ( s − τ ) | ) ds + (¯ p − E Z t | x ∆ ( s ) | ¯ p ds + 2 E Z t | x ∆ ( s ) − ¯ x ∆ ( s ) | ¯ p/ | f ∆ (¯ x ∆ ( s ) , ¯ x ∆ ( s − τ )) | ¯ p/ ds ≤ C + C Z t (cid:0) E | x ∆ ( s ) | ¯ p + E | ¯ x ∆ ( s ) | ¯ p + E | ¯ x ∆ ( s − τ ) | ¯ p (cid:1) ds + 2 E Z T | x ∆ ( s ) − ¯ x ∆ ( s ) | ¯ p/ | f ∆ (¯ x ∆ ( s ) , ¯ x ∆ ( s − τ )) | ¯ p/ ds. But, by Lemma 2.5 with p = ¯ p and inequalities (2.8) and (2.6), we have E Z T | x ∆ ( s ) − ¯ x ∆ ( s ) | ¯ p/ | f ∆ (¯ x ∆ ( s ) , ¯ x ∆ ( s − τ )) | ¯ p/ ds ≤ ( h (∆)) ¯ p/ Z T E ( | x ∆ ( s ) − ¯ x ∆ ( s ) | ¯ p/ ) ds ≤ ( h (∆)) ¯ p/ Z T ( E | x ∆ ( s ) − ¯ x ∆ ( s ) | ¯ p ) / ds ≤ c ¯ p T ( h (∆)) ¯ p ∆ ¯ p/ ≤ c ¯ p T. (4.5)We therefore have E | x ∆ ( t ) | ¯ p ≤ C + C Z t (cid:0) E | x ∆ ( s ) | ¯ p + E | ¯ x ∆ ( s ) | ¯ p + E | ¯ x ∆ ( s − τ ) | ¯ p (cid:1) ds ≤ C + C Z t (cid:16) sup ≤ u ≤ s E | x ∆ ( u ) | ¯ p (cid:17) ds. As this holds for any t ∈ [0 , T ] while the sum of the RHS terms is non-decreasing in t , we then seesup ≤ u ≤ t E | x ∆ ( u ) | ¯ p ≤ C + C Z t (cid:16) sup ≤ u ≤ s E | x ∆ ( u ) | ¯ p (cid:17) ds. The well-known Gronwall inequality yields thatsup ≤ u ≤ T E | x ∆ ( u ) | ¯ p ≤ C. As this holds for any ∆ ∈ (0 , ∆ ∗ ] while C is independent of ∆, we see the required assertion (4.4). ✷ The following two lemmas are the analogues of Lemmas 3.2 and 3.3.
Lemma 4.5
Let Assumptions 2.1 and 4.1 hold. For any real number
R > k ξ k , define the stoppingtime τ R = inf { t ≥ | x ( t ) | ≥ R } . Then P ( τ R ≤ T ) ≤ CR ¯ p . (4.6)14 emma 4.6 Let Assumptions 2.1 and 4.1 hold. For any real number
R > k ξ k and ∆ ∈ (0 , ∆ ∗ ] ,define the stopping time ρ ∆ ,R = inf { t ≥ | x ∆ ( t ) | ≥ R } . Then P ( ρ ∆ ,R ≤ T ) ≤ CR ¯ p . (4.7)Their proofs are similar to those of Lemmas 3.2 and 3.3, respectively, so are omitted. We cannow state our main result in this section. Theorem 4.7
Let Assumptions 2.1, 3.4 and 4.1 hold. Then, for any q ∈ [2 , ¯ p ) , lim ∆ → E | x ∆ ( T ) − x ( T ) | q = 0 and lim ∆ → E | ¯ x ∆ ( T ) − x ( T ) | q = 0 . (4.8) Proof . We use the same notation as in the proof of Theorem 3.5. Fix any q ∈ [2 , ¯ p ). Using theYoung inequality, we can show that for any δ > E | e ∆ ( T ) | q ≤ E (cid:16) | e ∆ ( T ) | q I { θ ∆ ,R >T } (cid:17) + qδ ¯ p E | e ∆ ( T ) | ¯ p + ¯ p − q ¯ pδ q/ (¯ p − q ) P ( θ ∆ ,R ≤ T ) . (4.9)By Lemmas 4.3 and 4.4, we have E | e ∆ ( T ) | ¯ p ≤ C, (4.10)while by Lemmas 4.5 and 4.6, P ( θ ∆ ,R ≤ T ) ≤ P ( τ R ≤ T ) + P ( ρ ∆ ,R ≤ T ) ≤ CR ¯ p . (4.11)Using these and (3.20) (please recall Remark 3.6), we obtain E | e ∆ ( T ) | q ≤ H ∆ q (0 . ∧ γ ) + Cqδ ¯ p + C (¯ p − q )¯ pR ¯ p δ q/ (¯ p − q ) . (4.12)Now, for any ε >
0, we first choose δ sufficiently small for Cqδ/ ¯ p ≤ ε/ R sufficiently large for C (¯ p − q )¯ pR ¯ p δ q/ (¯ p − q ) ≤ ε , and further then choose ∆ sufficiently small for H ∆ q (0 . ∧ γ ) ≤ ε/ E | e ∆ ( T ) | q ≤ ε. (4.13)In other words, we have shown that lim ∆ → E | e ∆ ( T ) | q = 0 . This, along with Lemma 2.5, implies another assertionlim ∆ → E | x ( T ) − ¯ x ∆ ( T ) | q = 0 . The proof is therefore complete. ✷ Let us now discuss an example to illustrate this theorem before we study the convergencerates. 15 xample 4.8
