Explicit numerical approximations for stochastic differential equations in finite and infinite horizons: truncation methods, convergence in p

Transcription

1 Li, Xiaoye and Mao, Xerong and Yin, George 018 xplicit nmerical approximations for stochastic differential eqations in finite and infinite horizons : trncation methods, convergence in pth moment, and stability. IMA Jornal of Nmerical Analysis. ISSN , This version is available at Strathprints is designed to allow sers to access the research otpt of the University of Strathclyde. Unless otherwise explicitly stated on the manscript, Copyright and Moral Rights for the papers on this site are retained by the individal athors and/or other copyright owners. Please chec the manscript for details of any other licences that may have been applied. Yo may not engage in frther distribtion of the material for any profitmaing activities or any commercial gain. Yo may freely distribte both the rl and the content of this paper for research or private stdy, edcational, or not-for-profit prposes withot prior permission or charge. Any correspondence concerning this service shold be sent to the Strathprints administrator: strathprints@strath.ac. The Strathprints instittional repository is a digital archive of University of Strathclyde research otpts. It has been developed to disseminate open access research otpts, expose data abot those otpts, and enable the management and persistent access to Strathclyde's intellectal otpt.

2 IMA Jornal of Nmerical Analysis , 1 46 doi: /imanm/dry015 xplicit nmerical approximations for stochastic differential eqations in finite and infinite horizons: trncation methods, convergence in pth moment and stability Xiaoye Li School of Mathematics and Statistics, Northeast Normal University, Changchn, Jilin 13004, China Xerong Mao Department of Mathematics and Statistics, University of Strathclyde, Glasgow G1 1XH, UK Corresponding athor: x.mao@strath.ac. and George Yin Department of Mathematics, Wayne State University, Detroit MI 480, USA [Received on 17 Janary 017; revised on 10 Febrary 018] Solving stochastic differential eqations SDs nmerically, explicit ler Maryama M schemes are sed most freqently nder global Lipschitz conditions for both drift and diffsion coefficients. In contrast, withot imposing the global Lipschitz conditions, implicit schemes are often sed for SDs bt reqire additional comptational effort; along another line, tamed M schemes and trncated M schemes have been developed recently. Taing advantages of being explicit and easily implementable, trncated M schemes are proposed in this paper. Convergence of the nmerical algorithms is stdied, and pth moment bondedness is obtained. Frthermore, asymptotic properties of the nmerical soltions sch as the exponential stability in pth moment and stability in distribtion are examined. Several examples are given to illstrate or findings. Keywords: local Lipschitz condition; explicit M scheme; finite horizon; infinite horizon; pth moment convergence; moment bond; stability; invariant measre. 1. Introdction In this paper, we stdy nmerical soltions of d-dimensional stochastic differential eqations SDs of the form dxt = fxt dt + gxt dbt, t 0, x0 = x 0, 1.1 where Bt isanm-dimensional Brownian motion and f : R d R d, g : R d R d m, which satisfy a local Lipschitz condition, namely, for any N > 0 there is a constant C N sch that fx fy gx gy CN x y 1. for any x, y R d with x y N. Clearly,iff, g C 1, they satisfy the local Lipschitz condition. Or primary objective is to constrct easily implementable nmerical soltions and prove that they The Athors 018. Pblished by Oxford University Press on behalf of the Institte of Mathematics and its Applications. This is an Open Access article distribted nder the terms of the Creative Commons Attribtion License which permits nrestricted rese, distribtion, and reprodction in any medim, provided the original wor is properly cited.

3 X. LI T AL. converge to the tre soltion of the nderlying SDs. In addition to obtaining the asymptotic pth moment convergence and moment bondedness we consider the approximations to the invariant distribtions in infinite horizon. xplicit ler Maryama M schemes are most poplar for approximating the soltions of SDs nder global Lipschitz continosly; see, for example, Kloeden & Platen 199 and Higham et al. 00. However, many important SD models satisfy only local Lipschitz conditions or have growth rates faster than linear. For sch SDs, the classical strong convergence for classical M methods does not hold. Htzenthaler et al. 011 showed that the pth moments of the M approximation for a large class of SDs with coefficients satisfying sper-linear growth diverge to infinity for all p [1,. Implicit methods were developed to approximate the soltions of these SDs. Higham et al. 00 showed that the bacward M schemes converge if the diffsion coefficients are globally Lipschitz while the drift coefficient satisfies a one-sided Lipschitz condition. More details on the implicit methods can be fond in Kloeden & Platen 199, Saito & Mitsi 1993, H 1996, Milstein et al. 1998, Brrage & Tian 00, Appleby et al. 010 and Szprch et al However, additional comptational effort is reqired for the implementation of the implicit methods. Since explicit nmerical methods have advantages, a cople of modified M methods have recently been developed for nonlinear SDs. Htzenthaler et al. 01 proposed tamed M schemes to approximate SDs with the global Lipschitz diffsion coefficient and one-sided Lipschitz drift coefficient. Sabanis 013, 016 developed tamed M schemes for SDs with nonlinear growth coefficients. Moreover, stopped M schemes Li & Mao, 013, trncated M schemes Mao, 015, mltilevel M schemes Anderson et al., 016 and their variants have also been developed to deal with the strong convergence problem for nonlinear SDs. However, to the best of or nowledge, these modified M methods still cannot handle the convergence of a large class of SDs with nonlinear drift and diffsion coefficients, for example, the constant elasticity of volatility model CV model arising in finance for an asset price of the form Lewis, 000 drt = β 0 β 1 rt dt + σ rt 3/ dbt, 1.3 where β 0,β 1, σ are positive constants. Based on the motivation above, we constrct easily implementable explicit M schemes for SDs with only local Lipschitz drift and diffsion coefficients and establish their convergence. In the process of establishing the strong mean sqare convergence theory conditionally, Higham et al., 00, p.1060 posed an open problem and noted that in general, it is not clear when sch moment bonds can be expected to hold for explicit methods with f, g C 1. Despite recent progress in the nmerical methods for nonlinear SDs this problem remains open to date. In this paper, we answer the qestion of Higham et al. positively by reqiring only that the drift and diffsion coefficients are locally Lipschitz and satisfy a strctre condition Assmption.1 forthepth moment bondedness of the exact soltion for some p 0, +. Talay & Tbaro 1990 investigated the probability law of approximation sing the M scheme for SD with smooth f and g whose derivatives of any order are bonded. Frthermore, Bally & Talay 1996 expanded the error in power of the step size. Gyöngy 1998 analysed the almost sre convergence. Here we focs on the moment convergence. Higham et al. 00 and Htzenthaler et al. 01 provided the 1/-order rate of convergence in moment sense for the bacward scheme and the tamed M scheme nder a one-sided Lipschitz condition and polynomial growth for f and global Lipschitz condition for g, respectively. Recently, Sabanis 016 developed a tamed M scheme with 1/-order rate of convergence. In this paper, we propose a trncation algorithm to relax the restrictions in the stdies by Higham et al. 00 and Htzenthaler et al. 01. We demonstrate the convergence

