Convergence Rate of Euler-Maruyama Scheme for SDDEs of Neutral Type

Transcription

1 arxiv: v2 [mat.pr] 22 Mar 216 Convergence Rate of uler-maruyama Sceme for SDDs of Neutral Type Yanting Ji, Jianai Bao, Cenggui Yuan Abstract In tis paper, we are concerned wit convergence rate of uler-maruyama M) sceme for stocastic differential delay equations SDDs) of neutral type, were te neutral term, te drift term and te diffusion term are allowed to be of polynomial growt. More precisely, for SDDs of neutral type driven by Brownian motions, we reveal tat te convergence rate of te corresponding M sceme is one alf; Wereas for SDDs of neutral type driven by jump processes, we sow tat te best convergence rate of te associated M sceme is close to one alf. AMS subject Classification: 65C3, 6H1. Key Words: stocastic differential delay equation of neutral type, polynomial condition, uler sceme, convergence rate, jump processes. 1 Introduction Tere is numerous literature concerned wit convergence rate of numerical scemes for stocastic differential equations SDs). nder a Hölder condition, Gyöngy-Rásonyi [8] provided a convergence rate of M sceme; nder te Kasminskii-type condition, Mao [11] revealed tat te convergence rate of te truncated M metod is close to one alf; Sabanis [16] recovered te classical rate of convergence i.e., one alf) for te tamed uler scemes, were, for te SD involved, te drift coefficient satisfies a one-side Lipscitz condition and a polynomial Lipscitz condition, and te diffusion term is Lipscitzian. Tere is also some literature on convergence rate of numerical scemes for stocastic functional differential equations SFDs). For example, under a log-lipscitz condition, Bao et al. [4] studied convergence rate of M approximation for a range of SFDs driven by jump processes; Bao-Yuan [3] investigated convergence rate of M approac for a class of SDDs, were te drift and diffusion coefficients are allowed to be polynomial growt wit respect to te delay variables; Gyöngy-Sabanis [7] discussed rate of almost sure convergence of uler approximations for SDDs under monotonicity conditions. Department of Matematics, Swansea niversity, Singleton Park, SA2 8PP, K, matjyt@gmail.com Scool of Matematics and Statistics, Central Sout niversity, Cina, jianaibao13@gmail.com Department of Matematics, Swansea niversity, Singleton Park, SA2 8PP, K, C.Yuan@swansea.ac.uk 1