Consider the scalar SDDE dx ( t ) = f ( x ( t ) , x ( t − τ )) dt + g ( x ( t ) , x ( t − τ )) dB ( t ) , t ≥ , (4.14)with the initial data { x ( θ ) : − τ ≤ θ ≤ } = ξ ∈ C ([ − τ, R ) which satisfy Assumption 3.4, where f ( x, y ) = a + a | y | / − a x and g ( x, y ) = a | x | / + a y, x, y ∈ R , and a , · · · , a are all real numbers with a >
0. Clearly, the coefficients f and g are locallyLipschitz continuous, namely, they satisfy Assumption 2.1. Moreover, for any ¯ p >
2, we have xf ( x, y ) + ¯ p − | g ( x, y ) | ≤ | a || x | + | a || x || y | / − a | x | + (¯ p − | a || x | + | a || y | ) . But, by the Young inequality, | x || y | / = ( | x | ) / ( | y | ) / ≤ | x | + | y | . We therefore have xf ( x, y ) + ¯ p − | g ( x, y ) | ≤ | a || x | + ( | a | + | a | (¯ p − | x | − a | x | + ( | a | + a (¯ p − | y | ≤ K (1 + | y | ) , where K = ( | a | + | a | (¯ p − ∨ K and K = sup u ≥ (cid:2) | a | u + ( | a | + | a | (¯ p − u − a u (cid:3) < ∞ . That is, Assumption 4.1 is satisfied for any ¯ p >
2. To apply Theorem 4.7, we still need to designfunctions µ and h satisfying (2.5) and (2.6). Note thatsup | x |≤ u ( | f ( x ) | ∨ | g ( x ) | ) ≤ ˆ au , ∀ u ≥ , where ˆ a = ( | a | + | a | + a ) ∨ ( | a | + | a | ). We can hence have µ ( u ) = ˆ au and its inverse function µ − ( u ) = ( u/ ˆ a ) / for u ≥
0. For ε ∈ (0 , / h (∆) = ∆ − ε for ∆ >
0. Letting ∆ ∗ ∈ (0 , x ( t ) in the sense thatlim ∆ → E | x ∆ ( T ) − x ( T ) | q = 0 and lim ∆ → E | ¯ x ∆ ( T ) − x ( T ) | q = 0for every q ≥ In the previous sections, we showed the convergence in L q of the truncated EM solutions to thetrue solution. However, the convergence was in the asymptotic form without the convergence rate.In this section we will discuss the rate. To avoid the notation becoming too complicated, we willonly discuss the convergence rate in L but the technique developed here can certainly be appliedto study the rate in L q . Recall that we use two functions µ ( · ) and h ( · ) to define the truncated EMmethod. The choices of these functions are independent as long as they satisfy (2.5) and (2.6),16espectively. It is interesting to see that they will satisfy a related condition in order for us toobtain the convergence rate.We need an additional condition. To state it, we need a new notation. Let U denote thefamily of continuous functions U : R n × R n → R + such that for each b >
0, there is a positiveconstant κ b for which U ( x, ¯ x ) ≤ κ b | x − ¯ x | , ∀ x, ¯ x ∈ R n with | x | ∨ | ¯ x | ≤ b. Assumption 5.1
Assume that there is a positive constant H and a function U ∈ U such that ( x − ¯ x ) T ( f ( x, y ) − f (¯ x, ¯ y )) + 12 | g ( x, y ) − g (¯ x, ¯ y ) | ≤ H ( | x − ¯ x | + | y − ¯ y | ) − U ( x, ¯ x ) + U ( y, ¯ y ) (5.1) for all x, y, ¯ x, ¯ y ∈ R n . Let us first present a key lemma.