4 XPLICIT NUMRICAL APPROXIMATIONS FOR SDS IN FINIT AND INFINIT HORIZONS 3 of the algorithm nder weaer conditions compared with what is nown in the literatre. Then nder slightly stronger conditions similar to the stdy by Sabanis 016 we prove the convergence rate is optimal for the explicit schemes. While asymptotic properties of the nmerical soltions attract more and more attentions see the stdies by Roberts & Tweedie, 1996, Mattingly et al., 00, Higham et al., 003 and Zong et al., 016 the moment bondedness of the nmerical soltions is also often desirable becase its connection to the tightness and ergodicity. However, the classical M method fails to preserve the asymptotic bondedness for many nonlinear SDs. For example, Higham et al. 003 showed that for the nonlinear scalar SD dxt = [ ] xt x 3 t dt + xt dbt, 1.4 the second moment of the classical M nmerical soltion diverges to infinity in an infinite time interval for any given step size and an initial vale dependent on the step size. In this paper, as their conterparts of analytic soltions, we show that or explicit schemes will preserve the asymptotic moment bondedness as well as asymptotic stability for a large class of nonlinear SDs inclding 1.3 and 1.4 nder Assmptions 5.1, 6.1, 7.1. Frthermore, we consider asymptotic properties of or nmerical algorithms and demonstrate exponential stability and stability in distribtion. In this paper, adopting the trncation idea from the stdy by Mao 015 and sing a novel approximation techniqe, we constrct several explicit schemes nder certain assmptions on the coefficients of the SDs and derive convergence reslts in both finite and infinite time intervals. The nmerical soltions at the grid points are modified before each iteration according to the growth rates of the drift and diffsion coefficients sch that the nmerical soltions will preserve the properties of the exact soltion nicely. We approximate the exact soltion by piecewise constant interpolation directly, which is different from that of the stdies by Higham et al. 00, Htzenthaler et al. 01, Sabanis 013, Mao 015 and Bao et al Or main contribtions are as follows: An easily implementable scheme is proposed sch that its nmerical soltions converge to the exact soltion in a finite time interval. The rate of convergence is also stdied nder slightly stronger conditions. The open qestion posed in the stdy by Higham et al. 00, p.1060 is answered positively. The pth moment of or explicit nmerical soltion is bonded for the SDs with only local Lipschitz drift and diffsion coefficients. Appropriate trncation techniqes and approximation techniqes are tilized sch that properties of the exact soltion are preserved. The nmerical soltions preserve the pth moment bondedness property of the exact soltion almost completely, not only in a finite time interval bt also in an infinite time interval for some p > 0. Different schemes are constrcted to approximate different stochastic dynamical systems that are exponentially stable and/or stable in distribtion. The rest of the paper is organized as follows. Section gives some preliminary reslts on certain properties of the exact soltions. Section 3 begins to constrct an explicit scheme and demonstrate convergence in a finite time interval. Section 4 provides the rate of convergence. Section 5 goes frther to obtain the pth moment bondedness in an infinite time interval for some p > 0. Section 6 reconstrcts an explicit scheme to approximate the exponential stability. Section 7 analyses the stability of the SD

5 4 X. LI T AL. 1.1 in distribtion yielding an invariant measre μ. Then another explicit scheme is constrcted preserving the stability in distribtion and a nmerical invariant measre, which tends to μ as the step size tends to 0. Section 8 presents a cople of examples to illstrate or reslts. Section 9 gives frther remars to conclde the paper.. Preliminaries Throghot this paper, let Ω,F, { } F t t 0,P be a complete filtered probability space with { F t }t 0 satisfying the sal conditions that is, it is right continos and F 0 contains all P-nll sets. Let Bt = B 1 t,..., B m t T be an m-dimensional Brownian motion defined on the probability space. Let denote both the clidean norm in R d and the Frobenis norm in R d m.alsoletc denote a generic positive constant whose vale may change in different appearances. Moreover, let C,1 R d R + ;R + denote the family of all non-negative fnctions Vx, t onr d R +, which are continosly twice differentiable in x and once differentiable in t. For each V C,1 R d R + ;R +, define an operator LV fromr d R + torby where LVx, t = V t x, t + V x x, tfx + 1 ] [g trace T xv xx x, tgx, V x x, t = Vx, t x 1,..., Vx, t, V xx x, t = x d Vx, t x j x l For the reglarity and pth moment bondedness of the exact soltion we mae the following assmption. Assmption.1 There exists a pair of positive constants p and λ sch that lim sp x 1 + x x T fx + gx p x T gx. d d x 4 λ..1 Remar. We highlight that the family of drift and diffsion fnctions satisfying Assmption.1 is large. Denote by C a positive constant. a If there are positive constants a, ε and λ sch that x T gx a x 4 ε + C and that x T fx + gx λ x + C then Assmption.1 holds for any p > 0. b If there are positive constants a, ε and λ sch that x T gx λ x 4 + C and that x T fx + gx a x ε + C then Assmption.1 holds for any 0 < p <. c If there exists a positive constant λ sch that x T fx + gx λ x + C then Assmption.1 holds for p =. d If there are positive constants a, λ and > v + sch that x T gx λ x + C and that x T fx + gx a x v + C then Assmption.1 holds for 0 < p <. e If there are positive constants a, ε and sch that x T gx a x + + C and that x T fx + gx a ε x + C then Assmption.1 holds for 0 < p 1.

6 XPLICIT NUMRICAL APPROXIMATIONS FOR SDS IN FINIT AND INFINIT HORIZONS 5 Now we prepare the reglarity and moment bondedness of the exact soltion. Theorem.3 Under Assmption.1 with some p > 0 the SD 1.1 with any initial vale x 0 R d has a niqe reglar soltion xt satisfying sp xt p C T 0.. 0tT Proof. It follows from.1 that lim sp x 1 + x x T fx + gx p x T gx 1 + x λ. Then for any 0 <κ p λ /, there exists a constant M > 0 sch that 1 + x x T fx + gx p x T gx λ + κ 1 + x p x > M. By the continity of the fnctions f and g, 1 + x x T fx + gx p x T gx λ + κ p 1 + x + C x R d..3 It follows from the definition of operator L that L 1 + x p = p 1 + x p [ 1 + x x T fx + gx p x T gx ] p 1 + x p [ λ + κ 1 + x ] + C p = pλ + κ 1 + x p + C 1 + x p..4 If 0 < p 4 then 1 + x p 1 for any x R d, while if 4 < p then it follows from Yong s ineqality that for any given ε>0, for any x R d, 1 + x p [ ] 4 [ 1 p = ε ε p x p ] p 4 p 4 pε p εp 4 p 1 + x p.

7 6 X. LI T AL. Taing ε = κp Cp 4 we have 1 + x p 4 p [ Cp 4 κp ] p κ 1 + x p C for any x R d. Ths, for any p > 0, 1 + x p 4 p [ Cp 4 κp ] p κ 1 + x p C for any x R d..5 Therefore, it follows from.4 and.5 that L 1 + x p pλ + κ 1 + x p + C..6 The above ineqality and Assmption.1 garantee the existence of the niqe reglar soltion xt see the so-called Khasminsii test in the stdy by Mao & Rassias, 005. Using Itô s formla, for any 0 t T, 1 + xt p 1 + x 0 p pλ t + C + + κ 1 + xs p ds. 0 By Gronwall s ineqality we have 1 + xt p C + p/ x 0 p pλ e +κ T,.7 which implies the desired ineqality.. Remar.4 Assmption.1 garantees the existence of global soltions, their reglarity and their pth moment bondedness. This is an alternative to Khasminsii s condition that there exist positive constants α,β sch that LV p αv p + β with V = 1 + x 1/. Different from the stability analysis, woring with nmerical schemes, it is more preferable to se verifiable conditions. As a reslt, it is more feasible to pt conditions on the coefficients of the eqations rather than to se an axiliary fnction. Lemma.5 Let Assmption.1 hold. For each positive integer N > x 0 define τ N =:inf { t [0, + : xt N }..8 Then for any T > 0, P{τ N T} C N p,.9 where C is a generic positive constant dependent on T, p and x 0 and independent of N.