2 Increasingly real-world systems are modelled by SFDs of neutral type, as tey represent systems wic evolve in a random environment and wose evolution depends on te past states and derivatives of states of te systems troug eiter memory or time delay. In te last decade, for SFDs of neutral type, tere are a large number of papers on, e.g., stocastic stabilitysee, e.g., [5, 12, 13]), on large fluctuationssee, e.g., [1]), on large deviation principle see, e.g., [6]), on transportation inequality see, e.g., [2]), to name a few. So far, te topic on numerical approximations for SFDs of neutral type as also been investigated considerably. For instance, under a global Lipscitz condition, Wu-Mao [18] revealed convergence rate of te M sceme constructed is close to one alf; under a log- Lipscitz condition, Jiang et al. [9] generalized [2] by Yuan and Mao to te neutral case; under te Kasminskill-type condition, following te line of Yuan-Glover[21], [15, 22] studied convergence in probability of te associated M sceme; for preserving stocastic stability of te exact solutions) of variable numerical scemes, we refer to, e.g., [1, 19, 23, 24] and references terein. We remark tat most of te existing literature on convergence rate of M sceme for SFDs of neutral type is dealt wit under te global Lipscitz condition, were, in particular, te neutral term is contractive. Consider te following SDD of neutral type d{xt) X 2 t τ)} = {axt)bx 3 t τ)}dtcx 2 t τ)dbt), t, 1.1) in wic a,b,c R, τ > is some constant, and Bt) is a scalar Brownian motion. Observe tat all te neutral, drift and diffusion coefficients in 1.1) are igly linear wit respect to te delay variable so tat te existing results on convergence rate of M scemes associated wit SFDs of neutral type cannot be applied to te example above. So in tis paper we intend to establis te teory on convergence rate of M sceme for a class of SDDs of neutral type, were, in particular, te neutral term is of polynomial growt, so tat it covers more interesting models. Trougout te paper, te sortand notation a b is used to express tat tere exists a positive constant c suc tat a cb, were c is a generic constant wose value may cange from line to line. Let Ω,F,P) be a complete probability space wit a filtration F t ) t satisfying te usual conditions i.e., it is rigt continuous and F contains all P-null sets). For eac integer n 1, let R n,,, ) be an n-dimensional uclidean space. For A R n R m, te collection of all n m matrices, A stands for te Hilbert-Scmidt norm, i.e., A = m i=1 Ae i 2 ) 1/2, were e i ) i 1 is te ortogonal basis of R m. For τ >, wic is referred to as delay or memory, C := C[ τ,];r n ) means te space of all continuous functions φ : [ τ,] R n wit te uniform norm φ := τ θ φθ). Let Bt)) t be a standard m-dimensional Brownian motion defined on te probability space Ω,F,F t ) t,p). To begin, we focus on an SDD of neutral type on R n,,, ) in te form d{xt) GXt τ))} = bxt),xt τ))dtσxt),xt τ))dbt), t > 1.2) wit te initial value Xθ) = ξθ) for θ [ τ,], were G : R n R n, b : R n R n R n, σ : R n R n R n m. We assume tat tere exist constants L > and q 1 suc tat, for x,y,x,y R n, 2

3 A1) Gy) Gy) L1 y q y q ) y y ; A2) bx,y) bx,y) σx,y) σx,y) L x x L1 y q y q ) y y, were stands for te Hilbert-Scmidt norm; A3) ξt) ξs) L t s for any s,t [ τ,]. Remark 1.1. Tere are some examples suc tat A1) and A2) old. For instance, if Gy) = y 2,bx,y) = σx,y) = axy 3 for any x,y R and some a R, Ten bot A1) and A2) old by taking Vy,y) = y2 3 2 y2 for arbitrary y,y R. By following a similar argument to [12, Teorem 3.1, p.21], 1.2) as a unique solution {Xt)} under A1) and A2). In te sequel, we introduce te M sceme associated wit 1.2). Witout loss of generality, we assume tat = T/M = τ/m, 1) for some integers M,m > 1. For every integer k = m,,, set Y k) := ξk), and for eac integer k = 1,,M 1, we define Y k1) GY k1 m) ) = Y k) GY k m) )by k),y k m) )σy k) k m),y ) B k), 1.3) were B k) := Bk1)) Bk). For any t [k,k1)), set Yt) := Y k). To avoid te complex calculation, we define te continuous-time M approximation solution Yt) as below: For any θ [ τ,], Yθ) = ξθ), and Yt) =GYt τ))ξ) Gξ τ)) σys),ys τ))dbs), bys),ys τ))ds t [,T]. 1.4) A straigtforward calculation sows tat te continuous-time M approximate solution Yt) coincides wit te discrete-time approximation solution Yt) at te grid points t = n. Te first main result in tis paper is stated as below. Teorem 1.1. nder te assumptions A1)-A3), Xt) Yt) p p/2, p ) t T So te convergence rate of te M sceme i.e., 1.4)) associated wit 1.2) is one alf. Next, we move to consider te convergence rate of M sceme corresponding to a class of SDDs of neutral type driven by pure jump processes. More precisely, we consider an SDDs of neutral type d{xt) GXt τ))} = bxt),xt τ))dt gxt ),Xt τ) ),u)ñdu,dt) 1.6) wit te initial data Xθ) = ξθ),θ [ τ,]. Herein, G and b are given as in 1.2), g : R n R n R m, were BR); Ñdt,du) := Ndt,du) dtλdu) is te compensated 3