Lemma 5.2
Let Assumptions 2.1, 3.4 and 5.1 hold. Let
R > k ξ k be a real number and let ∆ ∈ (0 , ∆ ∗ ) be sufficiently small such that µ − ( h (∆)) ≥ R . Let θ ∆ ,R and e ∆ ( t ) be the same asdefined in Section 3. Then E | e ∆ ( T ∧ θ ∆ ,R ) | ≤ C (∆ γ ∨ [∆ / ( h (∆)) ]) , (5.2) where, as before, C is the generic constant independent of R and ∆ .Proof . We write θ ∆ ,R = θ for simplicity. The Itˆo formula shows that E | e ∆ ( t ∧ θ ) | = E Z t ∧ θ (cid:16) e T ∆ ( s )[ f ( x ( s ) , x ( s − τ )) − f ∆ (¯ x ∆ ( s ) , ¯ x ∆ ( s − τ ))]+ | g ( x ( s ) , x ( s − τ )) − g ∆ (¯ x ∆ ( s ) , ¯ x ∆ ( s − τ )) | (cid:17) ds (5.3)for 0 ≤ t ≤ T . We observe that for 0 ≤ s ≤ t ∧ θ , | ¯ x ∆ ( s ) | ∨ | ¯ x ∆ ( s − τ ) | ∨ | x ( s ) | ∨ | x ( s − τ ) | ≤ R. But we have the condition that µ − ( h (∆)) ≥ R , so | ¯ x ∆ ( s ) | ∨ | ¯ x ∆ ( s − τ ) | ∨ | x ( s ) | ∨ | x ( s − τ ) | ≤ µ − ( h (∆)) . Recalling the definition of the truncated functions f ∆ and g ∆ as well as (2.5), we hence have that f ∆ (¯ x ∆ ( s ) , ¯ x ∆ ( s − τ )) = f (¯ x ∆ ( s ) , ¯ x ∆ ( s − τ )) , g ∆ (¯ x ∆ ( s ) , ¯ x ∆ ( s − τ )) = g (¯ x ∆ ( s ) , ¯ x ∆ ( s − τ ))and | f ( x ( s ) , x ( s − τ )) | ∨ | f (¯ x ∆ ( s ) , ¯ x ∆ ( s − τ )) | ≤ h (∆) (5.4)17or 0 ≤ s ≤ t ∧ θ . It therefore follows from (5.3) that E | e ∆ ( t ∧ θ ) | = E Z t ∧ θ (cid:16) e T ∆ ( s )[ f ( x ( s ) , x ( s − τ )) − f (¯ x ∆ ( s ) , ¯ x ∆ ( s − τ ))]+ | g ( x ( s ) , x ( s − τ )) − g (¯ x ∆ ( s ) , ¯ x ∆ ( s − τ )) | (cid:17) ds (5.5)= E Z t ∧ θ (cid:16) x ( s ) − ¯ x ∆ ( s )) T [ f ( x ( s ) , x ( s − τ )) − f (¯ x ∆ ( s ) , ¯ x ∆ ( s − τ ))]+ | g ( x ( s ) , x ( s − τ )) − g (¯ x ∆ ( s ) , ¯ x ∆ ( s − τ )) | (cid:17) ds + E Z t ∧ θ x ∆ ( s ) − x ∆ ( s )) T [ f ( x ( s ) , x ( s − τ )) − f (¯ x ∆ ( s ) , ¯ x ∆ ( s − τ ))] ds. By Assumption 5.1 and (5.4), we then derive that E | e ∆ ( t ∧ θ ) | ≤ H E Z t ∧ θ (cid:16) | x ( s ) − ¯ x ∆ ( s ) | + | x ( s − τ ) − ¯ x ∆ ( s − τ ) | (cid:17) ds + E Z t ∧ θ (cid:16) − U ( x ( s ) , ¯ x ∆ ( s )) + U ( x ( s − τ ) , ¯ x ∆ ( s − τ )) (cid:17) ds + 4 h (∆) E Z t ∧ θ | ¯ x ∆ ( s ) − x ∆ ( s ) | ds. (5.6)But, by Assumption 3.4 and Lemma 2.5, we derive that E Z t ∧ θ (cid:16) | x ( s ) − ¯ x ∆ ( s ) | + | x ( s − τ ) − ¯ x ∆ ( s − τ ) | (cid:17) ds ≤ E Z t ∧ θ (cid:16) | e ∆ ( s ) | + | e ∆ ( s − τ ) | + | x ∆ ( s ) − ¯ x ∆ ( s ) | + | x ∆ ( s − τ ) − ¯ x ∆ ( s − τ ) | (cid:17) ds ≤ E Z t | e ∆ ( s ∧ θ ) | ds + 4 Z T E | x ∆ ( s ) − ¯ x ∆ ( s ) | ds + Z − τ | ξ ( s ) − ξ ( ⌊ s/ ∆ ⌋ ∆) | ds ≤ Z t E | e ∆ ( s ∧ θ ) | ds + C ∆( h (∆)) + τ K ∆ γ . (5.7)Moreover, by the property of the U -class function U and Assumption 3.4, we have E Z t ∧ θ (cid:16) − U ( x ( s ) , ¯ x ∆ ( s )) + U ( x ( s − τ ) , ¯ x ∆ ( s − τ )) (cid:17) ds ≤ Z − τ U ( ξ ( s ) , ξ ( ⌊ s/ ∆ ⌋ ∆)) ds ≤ Z − τ κ b | ξ ( s ) − ξ ( ⌊ s/ ∆ ⌋ ∆) | ds ≤ τ κ b K ∆ γ , (5.8)where b = k ξ k . Furthermore, by Lemma 2.5, E Z t ∧ θ | ¯ x ∆ ( s ) − x ∆ ( s ) | ds ≤ Z T E | ¯ x ∆ ( s ) − x ∆ ( s ) | ds ≤ C ∆ / h (∆) . (5.9)Substituting (5.7)-(5.9) into (5.6), we get E | e ∆ ( t ∧ θ ) | ≤ H Z t E | e ∆ ( s ∧ θ ) | ds + C (∆ γ ∨ [∆ / ( h (∆)) ]) .
18y the Gronwall inequality, we obtain the required assertion (5.2). ✷ Let us now state our first result on the convergence rate, where we reveal a strong relationbetween functions µ ( · ) and h ( · ), which are used to define the truncated EM method. Theorem 5.3
Let Assumptions 2.1, 5.1, 4.1 and 3.4 hold. Assume that h (∆) ≥ µ (cid:0) (∆ γ ∨ [∆ / ( h (∆)) ]) − / (¯ p − (cid:1) (5.10) for all sufficiently small ∆ ∈ (0 , ∆ ∗ ) . Then, for every such small ∆ , E | x ( T ) − x ∆ ( T ) | ≤ C (∆ γ ∨ [∆ / ( h (∆)) ]) (5.11) and E | x ( T ) − ¯ x ∆ ( T ) | ≤ C (∆ γ ∨ [∆ / ( h (∆)) ]) . (5.12) Proof . We use the same notation as in the proof of Theorem 4.7. It follows from (4.9)-(4.11) with q = 2 that the inequality E | e ∆ ( T ) | ≤ E (cid:16) | e ∆ ( T ∧ θ ∆ ,R ) | (cid:17) + 2 Cδ ¯ p + C (¯ p − pR ¯ p δ / (¯ p − (5.13)holds for any ∆ ∈ (0 , ∆ ∗ ), R > k ξ k and δ >
0. In particular, choosing δ = ∆ γ ∨ [∆ / ( h (∆)) ] and R = (∆ γ ∨ [∆ / ( h (∆)) ]) − / (¯ p − , we get E | e ∆ ( T ) | ≤ E | e ∆ ( T ∧ θ ∆ ,R ) | + C (∆ γ ∨ [∆ / ( h (∆)) ]) . (5.14)But, by condition (5.10), we have µ − ( h (∆)) ≥ (∆ γ ∨ [∆ / ( h (∆)) ]) − / (¯ p − = R. We can hence apply Lemma 5.2 to obtain E | e ∆ ( T ∧ θ ∆ ,R ) | ≤ C (∆ γ ∨ [∆ / ( h (∆)) ]) . (5.15)Substituting this into (5.14) yields the first assertion (5.11) . The second assertion (5.12) followsfrom (5.11) and Lemma 2.5. ✷ Let us discuss an example to illustrate Theorem 5.3 and to motivate our further results onthe convergence rates.