8 XPLICIT NUMRICAL APPROXIMATIONS FOR SDS IN FINIT AND INFINIT HORIZONS 7 Proof. By virte of Dynin s formla it follows from.6 that 1 + xt τ N p 1 + x 0 p + κ + pλ t τn 1 + xs p ds + CT 0 for any 0 t T. Gronwall s ineqality implies N p P { τ N T } xt τ N p 1 + x t τ N p C as desired. 3. xplicit scheme and convergence in pth moment In this section or aim is to constrct an easily implementable nmerical method and establish its strong convergence theory nder Assmption.1. To define the appropriate nmerical scheme we first estimate thegrowthrateoff and g. Choose a strictly increasing continos fnction ϕ : R + R + sch that ϕr as r and sp x r fx 1 + x gx ϕr r > x Denote by ϕ 1 the inverse fnction of ϕ; obviosly ϕ 1 :[ϕ0, R + is a strictly increasing continos fnction. We also choose a nmber 0, 1 and a strictly decreasing h : 0, ] 0, sch that h ϕ x 0, lim h = and 1 h K, 0, ], 3. 0 where K is a positive constant independent of. For a given 0, ] let s define the trncation mapping π : R d R d by π x = where we se the convention x x = 0 when x = 0. Clearly, x ϕ 1 h x x, 3.3 f π x h 1 + π x, gπ x h π x, x R d. 3.4 Next we propose or nmerical method to approximate the exact soltion of the SD 1.1. For any given step size 0, ] define y 0 = x 0, ỹ +1 = y + f y +gy B, 3.5 y +1 = π ỹ+1,

9 8 X. LI T AL. where t =, B = B t +1 B t. We refer to the nmerical method as a trncated M scheme. The nmerical soltions y are obtained by trncating the intermediate terms ỹ according to the growth rate of the drift and diffsion coefficients to avoid their possible large excrsions de to the nonlinearities of the coefficients and the Brownian motion increments. Conseqently, we have the following nice linear property fy h 1 + y, gy h y, Moreover, the trncated M method is an explicit one so it is easy to se. To proceed, we define ỹt and ytby ỹt := ỹ, yt := y, t [t, t Lemma 3.1 Under Assmption.1, the trncation scheme defined by 3.5 has the property sp sp 0< 0 T y p C T > Proof. For any integer 0wehave ỹ +1 = y + fy +gy B = y + y T f y + gy B + y T gy B + f y + f T y gy B. 3.9 Then 1 + ỹ +1 p = 1 + y p 1 + ξ p, 3.10 where ξ = yt f y + gy B + y T g y B + f y + f T y g y B 1 + y Thans to the Taylor formla, applying the recrsion with > 1, we have { 1 + p 1 + p + pp 8 + pp p , 0< p, 1 + p + pp P i, i < p i + 1, 3.1 where P i represents an ith-order polynomial of with coefficients depending only on p, and i is an integer. We will prove the reslt when 0 < p only; the other cases can be done similarly. It follows

10 XPLICIT NUMRICAL APPROXIMATIONS FOR SDS IN FINIT AND INFINIT HORIZONS 9 from 3.10 that 1 + ỹ p +1 F t 1 + y p [ 1 + p pp ξ F t + 8 The fact that B is independent of F t implies that ξ F t + pp p 4 ] ξ 3 48 F t B F t = B = 0, A B F t = A B = A, A R d m. This together with 3. and 3.6 implies ξ F t = 1 + y 1 [ y T fy + gy + fy ] 1 + y 1 [y T fy + gy +1 + y h ] 1 + y 1 y T fy + gy +K Using A B i 1 Ft = 0 and A B i F t = C i, A R 1 m, i 1, 3.15 we have ξ F t = 1 + y [ y T fy + gy B + y T gy B + fy + f T Ft ] y gy B 1 + y [ y T gy B + y T gy B T y T fy + gy B + fy + f T y gy B Ft ] y y T gy y y fy gy y y T gy y y 1 + y 3 h y y T gy 4K 3.16

11 10 X. LI T AL. and ξ 3 F t = 1 + y 3 [ y T fy + gy B + fy + y T gy B + f T 3Ft ] y gy B = 1 + y 3 [ y T fy + gy B + fy 3 + y T fy + gy B + fy ] y T gy B +f T y gy B F t 1 + y 3 [ 7 y T fy gy 6 B 6 + 9fy y 3 fy gy B +8 y gy 4 B y fy gy B + 16 y fy 3 gy B fy gy 4 B fy 4 gy B 4 ] F t C 1 + y 3 y 3 fy gy fy y 3 fy gy + y gy 4 + y fy gy 3 + y fy 3 gy 4 + fy gy fy 4 gy 5 C h h h h + h + h h h h 5 5 C Also we can prove that, for any i > 3, ξ i F t = O. Combining and sing.1 in Assmption.1, for any 0, 1 + ỹ p +1 F t [ 1 + y p 1 + y y T 1 + C +p fy + gy + p y T gy ] 1 + y 1 + y p 1 + C Thans to the trncated M scheme 3.5, for any integer satisfying 0 T, we obtain 1 + y p 1 + ỹ p [ = 1 + ỹ p ] F t C 1 + y p

12 XPLICIT NUMRICAL APPROXIMATIONS FOR SDS IN FINIT AND INFINIT HORIZONS 11 Solving the above linear first-order difference ineqality, we obtain 1 + y p 1 + C 1 + y 0 p e C 1 + y 0 p e CT 1 + y 0 p. Therefore, we get the desired reslt that sp sp 0< 0 T y p sp sp 0< 0 T 1 + y p C. The proof is complete. Lemma 3. Let Assmption.1 hold. For any 0, ] define ρ =:inf { t 0: ỹt ϕ 1 h }. 3.0 Then for any T > 0, P{ρ T} C ϕ 1 h p, 3.1 where C is a positive constant independent of. Proof. We write ρ = ρ for simplicity. Then ρ = β, where β =:inf { 0: ỹ ϕ 1 h }. Clearly, ρ and β are F t and F t stopping times, respectively. For ω { β + 1 } we have ỹ <ϕ 1 h and y =ỹ, whence it follows from 3.5 that ỹ +1 β =ỹ +1 =ỹ + [ f ] ỹ +g ỹ B =ỹ β + [ ] fỹ +gỹ B I[[0,β ]] + 1. On the other hand, for ω { β < + 1 },wehaveβ and hence ỹ +1 β =ỹ β =ỹ β + [ fỹ +gỹ B ] I[[0,β ]] + 1. In other words, we always have ỹ +1 β =ỹ β + [ fỹ +gỹ B ] I[[0,β ]] Then 1 + ỹ +1 β p = 1 + ỹ β p 1 + ξ I [[0,β ]] + 1 p, 3.3