4 Poisson measure associated wit te Poisson counting measure Ndt, du) generated by a stationary F t -Poisson point process {pt)} t on R wit caracteristic measure λ ), i.e., Nt,) = s DP),s t I ps)) for BR); Xt ) := lim s t Xs). We assume tat b and G suc tat A1) and A2) wit σ n m terein. We furter pose tat tere exist L,r > suc tat for any x,y,x,y R n and u, A4) gx,y,u) gx,y,u) L x x 1 y q y q ) y y ) u r and g,,u) u r, were q 1 is te same as tat in A1). A5) u p λdu) < for any p 2. Remark 1.2. Te jump coefficient may also be igly non-linear wit respect to te delay argument; for example, x,y R, u and q 1, gx,y,u) = xy q )u satisfies A5). By following te procedures of 1.3) and 1.4), te discrete-time M sceme and te continuous-time M approximation associated wit 1.6) are defined respectively as below: Y n1) GY n1 m) ) = Y n) GY n m) )by n) were Ñn := Ñn1),) Ñn,), and,y n m) )gy n),y n m),u) Ñn, 1.7) Yt) =GYt τ))ξ) Gξ τ)) bys),ys τ))ds 1.8) gys ),Ys τ) ),u)ñdu,ds), were Y is defined similarly as in 1.4). Our second main result in tis paper is presented as follows. Teorem 1.2. ndera1)-a5) wit σ n m terein, for any p 2 and θ,1) Xt) Yt) p 1 1θ) [T/τ]. 1.9) t T So te best convergence rate of M sceme i.e., 1.8)) associated wit 1.6) is close to one alf. Remark 1.3. By a close inspection of te proof for Teorem 1.2, te conditions A4) and A5) can be replaced by: For any p > 2 tere exists K p,k > and q > 1 suc tat gx,y,u) p λdu) K p 1 x p y q ); gx,y,u) gx,y,u) p λdu) K p [ x x p 1 y q y q ) y y p ]; gx,y,u) 2 λdu) K 1 x 2 y q ); gx,y,u) gx,y,u) 2 λdu) K [ x x 2 1 y q y q ) y y 2 ] for any x,y,x,y R n. 4

5 2 Proof of Teorem 1.1 Te lemma below provides estimates of te p-t moment of te solution to 1.2) and te corresponding M sceme, alongside wit te p-t moment of te displacement. Lemma 2.1. nder A1) and A2), for any p 2 tere exists a constant C T > suc tat Xt) ) p Yt) p C T, 2.1) t T t T and were Γt) := Yt) Yt). t T Γt) p ) p/2, 2.2) Proof. We focus only on te following estimate Yt) p C T 2.3) t T for some constant C T > since te uniform p-t moment of Xt) in a finite time interval can be done similarly. From A1) and A2), one as and Gy) 1 y 1q, 2.4) bx,y) σx,y) 1 x y 1q 2.5) for any x,y R n. By te Hölder inequality, te Burkold-Davis-Gundy B-D-G) inequality see, e.g., [12, Teorem 7.3, p.4]), we derive from 2.4) and 2.5) tat Ys) p 1 ξ p1q) Ys) p1q) τ s t τ s t τ { Ys) p Ys τ) p1q) }ds 1 ξ p1q) Ys) p1q) T τ r s τ s t τ Yr) p )ds, were we ave used Yk) = Yk) in te last display. Tis, togeter wit Gronwall s inequality, yields tat Ys) p 1 ξ p1q) Xs) ), p1q) s t s t τ) 5