Example 5.4
Consider the same SDE in Example 4.8. We need to verify Assumption 5.1. For x, y, ¯ x, ¯ y ∈ R , it is easy to show that( x − ¯ x )( f ( x, y ) − f (¯ x, ¯ y )) ≤ a | x − ¯ x | + ( | y | / − | ¯ y | / ) − . a | x − ¯ x | ( x + ¯ x ) . (5.16)But, by the mean value theorem,( | y | / − | ¯ y | / ) ≤ | y − ¯ y | ( | y | / + | ¯ y | / ) ≤ | y − ¯ y | ( | y | / + | ¯ y | / ) . Let a := sup u ≥ (8 u / − . a u ). Then 0 ≤ a < ∞ and( | y | / − | ¯ y | / ) ≤ a | y − ¯ y | + 0 . a | y − ¯ y | ( y + ¯ y ) . x − ¯ x )( f ( x, y ) − f (¯ x, ¯ y )) ≤ ( a ∨ a )( | x − ¯ x | + | y − ¯ y | ) − . a | x − ¯ x | ( x + ¯ x ) + 0 . a | y − ¯ y | ( y + ¯ y ) . (5.17)Similarly, we can show that0 . | g ( x, y ) − g (¯ x, ¯ y ) | ≤ ( a ∨ a )( | x − ¯ x | + | y − ¯ y | ) + 0 . a | x − ¯ x | ( x + ¯ x ) , (5.18)where a := sup u ≥ (9 a u − . a u ) ∈ (0 , ∞ ). It then follows from (5.17) and (5.18) that( x − ¯ x )( f ( x, y ) − f (¯ x, ¯ y )) + 0 . | g ( x, y ) − g (¯ x, ¯ y ) | ≤ H ( | x − ¯ x | + | y − ¯ y | ) − U ( x, ¯ x ) + U ( y, ¯ y ) , (5.19)where H = ( a ∨ a ) + ( a ∨ a ) and U ( x, ¯ x ) = 0 . a | x − ¯ x | ( x + ¯ x ). It is obvious that U ∈ U .In other words, we have shown that Assumption 5.1 is satisfied too. To apply Theorem 5.3, weuse the same functions µ ( · ) and h ( · ) as defined in Example 4.8. We observe that inequality (5.10)becomes ∆ − ε ≥ ˆ a ∆ − γ ) ∧ (1 / − ε )] / (¯ p − . (5.20)But, for any ε ∈ (0 , / p sufficiently large such that ε > γ ) ∧ (1 / − ε )] / (¯ p − E | x ( T ) − x ∆ ( T ) | = O (∆ (2 γ ) ∧ (1 / − ε ) ) and E | x ( T ) − ¯ x ∆ ( T ) | = O (∆ (2 γ ) ∧ (1 / − ε ) ) . (5.21)It is known that for every α ∈ (0 , . α -H¨older continuous (see, e.g.,[9]). If we regard the initial data ξ ( u ), u ∈ [ − τ,
0] as an observation of the state during the timeinterval [ − τ, γ ∈ (0 , . γ is close to 0.5, then (5.21) showsthe order of convergence is close to 0.25. Can we improve the order? The answer is yes thoughwe need stronger conditions. Assumption 5.5
Assume that there are positive constants α and H and a function U ∈ U suchthat ( x − ¯ x ) T ( f ( x, y ) − f (¯ x, ¯ y )) + 1 + α | g ( x, y ) − g (¯ x, ¯ y ) | ≤ H ( | x − ¯ x | + | y − ¯ y | ) − U ( x, ¯ x ) + U ( y, ¯ y ) (5.22) for all x, y, ¯ x, ¯ y ∈ R n . Assumption 5.6