13 1 X. LI T AL. where ξ = ỹt f ỹ + g ỹ B + ỹ T g ỹ B + f ỹ + f T ỹ g ỹ B 1 +. ỹ As in the proof of Lemma 3.1 we prove the assertion only for the case when 0 < p ; when p > it can be done in the same way. Using the techniqe in the proof of Lemma 3.1 we can show that 1 + ỹ +1 β p F t β 1 + ỹ β p [ 1 + p ξ I [[0,β ]] + 1 Ft β + pp ξ 8 I [[0,β ]]+1 F t β + pp p 4 ] ξ 3 48 I [[0,β ]]+1 F t β. 3.4 Note that B I [[0,β ]]+1 = Bt +1 β Bt β. Since Bt is a continos martingale, by virte of the Doob martingale stopping time theorem, we see that B I [[0,β ]] + 1 F t β = 0, and for any A R d m, A B I [[0,β ]]+1 F t β = A t +1 β t β F t β = A I [[0,β ]]+1 F t β. This together with 3. and 3.6 implies ξ I [[0,β ]] + 1 F t β = 1 + ỹ 1 [ ỹ T fỹ + gỹ + fỹ ] I [[0,β ]] + 1 Ft β 1 + ỹ 1 [ ỹ T fỹ + gỹ ỹ h ] I [[0,β ]] + 1 Ft β [ 1 + ỹ 1 ỹ T fỹ + gỹ ] +K 3 I [[0,β ]] + 1 Ft β. 3.5 Using A B i I [[0,β ]] + 1 F t β = C i I [[0,β ]] + 1 Ft β, A B i+1i[[0,β ]] + 1 Ft β = 0 A R 1 m, i 1,

14 XPLICIT NUMRICAL APPROXIMATIONS FOR SDS IN FINIT AND INFINIT HORIZONS 13 we have ξ I [[0,β ]] + 1 Ft β = 1 + ỹ [ ỹ T f ỹ + g ỹ B + ỹ T g ỹ B + f ỹ + f T ỹ g ỹ B I[[0,β ]] + 1 F t β 1 + ỹ [ 4ỹ T g ỹ 8 ỹ f ỹ gỹ ] I [[0,β ]] + 1 Ft β [ ỹ ỹ T g ] ỹ 4K I [[0,β ]] + 1 Ft β ] 3.6 and ξ 3 I [[0,β ]] + 1 F t β = 1 + ỹ 3 [ ỹ T fỹ + gỹ B + ỹ T gỹ B 3 + fỹ + f T ỹ gỹ B I[[0,β ]] + 1 F t β ] 1 + y 3 [[ 7 y T fy gy 6 B fy y 3 fy gy B +8 y gy 4 B y fy gy B + 16 y fy 3 gy B fy gy 4 B fy 4 gy B 4] I [[0,β ]] + 1 Ft β ] C 1 + y 3[ y 3 fy gy fy y 3 fy gy + y gy 4 + y fy gy 3 + y fy 3 gy 4 + fy gy fy 4 gy 5] [I [[0,β ]] + 1 ] Ft β [ C h h h h + h + h h h h 5 5] [I [[0,β ]] + 1 ] Ft β C I [[0,β ]] + 1 Ft β. 3.7

15 14 X. LI T AL. We can also prove that for any i > 3, ξ i F t = O I [[0,β ]] + 1 F t β. Combining , sing.1 in Assmption.1, for any 0, p 1 + ỹ +1 β F t β 1 + ỹ β p [ 1 + C + + p 1 + ỹ ỹ T f ỹ + g ỹ + p ỹt g ỹ ] 1 + ỹ I [[0,β ]] + 1 F t β 1 + p ỹ β 1 + C I [[0,β ]] + 1 F t β. 3.8 For any integer 1 T/ we obtain 1 + ỹ β p = 1 + ỹ β p Ft 1 β [ 1 + ỹ 1 β p 1 + C I [[0,β ]] ] Ft 1 β 1 + C 1 + ỹ 1 β p. 3.9 Solving the above first-order linear ineqality leads to 1+ ỹ β p 1 + C 1+ y 0 p e C 1+ y 0 p e CT 1+ y 0 p. Therefore, the desired assertion follows from pp ϕ 1 { } h ρ T ỹt ρ p = ỹ[t/ ] β p 1 + p ỹ[t/ ] β C. The proof is complete. The following theorem presents the pth moment convergence of the trncated nmerical soltions. Theorem 3.3 Under Assmption.1, for any q 0, p, lim 0 yt xt q = 0 T

16 XPLICIT NUMRICAL APPROXIMATIONS FOR SDS IN FINIT AND INFINIT HORIZONS 15 Proof. Let τ N and ζ be the same as before. Define θ N, = τ N ρ, e T = xt ȳt. Using Yong s ineqality, for any δ>0, we have e T q = e T q I {θn, >T} + e T q I {θn, T} e T q qδ I {θn, >T} + p e T p + p q pδ q/p qp{ θ N, T } It follows from the reslts of Theorem.3 and Lemma 3.1 that e T p p xt p + p yt p C. Now let ε>0 be arbitrary. Choose δ>0 sfficiently small for Cqδ/p ε/3; then we have Choose N > 1 sfficiently large sch that qδ p e T p ε Cp q N p pδ q/p q ε 6. Choose > 0 sfficiently small sch that ϕ 1 h N It follows from the reslts of Lemmas.5 and 3. that for any 0, ], p q pδ q/p qp{ θ N, T } p q { P τn pδ q/p q T } +P { ρ T } p q C pδ q/p q N p + C ϕ 1 h p Cp q N p pδ q/p q ε Combining 3.31, 3.3 and 3.34, we now that for the chosen N and all 0, ], e e T q T q I { } + ε θn, >T 3. If we can show that e lim T q I { } = 0, θ N, >T the desired assertion follows. For this prpose, we define the trncation fnctions f N x = f x N x x and x N x g N x = g, x R d. x

17 16 X. LI T AL. Consider the trncated SD dyt = f N yt dt + gn yt dbt 3.36 with the initial vale z0 = x 0.By1. in Assmption.1, f N and g N are globally Lipschitz continos with the Lipschitz constant C N. Therefore, SD 3.36 has a niqe reglar soltion yt on t 0 satisfying xt τ N = yt τ N a.s., t On the other hand, for each 0, ], we apply the M method to SD 3.36 and we denote by t the piecewise constant M soltion see Kloeden & Platen, 199; Higham et al., 00 that has the property sp 0tT yt t q C q/ T It follows from 3.5 thatforall 0, ], y t θ N, =ỹ t θn, = t θn, a.s., t Using , e T q I {θn, >T} = e T θn, q I {θn, >T} x T θn, y T θn, q y t θn, T θn, q y t θn, t θn, q = sp 0tT sp yt t q 0tT θ N, yt t q sp 0tT C q/. Therefore, 3.35 holds and the desired assertion follows. 4. Convergence rate In this section, or aim is to establish a rate of convergence reslt nder Assmption.1 and additional conditions on f and g. The rate is optimal, similar to the standard reslts for the explicit M scheme

18 XPLICIT NUMRICAL APPROXIMATIONS FOR SDS IN FINIT AND INFINIT HORIZONS 17 with globally Lipschtiz f and g. The wor of Higham et al. 00 gives the optimal rate in qth moment for the implicit M scheme for q with global Lipschitz g and a one-sided Lipschitz f together with polynomial growth. Using a similar condition to the stdy by Higham et al. 00, the rate for the tamed ler was obtained Htzenthaler et al., 01.TheworofSabanis 016 developed the tamed M scheme, then obtained the convergence rate nder a condition similar to ors. To obtain the rates of convergence we need somewhat stronger conditions compared with the convergence alone, which are stated as follows. Assmption 4.1 There exist positive constants p 0 >, L and l sch that x y T fx fy + p 0 1 gx gy L x y, 4.1 fx fy L 1 + x l + y l x y, x, y R d. 4. Remar 4. One observes that if Assmption 4.1 holds then gx gy C 1 + x l + y l x y. 4.3 In addition, fx fx f0 + f0 L 1 + x l x + f0 C 1 + x l+1, 4.4 and by Yong s ineqality, gx C [ x + x 1 + x l+1] 1/ + g0 C 1 + x l/ Remar 4.3 Under Assmption 4.1, we may define ϕ in 3.1 byϕr = C 1 + r l for any r > 0. Then ϕ 1 r = r/c 1 1/l for all r > C. In order to obtain the rate, we specify h = K for all 0, ], where 0, 1/] will be specified in the proof of Lemma 4.7. Ths, π x = x K /C 1 1/l x/ x for any x R d. Maing se of scheme 3.5 we define an axiliary approximation process by ȳt = y + f y t t + g y Bt B t Note that ȳ t = y t = y, that is, ȳt and yt coincide at the grid points. t [ t, t Lemma 4.4 If Assmptions.1 and 4.1 hold with l + 1 p, for any q 0 [, p/l + 1], for the process given by 4.6, sp ȳt yt q 0 C q 0 T > 0, 0, ], 4.7 0tT where C is a positive constant independent of.