6 wic furter implies tat t τ Xt) p ) 1 ξ p1q), and tat t 2τ Xt) p ) 1 ξ p1q) Tus 2.3) follows from an inductive argument. t τ Xt) p1q) ) 1 ξ p1q)2. mploying Hölder s inequality and BDG s inequality, we deduce from1.4) and2.5) tat t T ) { Γt) p k M 1 k t k1) n t k1) k k1) { p 1 k M 1 k1) p 2 1 p 2 1 p 2. k k M 1 k k bys),ys τ))ds p ) σys),ys τ))dbs) p )} bys),ys τ)) p ds } σys),ys τ)) p ds { k1) 1 Ys) p Ys τ) pq1)) ds} k were in te last step we ave used 2.3). Te desired assertion is terefore complete. Wit Lemma 2.1 in and, we are now in te position to finis te proof of Teorem 1.1. Proof of Teorem 1.1. We follow te Yamada-Watanabe approac see, e.g., [3]) to complete te proof of Teorem 1.1. For fixed κ > 1 and arbitrary ε,1), tere exists a continuous non-negative function ϕ κε ) wit te port [ε/κ,ε] suc tat Set ε ε/κ ϕ κε x)dx = 1 and ϕ κε x) 2 xlnκ, x >. φ κε x) := It is readily to see tat φ κε ) suc tat x y ϕ κε z)dzdy, x >. x ε φ κε x) x, x >. 2.6) Let V κε x) = φ κε x ), x R n. 2.7) 6

7 By a straigtforward calculation, it olds and V κε )x) = φ κε x ) x 1 x, x R n 2 V κε )x) = φ κε x ) x 2 I x x) x 3 x 2 φ κε x )x x, x Rn, were and 2 stand for te gradient and Hessian operators, respectively, I denotes te identical matrix, and x x = xx wit x being te transpose of x R n. Moreover, we ave V κε )x) 1 and 2 V κε )x) 2n 1 1 ) 1 lnκ x 1 [ε/κ,ε] x ), 2.8) were 1 A ) is te indicator function of te subset A R. For notation simplicity, set Zt) := Xt) Yt) and Λt) := Zt) GXt τ))gyt τ)). 2.9) In te sequel, let t [,T] be arbitrary and fix p 2. Due to Λ) = R n and V κε ) =, an application of Itô s formula gives were and Set V κε Λt)) = V κε )Λs)),Γ 1 s) ds 1 2 V κε )Λs)),Γ 2 s)dbs) =: I 1 t)i 2 t)i 3 t), trace{γ 2 s)) 2 V κε )Λs))Γ 2 s)}ds Γ 1 t) := bxt),xt τ)) byt),yt τ)) 2.1) Γ 2 t) := σxt),xt τ)) σyt),yt τ)). Vx,y) := 1 x q y q,x,y R n. 2.11) According to 2.1), for any q 2 tere exists a constant C T > suc tat ) VXt τ),yt τ)) q C T. 2.12) t T Noting tat Xt) Yt) = Λt)Γt)GXt τ)) GYt τ)), 2.13) 7

8 and using Hölder s inequality and B-D-G s inequality, we get from 2.8) and A1)-A2) tat Θt) : = I 1 s) ) p I 3 s) p s t s t { Γ 1 s) p Γ 2 s) p }ds Xs) Ys) p ds τ τ { Λs) p Γs) p }ds VXs),Ys)) p Xs) Ys) p )ds τ τ VXs),Ys)) p Xs) Ys) p )ds. Also, by Hölder s inequality, it follows from 2.2), A3) and 2.12) tat 2.14) Θt) { Λs) p Γs) p VXs τ),ys τ)) 2p ) 1/2 Zs τ) 2p ) 1/2 Γs τ) 2p ) 1/2 }ds { Λs) p Γs) p Zs τ) 2p ) 1/2 Γs τ) 2p ) 1/2 }ds { Λs) p Zs τ) 2p ) 1/2 p/2 }ds. In te ligt of 2.8)-2.13), we derive from A1) tat I 2 s) p s t 2 V κε )Λs)) p Γ 2 s) 2p ds 1 Λs) p{ Xs) Ys) 2p VXs τ),ys τ)) 2p Xs τ) Ys τ) 2p )}I [ε/κ,ε] Λs) )ds 1 Λs) p{ Λs) 2p Γs) 2p GXs τ)) GYs τ)) 2p VXs τ),ys τ)) 2p Xs τ) Ys τ) 2p )}I [ε/κ,ε] Λs) )ds 1 Λs) p{ Λs) 2p Γs) 2p VXs τ),ys τ)) 2p Xs τ) Ys τ) 2p )}I [ε/κ,ε] Λs) )ds { Λs) p ε p Γs) 2p ε p V 2p Xs τ),ys τ)) Xs τ) Ys τ) 2p )}ds {ε p p Λs) p ε p Zs τ) 4p )) 1/2 }ds, 2.15) 2.16) 8