Assume that there is a pair of positive constants r and H such that | f ( x, y ) − f (¯ x, ¯ y ) | ∨ | g ( x, y ) − g (¯ x, ¯ y ) | ≤ H ( | x − ¯ x | + | y − ¯ y | )(1 + | x | r + | ¯ x | r + | y | r + | ¯ y | r ) (5.23) for all x, y, ¯ x, ¯ y ∈ R n . Lemma 5.7
Let Assumptions 2.1, 3.4, 4.1, 5.5 and 5.6 hold and ¯ p > r . Let R > k ξ k be a realnumber and let ∆ ∈ (0 , ∆ ∗ ) be sufficiently small such that µ − ( h (∆)) ≥ R . Let θ ∆ ,R and e ∆ ( t ) bethe same as defined in Section 3. Then E | e ∆ ( T ∧ θ ∆ ,R ) | ≤ C (∆ γ ∨ [∆( h (∆)) ]) . (5.24)20 roof . We use the same notation as in the proof of Lemma 5.2. It follows from (5.5) that E | e ∆ ( t ∧ θ ) | ≤ E Z t ∧ θ (cid:16) e T ∆ ( s )[ f ( x ( s ) , x ( s − τ )) − f ( x ∆ ( s ) , x ∆ ( s − τ ))]+ (1 + α ) | g ( x ( s ) , x ( s − τ )) − g ( x ∆ ( s ) , x ∆ ( s − τ )) | + 2 e T ∆ ( s )[ f ( x ∆ ( s ) , x ∆ ( s − τ )) − f (¯ x ∆ ( s ) , ¯ x ∆ ( s − τ ))]+ (1 + α − ) | g ( x ∆ ( s ) , x ∆ ( s − τ )) − g (¯ x ∆ ( s ) , ¯ x ∆ ( s − τ )) | (cid:17) ds. (5.25)By Assumptions 3.4, 5.5 and 5.6, we can then show E | e ∆ ( t ∧ θ ) | ≤ (4 H + 1) Z t E | e ∆ ( s ∧ θ ) | ds + 2 τ κ b K ∆ γ + J, (5.26)where (5.8) has been used and J := E Z t ∧ θ H (2 + α − )( | x ∆ ( s ) − ¯ x ∆ ( s ) | + | x ∆ ( s − τ ) − ¯ x ∆ ( s − τ ) | ) × (1 + | x ∆ ( s ) | r + | ¯ x ∆ ( s ) | r + | x ∆ ( s − τ ) | r + | ¯ x ∆ ( s − τ ) | r ) ds. But, by the H¨older inequality, Lemmas 2.5 and 4.3 and Assumption 3.4, we can derive that J ≤ C Z T (cid:16) E | x ∆ ( s ) − ¯ x ∆ ( s ) | p/ (¯ p − r ) + E | x ∆ ( s − τ ) − ¯ x ∆ ( s − τ ) | p/ (¯ p − r ) (cid:17) (¯ p − r ) / ¯ p × (cid:16) E | x ∆ ( s ) | ¯ p + E | ¯ x ∆ ( s ) | ¯ p + E | x ∆ ( s − τ ) | ¯ p + E | ¯ x ∆ ( s − τ ) | ¯ p (cid:17) r/ ¯ p ds ≤ C (∆ γ ∨ [∆( h (∆)) ]) . Substituting this into (5.26) gives E | e ∆ ( t ∧ θ ) | ≤ (4 H + 1) Z t E | e ∆ ( s ∧ θ ) | ds + C (∆ γ ∨ [∆( h (∆)) ]) , which implies the required assertion (5.24). ✷ The following theorem gives a better convergence rate than Theorem 5.3.