19 18 X. LI T AL. Proof. For any t [0, T] there is a non-negative integer sch that t [ t, t +1. Then ȳt yt q 0 = ȳt yt q 0 q fy q 0 q 0 + q 0 gy q 0 Bt Bt q 0 C fy q0 q 0 + gy q0 q 0. De to 4.4, 4.5 and Lemma 3.1, ȳt yt q 0 C 1 + y l+1 q 0 q 0 + C 1 + y l +1 q 0 q 0 C + C y p l+1q 0 p q 0 + C y p l+q 0 p q 0 C q 0. The reqired assertion follows. Using techniqes in the proofs of Lemmas 3.1 and 3., we obtain the following lemmas. Lemma 4.5 Under Assmption.1, for the nmerical soltion of scheme 4.6, sp sp 0< 0tT ȳt p C T > Lemma 4.6 Let Assmption.1 hold. For any 0, ] define { } ζ := inf t 0: ȳt ϕ 1 h. 4.9 Then for any T > 0, P { ζ T } C ϕ 1 h p, 4.10 where C is a positive constant independent of. Lemma 4.7 If Assmptions.1 and 4.1 hold with 4l+1 p then for any q [, p 0 [, p/l+1], for the nmerical soltion defined by 3.5 and 4.6 with = lq/p q, ȳt xt q C q T > Proof. Define θ = τ ϕ 1 h ρ ζ, Ω 1 := { ω : θ > T }, ēt = xt ȳt, for any t [0, T], where τ N, ρ and ζ are defined by.8, 3.0 and 4.9, respectively. Using Yong s ineqality, for

20 XPLICIT NUMRICAL APPROXIMATIONS FOR SDS IN FINIT AND INFINIT HORIZONS 19 any κ>0, we have ēt q = ēt q I Ω1 + ēt q I Ω c 1 Theorem.3 and Lemma 4.5 yield ēt q q κ I Ω1 + p ēt p + p q p κq/p qp Ω1 c. 4.1 q κ It follows from the reslts of Lemmas.5, 3. and 4.5 that p ēt p C κ p q p κq/p qp Ω1 c p q { P τϕ p κq/p q 1 h T} +P { ρ T } +P { ζ T } p q 3C p κq/p q ϕ 1 h p C p l κq p q On the other hand, for any t [0, T], ēt = The Itô formla leads to t t fxs fys ds + gxs gys dbs. 0 0 ēt q = t 0 q ēs q 4 [ ēs ē T s fxs fys + gxs gys + q ēt s gxs gys ] ds + Mt t 0 q ēs q ē T s fxs fys + q 1 gxs gys ds + Mt, where Mt = t q 0 ēs q ē T s gxs gys dbs is a local martingale with initial vale 0. This implies ēt θ q q t θ ēs q [ ē T s fxs fys + q 1 ] g xs gys ds

21 0 X. LI T AL. De to q [, p 0 we choose a small constant ι>0 sch that 1 + ι q 1 p0 1. It follows from Assmption 4.1 that for any 0 s t θ, ē T s fxs fys + q 1 g xs g ys ē T s fxs fȳs + ē T s f ȳs fys +1 + ι q 1 gxs gȳs q 1 gȳs gys ι L ēs + ēs fȳs fys q 1 gȳs gys. ι Inserting the above ineqality into 4.15 wehave ē t θ q q + t θ ι L ēs q + ēs q 1 fȳs fys q 1 ēs q gȳs gys ds. Then an application of Yong s ineqality together with Assmption 4.1 leads to ēt θ q t θ C ēs q + fȳs fys q + gȳs gys q ds 0 t θ C 0 + C ēs q ȳs l + ys l q ȳs ys q 1 + ȳs l + ys l q ȳs ys q ds t 0 ē s θ q ds + C T 0 [ 1 + ȳs lq + ys lq ȳs ys q] ds. Using Hölder s eqality and Jensen s eqality, and then Lemmas 4.4 and 4.5, wehave T 0 T C 0 T C 0 [ 1 + ȳs lq + ys lq ȳs ys q] ds [ 1 + ȳs lq + ys lq ] 1 [ 1 ȳs ys q] ds [ 1 + ȳs p lq p + ys p ] 1 lq [ ] p ȳs ys p l+1q p l+1 ds 4.16 C q. 4.17

22 XPLICIT NUMRICAL APPROXIMATIONS FOR SDS IN FINIT AND INFINIT HORIZONS 1 Inserting 4.17 into4.16 and applying Gronwall s ineqality we obtain ēt q I Ω1 ē T θ q C q Inserting 4.13, 4.14 and 4.18 into4.1 yields ēt q C q + C κ + C p l κq p q Let q = κ = p κq l p q, which implies = lq p q, κ = q. Therefore, the desired assertion follows. Therefore, by virte of Lemmas 4.4 and 4.7, we get or desired rate of convergence. Theorem 4.8 If Assmptions.1 and 4.1 hold with 4l+1 p then, for any q [, p 0 [, p/l+1], for the nmerical soltion defined by 3.5 with = lq/p q, yt xt q C q T > Remar 4.9 Higham et al. 00 and Htzenthaler et al. 01 obtained the optimal rate 1/ for the bacward M scheme and the tamed M scheme of strong convergence nder the following condition: the fnctions f and g are C 1, and there exists a constant c sch that x y T fx fy c x y, gx gy c x y, fx fy c 1 + x l + y l x y, x, y R d. Note that the above condition implies that Assmptions.1 and 4.1 hold for any p > and any p 0 >. Ths, nder sch a condition, in view of Theorem 4.8, the convergence rate of or trncated scheme is optimal. Note that a similar convergence rate reslt was also obtained by Sabanis 016 for a modified tamed M scheme nder conditions similar to ors. 5. The pth moment bondedness in infinite time intervals Since the moment bondedness in an infinite time interval is related closely to the tightness of the nmerical soltion, as well as the ergodicity, we go frther to realize this property by or explicit nmerical soltion. Mattingly et al. 00 showed that for a class of nonlinear SDs the mean sqare of the M nmerical soltions in the infinite interval tends to infinity bt the mean sqare of the exact soltions is bonded. Ths, they had to approximate the SDs by the implicit scheme. Now approximating the exact soltions in an infinite time interval by or nmerical method will demonstrate its advantages. First, we give the moment bondedness reslt on the exact soltions. For convenience, we impose the following hypothesis.