9 were in te last step we ave used Hölder s inequality. Now, according to 2.6), 2.15) and 2.16), one as Λs) p ǫ p V κε Λs)) s t s t ǫ p Θt) I 3 s) p ε p Tus, Gronwall s inequality gives tat Λs) p ε p p/2 ε p p s t s t { p/2 ε p p r s Λr) p ) Zs τ) 2p )) 1/2 ε p Zs τ) 4p )) 1/2 }ds. t τ) p/2 { Zs) 2p )) 1/2 ε p Zs) 4p )) 1/2 }ds t τ) { Zs) 2p )) 1/2 ε p Zs) 4p )) 1/2 }ds, 2.17) by coosing ε = 1/2 and taking Zt) for t [ τ,] into account. Next, by A1) and 2.12) it follows from Hölder s inequality tat ) Zt) p Λt) ) p GXt)) GYt)) p t T t T τ t T τ ) Λt) ) p VXt),Yt)) p Xt) Yt) p ) 2.18) t T τ t T τ ) 1/2. Λt) ) p p/2 Zt) 2p t T Substituting 2.17) into 2.18) yields tat Zt) p p/2 t T Hence, we ave and t 2τ T τ) t T τ) t T τ) Zt) 2p )) 1/2 { Zt) 2p )) 1/2 ε p Zt) 4p )) 1/2 }dt. t τ Zt) p ) p/2 τ p/2 Zt) p ) p/2, t τ Zt) 2p )) 1/2 { Zt) 2p )) 1/2 ε p Zt) 4p )) 1/2 }dt )

10 by taking ε = 1/2. Tus, te desired assertion 1.5) follows from an inductive argument. 3 Proof of Teorem 1.2 Hereinafter, Xt)) is te strong solution to 1.6) and Yt)) is te continuous-time M sceme i.e., 1.8)) associated wit 1.6). Te lemma below plays a crucial role in revealing convergence rate of te M sceme. Lemma 3.1. nder A1)-A5) wit σ n m terein, Xt) ) p Yt) p C T, p 2 3.1) t T for some constant C T > and were Γt) := Yt) Yt). t T t T Γt) p ), p 2, 3.2) Proof. By carrying out a similar argument to tat of [12, Teorem 3.1, p.21], 1.6) admits a unique strong solution {Xt)} according to [17, Teorem 117, p.79]. On te oter and, since te proof of 3.1) is quite similar to tat of 2.1), we ere omit its detailed proof. In te sequel, we aim to sow 3.2). From A4), it follows tat gx,y,u) C1 x y 1q ) u r, x,y R n, u 3.3) for some C >. Applying B-D-G s inequality see, e.g., Lemma [14, Teorem 1]) and Hölder s inequality, we derive tat t T ) { Γt) p k M 1 k t k1) k t k1) k k bys),ys τ))ds p ) gys ),Ys τ) ),u)ñds,du) p)} { k1) p 1 bys),ys τ)) p k M 1 k ) } gys),ys τ),u) p λdu) ds k M 1 p 1 p, { k1) k 1 ) } u pr λdu) ds 1 τ r s Yr) p1q) ))