Theorem 5.8
Let Assumptions 2.1, 3.4, 4.1, 5.5 and 5.6 hold and ¯ p > r . Assume that h (∆) ≥ µ (cid:0) (∆ γ ∨ [∆( h (∆)) ]) − / (¯ p − (cid:1) (5.27) for all sufficiently small ∆ ∈ (0 , ∆ ∗ ) . Then, for every such small ∆ , E | x ( T ) − x ∆ ( T ) | ≤ C (∆ γ ∨ [∆( h (∆)) ]) (5.28) and E | x ( T ) − ¯ x ∆ ( T ) | ≤ C (∆ γ ∨ [∆( h (∆)) ]) . (5.29) Proof . We use the same notation as in the proof of Theorem 5.3. Choosing δ = ∆ γ ∨ [∆( h (∆)) ] and R = (∆ γ ∨ [∆( h (∆)) ]) − / (¯ p − ,
21e get from (5.13) that E | e ∆ ( T ) | ≤ E | e ∆ ( T ∧ θ ∆ ,R ) | + C (∆ γ ∨ [∆( h (∆)) ]) . (5.30)But, by condition (5.27), we have µ − ( h (∆)) ≥ (∆ γ ∨ [∆( h (∆)) ]) − / (¯ p − = R. We can hence apply Lemma 5.7 to obtain E | e ∆ ( T ∧ θ ∆ ,R ) | ≤ C (∆ γ ∨ [∆( h (∆)) ]) . (5.31)Substituting this into (5.30) yields the first assertion (5.28) . The second assertion (5.29) followsfrom (5.28) and Lemma 2.5. ✷ Example 5.9
Let us return to Example 4.8 once again. Instead of (5.18), we can have thefollowing alternative estimate | g ( x, y ) − g (¯ x, ¯ y ) | ≤ a ∨ a )( | x − ¯ x | + | y − ¯ y | ) + 0 . a | x − ¯ x | ( x + ¯ x ) , (5.32)where a := sup u ≥ (9 a u − . a u ) ∈ (0 , ∞ ). It then follows from (5.17) and (5.32) that( x − ¯ x )( f ( x, y ) − f (¯ x, ¯ y )) + | g ( x, y ) − g (¯ x, ¯ y ) | ≤ H ( | x − ¯ x | + | y − ¯ y | ) − U ( x, ¯ x ) + U ( y, ¯ y ) , (5.33)where H = ( a ∨ a ) + 2( a ∨ a ) and U ( x, ¯ x ) = 0 . a | x − ¯ x | ( x + ¯ x ). In other words, we haveshown that Assumption 5.5 is satisfied with α = 1. It is also straightforward to show that | f ( x, y ) − f (¯ x, ¯ y ) | ≤ a | y − ¯ y | (1 + | y | + | ¯ y | ) + 16 a | x − ¯ x | ( | x | + | ¯ x | ) . (5.34)We hence see from (5.32) and (5.34) that Assumption 5.6 is also satisfied with r = 4. In otherwords, we have shown that Assumptions 2.1, 4.1, 3.4, 5.5 and 5.6 hold for every ¯ p > r = 4. Let µ ( · ) and h ( · ) be the same as before. We can then conclude by Theorem 5.8 that the truncatedEM solutions of the SDE (4.14) satisfy E | x ( T ) − x ∆ ( T ) | = O (∆ (2 γ ) ∧ (1 − ε ) ) and E | x ( T ) − ¯ x ∆ ( T ) | = O (∆ (2 γ ) ∧ (1 − ε ) ) . (5.35)In particular, if γ is close to 0.5 (or bigger than half), this shows that the order of convergence isclose to 0.5. In this paper we have used the new explicit method, called the truncated EM method, to studythe strong convergence of the numerical solutions for nonlinear SDDEs. For a given stepsize ∆,we define the discrete-time truncated EM numerical solution and then form two versions of thecontinuous-time truncated EM solutions, namely the continuous-time step-process truncated EMsolution ¯ x ∆ ( t ) and the continuous-time continuous-process truncated EM solution x ∆ ( t ). Underthe local Lipschitz condition plus the generalized Khasminskii-type condition, we have successfullyshown the strong convergence of both continuous-time truncated EM solutions to the true solutionin the sense that lim ∆ → E | x ∆ ( T ) − x ( T ) | q = 0 and lim ∆ → E | ¯ x ∆ ( T ) − x ( T ) | q = 0for any T > q ∈ [1 , q ≥
2. We have also discussed the convergence ratesin L under some additional conditions. We have used several examples to illustrate our theorythroughout the paper. 22 cknowledgements The authors would like to thank the Leverhulme Trust (RF-2015-385), the Royal Society of London(IE131408), the Natural Science Foundation of China (11471216), the Natural Science Foundationof Shanghai (14ZR1431300), the Ministry of Education (MOE) of China (MS2014DHDX020) fortheir financial support.
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