23 X. LI T AL. Assmption 5.1 There exists a pair of positive constants p and λ sch that lim sp x 1 + x x T fx + gx p x T gx x 4 λ. 5.1 Theorem 5. Under Assmption 5.1, the soltion xt of the SD 1.1 satisfies sp xt p C. 5. 0t< Proof. For the given p > 0 and λ>0 choose 0 <κ pλ/. Using Itô s formla and.6 we obtain pλ t e κ 1 + xt p = 1 + x 0 p x 0 p x 0 p + t L e 0 t pλ κ 0 t 0 e Ce pλ κ pλ s κ 1 + xs p ds [ s pλ κ s ds 1 + x 0 p [ pλ ] t + C e κ xs p +L 1 + xs p ] ds Ths, 1 + xt p 1 + x 0 p e pλ κ t + C C. 5.3 This implies the desired ineqality. Remar 5.3 Althogh Assmption.1 holds directly from Assmption 5.1 we highlight that the family of the drift and diffsion fnctions satisfying Assmption 5.1 is large. We give the following examples as special cases in which Assmption 5.1 holds. a If there are positive constants a, ε and λ sch that x T gx a x 4 ε + C and that x T fx + gx λ x + C then Assmption 5.1 holds with any p > 0. b If there are positive constants a, ε and λ sch that x T gx λ x 4 + C and that x T fx + gx a x ε + C then Assmption 5.1 holds with any 0 < p <. c If there are positive constants a and ε<a sch that x T gx a x 4 + C and x T fx + gx a ε x + C then Assmption 5.1 holds with some 0 < p 1 and 1 λ<0. d If there is a positive constant λ sch that x T fx + gx λ x + C then Assmption 5.1 holds with p =.

24 XPLICIT NUMRICAL APPROXIMATIONS FOR SDS IN FINIT AND INFINIT HORIZONS 3 Remar 5.4 Assmption 5.1 garantees the asymptotically pth moment bondedness of exact soltions, which is also an alternative to Khasminsii s condition, which states that there exist positive constants α and β sch that LV p αv p + β with V = 1 + x 1/. Again, for nmerical schemes, it is preferable to pt conditions on the coefficients as mentioned before. In order to obtain the asymptotic moment bondedness of the trncated M scheme 3.5 we reqire the chosen fnction h : 0, ] 0, to satisfy for some θ 0, 1/. 1/ θ h K 0, ], 5.4 Theorem 5.5 Under Assmption 5.1 there is a 1 0, 1 sfficiently small sch that the nmerical soltions of the trncated M scheme 3.5 have the property that for any compact set K R d sp sp 0< 1 x 0 K sp 0< y p C. 5.5 Proof. Using the method of proof in Lemma 3.1 we now that 3.18 holds, that is, 1 + ỹ p +1 Ft 1 + y p [ 1 + o 1+θ + p 1+ y y T fy + gy + p y T gy 1+ y. 5.6 For any given ε 0, pλ/ it follows from Assmption 5.1 that 1 + x x T fx + gx p x T gx λ + ε 1 + x + C x R d. 3p From Yong s ineqality we now that pc 1 + x p C ε 1 + x p for any x R d, where C 1 is a positive constant. Choose 1 0, 1 sfficiently small sch that o θ 1 ε/3, 1 pλ ε 1 > 0. Inserting the above ineqalities into 5.6 yields, for any ] 0, 1, [ ] 1 + ỹ +1 p Ft 1 + y p pλ 1 ε + C qation 5.7 impliesthatforany 0, 1 ] and 0, 1 + y p ỹ p [ +1 = 1 + ỹ p ] +1 F t [ ] pλ 1 ε 1 + y p + C 1.

25 4 X. LI T AL. Solving the first-order nonhomogeneos ineqality yields 1 + y p [ ] pλ ε 1 + y 0 p + C y 0 p + C, i=0 [ ] pλ i 1 ε where C is independent of and. Ths, the desired ineqality follows. 6. xponential stability in pth moment In this section, we focs on the exponential stability in pth moment. First, we give a sfficient condition for exponential stability in pth moment of the exact soltion. Since stability describes the dynamical behavior more precisely than the bondedness, we will constrct a trncation mapping and an explicit scheme according to the sper-linear growth of the diffsion and drift coefficients. This scheme is sitable for the realization of stability for the nonlinear SDs. For convenience we impose the following hypothesis. Assmption 6.1 There exists a pair of positive constants p and λ sch that x x T fx + gx p x T gx λ x 4 x R d. 6.1 Theorem 6. Under Assmption 6.1, the soltion xt of the SD 1.1 satisfies xt p x 0 p e pλt/ t 0, 6. where p and λ are given in Assmption 6.1. That is, the trivial soltion of the SD 1.1 is exponentially stable in pth moment. Proof. It follows from the definition of operator L and Assmption 6.1 that [ L e pλ t x p = e pλ t x p pλ + p Ths, the desired assertion follows from Itô s formla. x x T fx + gx p x T gx ] x 4 0. Remar 6.3 Assmption 6.1 garantees the exponential stability of the exact soltions in pth moment, which is also an alternative to Khasminsii s condition, which states that there exists a positive constant α sch that LV p αv p with V = x. We se Assmption 6.1 becase it is on the coefficients of the SDs. Note that Assmption 5.1 is sfficient for the bondedness of the pth moment of the analytic soltions bt not enogh to force the soltions to tend to 0. Ths, for the desired stability, Assmption 6.1 is needed. It was pointed ot in the stdy by Higham et al. 003, p.99 that the reslt 6. forces f 0 = 0 and g0 = 0, in the SD 1.1. To define the trncation mapping for sper-linear diffsion and drift

26 XPLICIT NUMRICAL APPROXIMATIONS FOR SDS IN FINIT AND INFINIT HORIZONS 5 terms we first choose a strictly increasing continos fnction ϕ 1 : R + R + sch that ϕ 1 r as r and fx sp gx 0< x r x x ϕ 1 r r > Denote by ϕ 1 1 the inverse fnction of ϕ 1, obviosly ϕ 1 1 :[ϕ0, R + is a strictly increasing continos fnction. We also choose a nmber 0, 1 and a strictly decreasing h 1 : 0, ] 0, sch that h 1 ϕ 1 x 0, lim h 1 = and 1/ θ 1 h 1 K, 0, ] hold for some θ 1 0, 1/, where K is a positive constant independent of. For a given 0, ], let s define a trncation mapping π 1 : Rd R d by where we let x x = 0 when x = 0. Obviosly, f π 1 x h1 x, π 1 x = x ϕ1 1 h1 x x, 6.5 g π 1 x 1 h 1 x, x 0, x R d. 6.6 Remar 6.4 If fx gx C x, for all x R d,letϕ 1 r C for any r [0, ], and let ϕ1 1 for any [C, ; choose > 0 sch that h 1 C C. Ths, π 1 x = x, f π 1x h 1 x and g π 1x h 1 1 x hold always. Given a step size 0, ], applying the trncation mapping to the trncated M method yields the scheme 0 = x 0, ũ +1 = + f +g B, = π 1 ũ+1. To obtain the continos-time approximation we define tbyt := for all t [ t, t +1. The trncation mapping π 1 x satisfies 3.4. Ths, Lemma 3.1 and Theorems 3.3 and 5.5 hold for the nmerical soltion t of the scheme 6.7 nder Assmption 6.1. Moreover, π 1 x has the more precise property 6.6, which may reslt in the corresponding scheme realizing the exponential stability of the SD 1.1. Theorem 6.5 Under Assmption 6.1, for any ε 0, pλ, there is a 0, ], sch that for any 0, ], the nmerical soltion t of the trncated M scheme 6.7 satisfies t p x 0 p e pλ εt t