11 were we ave used A2) wit σ n m and 3.3) in te tird step, and 3.1) and A5) in te last two step, respectively. So 3.2) follows as required. Next, we go back to finis te proof of Teorem 1.2. Proof of Teorem 1.2. We follow te idea of te proof for Teorem 1.1 to complete te proof. Set Γ 3 t,u) := gxt),xt τ),u) gyt),yt τ),u). Applying Itô s formula as well as te Lagrange mean value teorem to V κε ), defined by 2.7), gives tat V κε Λt)) = V κε )Λs)),Γ 1 s) ds {V κε Λs)Γ 3 s)) V κε Λs)) V κε )Λs)),Γ 3 s) }λdu)ds {V κε Λs )Γ 3 s )) V κε Λs ))}Ñdu,ds) = V κε Λ)) V κε )Λs)),Γ 1 s) ds { 1 } V κε Λs)rΓ 3 s)) V κε Λs)),Γ 3 s) dr λdu)ds { 1 V κε Λs )rγ 3 s )),Γ 3 s ) dr}ñdu,ds) =: J 1 t)j 2 t)j 3 t), in wic Γ 1 is defined as in 2.1). By B-D-G s inequality see, e.g., Lemma [14, Teorem 1]), we obtain from 2.8), 2.14) wit σ n m terein, A4) and A5) tat Υt) := 3 i=1 s t J i s) p ) { Λs) p Γs) p }ds τ τ VXs),Ys)) p Xs) Ys) p )ds, were V, ) is introduced in 2.11). Observe from Hölder s inequality tat VXs),Ys)) p Ys) Ys) p VXs),Ys)) p1θ) θ ) θ 1θ Xs) Ys) p1θ) ) 1 1θ VXs),Ys)) p1θ) θ ) θ 1θ Zs) p1θ) Γs) p1θ) ) 1 1θ 1 Xs) pq1θ) θ Zs) p1θ) ) 1 1θ Γs) p1θ) ) 1 1θ 1 1θ Zs) p1θ) ) 1 1θ, θ >, Ys) pq1θ) θ ) θ 1θ Zs) p1θ) Γs) p1θ) ) 1 1θ 3.4) 11

12 in wic we ave used 3.1) in te penultimate display and 3.2) in te last display, respectively. So we arrive at Υt) 1 1θ { Λs) p Γs) p }ds Tis, togeter wit 2.6) and 3.2), implies Λt) p ǫ p V κε Λs)) s t s t τ ǫ p 1 1θ { Λs) p Γs) p }ds 1 1θ Λs) p ds τ τ τ Zs) p1θ) ) 1 1θ ds. τ τ Zs) p1θ) ) 1 1θ ds Zs) p1θ) ) 1 1θ ds by taking ε = 1 p1θ) in te last display. sing Gronwall s inequality, due to Zθ) = for θ [ τ,], one as t T ) T τ) Λt) p 1 1θ Zs) p1θ) ) 1 1θ ds. Next, observe from A1) and Hölder s inequality tat ) Zt) p Λt) ) p GXt)) GYt)) p t T t T τ t T τ ) Λt) ) p VXt),Yt)) p Xt) Yt) p ) t T τ t T τ Λt) p t T { 1 { 1 1θ τ t T τ t T τ ) Xt) pq1θ) θ Zt) ) p1θ) t T τ) τ t T τ t T Zt) p1θ) )) 1 1θ, t τ )} Yt) pq1θ) θ 1θ θ Γt) p1θ) )} 1 1θ were in te last step we ave utilized 3.1) and 3.2). So we find tat Zt) p 1 1θ, wic, in addition to 3.5), furter yields tat Zt) p 1 1θ t 2τ 1 1θ) 2 1 1θ 1 1θ) t τ Zt) p1θ) )) 1 1θ 3.5)

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