27 6 X. LI T AL. That is, the trncated M scheme 6.7 is exponentially stable in the pth moment. Proof. For any δ>0, we have δ + ũ +1 p/ = δ + p/ 1 + η p/, where η = T f + g B + T g B + f + f T g B δ +. Now we prove only the case when 0 < p < and the proofs for other cases are similar. Thans to ineqality 3.1, for 0 < p <, we have δ + ũ p +1 Ft δ + p [ Both 6.6 and 6.7 imply 1 + p pp η F t + 8 η F t + η F t = δ + 1 [ T f + gy + f ] pp p 4 ] η 3 48 F t. 6.9 δ + 1 [ T f + g + h 1 ] δ + 1 T f + g +K 1+θ Using 3.15, we have η F t = δ + [ T f + g B + T g B ] + f + f T g B F t δ + [ T T g B + T g B T f ] + g B + f + f T g B F t 4 δ + T g 8 δ + f g 4 δ + T g 8 δ + 4 h 1 4 δ + T g 4K 1+θ

28 XPLICIT NUMRICAL APPROXIMATIONS FOR SDS IN FINIT AND INFINIT HORIZONS 7 and η 3 F t = δ + [ 3 T f + g B + T g B ] 3 + f + f T Ft g B δ + 3 [ 7 T f g 6 B f f g B +8 g 4 B f g B + 16 f 3 g B f g 4 B f 4 g B 4 ] Ft C δ f g f f g + g 4 + f g 3 + f 3 g 4 + f g f 4 g 5 C h h h h 1 + h 1 + h h h h C 1+θ We can also prove that, for any i > 3, η i F t = o 1+θ 1. Combining implies δ + ũ p +1 F t δ + p [ 1 + o 1+θ 1 + p δ + T f + g + p T g ] δ +. For any given ε 0, pλ, choose 0, ] small sfficiently sch that o θ 1 ε/. Taing the expectation on both sides, by Assmption 6.1, we have for any 0, ], δ + ũ p ε [ δ + p ] pλ [ δ + p ] 4 = + p [δ δ + p T f + g ] 1 + ε [ δ + p ] pλ [ δ + p ] 4 [ + p δ δ+ p [ ] ] [ + T f p δ δ + p [ ] ] T f + p [δ δ + p g ].

29 8 X. LI T AL. Letting δ 0 and sing the theorem on monotone convergence we have ũ +1 p [ 1 pλ ε ] p Choose < /pλ ε; then, for any 0, ],wehave0< 1 pλ ε / < 1. It follows from 6.13 that, for any integer 0, +1 p ũ +1 p 1 pλ ε p. Ths, +1 p 1 pλ ε +1 x0 p. By the elementary ineqality 1 pλ ε e pλ ε we obtain +1 p x 0 p e pλ ε+1 / = x 0 p e pλ εt +1/ 0. Ths, the desired ineqality 6.8 for the case 0 < p < follows from the definition of t. The reqired ineqality for p can be proved similarly. Therefore, the proof is complete. 7. Stability in distribtion This section focses on asymptotic stability in distribtion of SD 1.1 and the nmerical approximation to the invariant measres. In past decades mch effort has been devoted to approximating invariant measres for ergodic stochastic processes. Talay 00 obtained convergence rates for approximation to the invariant measres sing an M implicit scheme for a stochastic Hamiltonian dissipative system with nonglobal Lipschitz coefficients and additive noise. Lamberton & Pagès 00, 003 stdied recrsive stochastic algorithms with decreasing step sizes to approximate the invariant distribtion for an ler scheme nder Lyapnov-type assmptions nder the provision of the existence of sch a Lyapnov fnction. Li & Mao 015 too advantage of the implicit bacward M scheme to approximate the invariant measre for nonlinear SDs with nonglobal Lipschitz coefficients. Mei & Yin 015 ascertained convergence rates for approximation to invariant measres sing M schemes with decreasing step sizes for switching diffsions. Approximation sing M schemes to the invariant measres for switching diffsions was also dealt with in the stdy by Bao et al In this paper, we first give sfficient conditions that garantee SD 1.1 is asymptotically stable in distribtion. Then we constrct a trncation mapping and explicit schemes that can approximate the invariant measre of SD 1.1 effectively. For convenience we impose the following hypothesis. Assmption 7.1 There exists a pair of positive constants ρ and ν sch that x y [ x y T fx fy + gx gy ] ρ x y T gx gy ν x y 4 x, y R d. 7.1 Lemma 7. Under Assmption 7.1, SD1.1 has the property lim xt; xt; t v ρ = 0 niformly in, v K, 7.

30 XPLICIT NUMRICAL APPROXIMATIONS FOR SDS IN FINIT AND INFINIT HORIZONS 9 for any compact sbset K R d, where ρ is given in Assmption 7.1 and xt; x 0 denotes the niqe global soltion of SD 1.1 with the initial vale x 0 R d. Proof. It follows from SD 1.1 that d xt; xt; v = fxt; fxt; v dt + gxt; gxt; v dbt. 7.3 By virte of the definition of the operator L, L x y ρ = ρ x y ρ 4 [ x y [ x y T fx fy + gx gy ] Using Itô s formla we obtain Then we have ρ x y T gx gy ] ρν x y ρ. 7.4 e ρν t xt; xt; v ρ v ρ + v ρ. Ths, the desired reslt follows. t 0 e ρμ s [ ρν xs; xs; v ρ +L xs; xs; v ρ] ds xt; xt; v ρ v ρ e ρν t t Remar 7.3 Assmption 7.1 garantees the attractivity of the analytic soltions, which is also an alternative to Khasminsii s condition, which states that there exists a positive constant α sch that L x y p α x y p holds for any x, y R d. As in the other conditions, we prefer to pt the conditions on the coefficients of the SDs for verification prposes. Theorem 7.4 Under Assmptions 5.1 and 7.1, SD1.1 is asymptotically stable in distribtion. Proof. We adopt the idea of Mao & Yan 006, Theorem The main difference is that we remove the linear growth reqirement of the drift and diffsion terms. Since the proof is technical we divide it into three steps. Step 1: Under Assmptions 5.1 and 7.1, SD1.1 has a niqe reglar soltion with an initial vale x 0 denoted by xt; x 0, which is a time-homogeneos Marov process. Let Pt; x 0, denote the transition probability of the process xt; x 0.LetP R d denote all probability measres on R d. Then for P 1, P P R d define a metric d L as d L P 1, P = sp l L lxp 1 dx lxp dx, R d R d

31 30 X. LI T AL. where L = {l : R d R : lx ly x y and l 1 }. Given any compact set K R d, for any, v K and l L, compte l xt; l xt; v xt; xt; v. 7.6 If Assmption 7.1 holds for ρ 1 then for any ε>0thereisat 1 > 0 sch that xt; xt; v xt; xt; v [ xt; xt; v ρ] 1 ρ < ε t T 1, niformly in, v K. For this ε, ifρ<1, by Assmption 7.1, there is a T 1 > 0 sch that xt; xt; v ρ < ε 8 t T 1, niformly in, v K. Hence, xt; xt; v P{ xt; xt; v } + I { xt; xt;v <} xt; xt; v 1 ρ xt; xt; v ρ + 1 ρ xt; xt; v ρ ρ xt; xt; v ρ < ε. In other words, for any ρ>0, there is a T 1 > 0 sch that xt; xt; v < ε for all t T 1, niformly in, v K. Itfollowsfrom7.6 that lxt; lxt; v < ε for all t T 1. Since l is arbitrary we have sp lxt; lxt; v ε l L t T Then d L Pt;,,Pt; v, ε <ε for all t T 1, namely, lim t d L Pt;,,Pt; v, = 0 niformly in, v K. Step : For any x 0 R d, {Pt; x 0, : t 0} is Cachy in the space P R d with metric d L, namely, there is a T > 0 sch that d L Pt + s; x 0,,Pt; x 0, ε t T, s > 0. This is eqivalent to sp l xt + s; x0 l xt; x 0 ε t T, s > l L

32 XPLICIT NUMRICAL APPROXIMATIONS FOR SDS IN FINIT AND INFINIT HORIZONS 31 Now for any l L and t, s > 0, compte l xt + s; x0 l xt; x 0 = lxt + s; x 0 F s lxt; x0 = R lxt; yps; x 0,dy lxt; x 0 d lxt; y lxt; x0 Ps; x0,dy R d P s; x 0, S c N + lxt; y lxt; x 0 Ps; x 0,dy, 7.9 S N where S N = { x R d : x N } and S c N = Rd S N.By5. of Theorem 5. there is a positive constant N > x 0 sfficiently large sch that P s; x 0, S c ε N < 4 s On the other hand, by 7.7 there is a T > 0 sch that sp lxt; y lxt; x0 ε l L t T, y S N Sbstitting 7.10 and 7.11 into7.9 yields lxt + s; x0 lxt; x 0 <εfor all t T, s > 0. Since l is arbitrary the desired ineqality 7.8 mst hold. Step 3: For a given x 0 R d, it follows from 7.10 that { Pt; x 0, } is tight. Since R d is complete and separable it is relatively compact see Billingsley, 1968, Theorems 6.1, 6.. Then any seqence { Ptn ; x 0, } t n as n has a wea convergent sbseqence denoted by { Pt n ; x 0, } with some notation abse. Assme its wea limit is an invariant measre μ ; then there is a positive integer N sch that t N > T and d L Pt n ; x 0,,μ < ε for all n N. Then it follows from 7.8 that d L Pt; x 0,, μ d L Pt n ; x 0,, μ + d L Pt n ; x 0,, Pt; x 0, < ε t T. Ths, lim t d L Pt; x 0,, μ = 0 and the invariant measre μ is niqe. For any y 0 R d, lim d L Pt; y 0,, μ lim d L Pt; y 0,,Pt; x 0, + lim d L Pt; x 0,, μ = 0. t t t Therefore, the desired reslt follows. In order to approximate the invariant measre μ of SD 1.1 we need to constrct a scheme sch that for any 0, ] the nmerical soltions are attractive in ρth moment and have a niqe nmerical invariant measre. However, the trncation mappings π x and π 1 x are not sitable for the attractive nmerical soltions. Ths, we constrct the trncation mapping π x according to the local Lipschitz growth of drift and diffsion coefficients. Then maing se of the appropriate trncation

33 3 X. LI T AL. mapping we give an explicit scheme. Finally, we show that it prodces a niqe nmerical invariant measre μ that tends to the invariant measre μ of SD 1.1 as 0. Under the local Lipschitz condition, to define the trncation mapping, we first choose a strictly increasing continos fnction ϕ : R + R + sch that ϕ r as r and fx fy gx gy sp x y r,x y x y x y ϕ r r > Denote by ϕ 1 the inverse fnction of ϕ ; obviosly ϕ 1 :[ϕ 0, R + is a strictly increasing continos fnction. We also choose a nmber 0, 1 and a strictly decreasing h : 0, ] 0, sch that h ϕ x 0 f0 g0, lim 0 h = and 1/ θ h K, 0, ] 7.13 holds for some θ 0, 1/, where K is a positive constant independent of. For a given 0, ] let s define another trncation mapping π : Rd R d by where we let x x = 0 when x = 0. Note that π x = x ϕ 1 h x x, 7.14 f π x f π y h π x π y, 7.15 g π x g π y 1 h π x π y, x, y R d We also have f π x h 1 + π x, g π x 1 h 1 + π x, x R d Remar 7.5 If fx fy gx gy C x y for all x, y R d,letϕ r C for any r [0, ], and let ϕ 1 for any [C, ; choose > 0 sch that h C C. Ths, π x = x, hold always. Given a step size 0, ], define the trncated M method scheme by w 0 = x 0, w +1 = w + fw +gw B, w +1 = π w

34 XPLICIT NUMRICAL APPROXIMATIONS FOR SDS IN FINIT AND INFINIT HORIZONS 33 To obtain the continos-time approximation we define wtby wt := w t [ t, t +1. Theorem 7.6 Under Assmption 7.1, for any ε 0, ρν, there is a constant 3 0, ] sch that the soltions of the trncated M scheme 7.18 satisfy sp w t w v t ρ v ρ e ρν εt/ t 0, , 3 ] where w and w v denote the nmerical soltions defined by 7.18 with different initial vales and v, respectively, and ρ and ν are given in Assmption 7.1. Proof. Becase the proof is rather technical we divide it into three steps. Step 1: For any integer 0wehave w +1 wv +1 = w wv + f w f w v + g w g w v B = w w v + w T wv f w f w v + g w g w v B + w T wv g w g w v B + f w f w v + f w f w v T g w g w v B. For any δ>0, δ + w +1 wv ρ/ +1 = δ + w wv ρ/ 1 + ζ ρ/, where ζ = w T wv f w f w v + g w g w v B δ + w w v + w T wv g w g w v δ + f w w w v B + f w v δ + w w v + f w f w v T g w g w v δ + w w v B.

35 34 X. LI T AL. We give the proof otline for the case 0 < p < and other cases can be prove similarly. Using the properties of the Brownian motion 7.13, 7.15, 7.16 and the elementary ineqality we can obtain δ + w +1 wv +1 ρ/ F t δ + w wv ρ/ 1 + o 1+θ + ρ w T wv f w f w v + g w g w v δ + w w v ρρ w T wv g w g w v + δ + w w v. For any given ε 0, ρν choose 0, ] sfficiently small sch that o θ ε/. It follows from Assmption 7.1 that, for any 0, ], δ + w +1 wv +1 ρ/ Ft δ + w w v [ ρ/ 1 + ε ρν + ρδ w w v δ + w w v w T wv f w f w v + g w g w v ] δ + w w v. Taing the expectation on both sides, letting δ 0, by the theorem on monotone convergence, we have w +1 w v +1 ρ 1 ρν ε w w v ρ. 7.0 Step : The ineqality π x π y x y x, y R d 7.1

36 XPLICIT NUMRICAL APPROXIMATIONS FOR SDS IN FINIT AND INFINIT HORIZONS 35 holds always. In fact, if x y ϕ 1 h, ϕ 1 h,7.1 holds obviosly. If x ϕ 1 h, y x y π x π y = x y x π y = x T y + y + x T π π y y = y π y x T y π y y π y x y π y = y π y x y ϕ 1 h y y = y π y x y ϕ 1 h = y π y x y ϕ 1 h = y π y x y π y = y π y + y π y x 0. Then 7.1 follows immediately. If x ϕ 1 h, y ϕ 1 h, 7.1 holds also by symmetry on x and y. Finally, if x y ϕ 1 h, x y π x π y = x π x + y π T y x T y π x π y = x π x + y π y x T y ϕ 1 h x y x T y x π x + y π y x y ϕ 1 h = x x y + y 0. Then 7.1 follows immediately. Ths, the desired ineqality 7.1 holds for all cases. Step 3: Choose 3 < /ρν ε, then for any 0, 3 ], we have 0 < 1 ρν ε / < 1. It follows from 7.0 and 7.1 that for any integer 0, w +1 w v +1 p w +1 w v +1 p 1 ρν ε w w v p.