Hindawi Publishing Corporation î e ScientiďŹc World Journal Volume 2014, Article ID 589167, 9 pages http://dx.doi.org/10.1155/2014/589167
Research Article Stochastic đ-Methods for a Class of Jump-Diffusion Stochastic Pantograph Equations with Random Magnitude Hua Yang1,2 and Feng Jiang3 1
School of Automation, Huazhong University of Science and Technology, Wuhan 430074, China School of Mathematics and Computer Science, Wuhan Polytechnic University, Wuhan 430023, China 3 School of Statistics and Mathematics, Zhongnan University of Economics and Law, Wuhan 430073, China 2
Correspondence should be addressed to Hua Yang;
[email protected] and Feng Jiang;
[email protected] Received 30 August 2013; Accepted 21 October 2013; Published 6 February 2014 Academic Editors: N. Ganikhodjaev and Y. Sun Copyright Š 2014 H. Yang and F. Jiang. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. This paper is concerned with the convergence of stochastic đ-methods for stochastic pantograph equations with Poisson-driven jumps of random magnitude. The strong order of the convergence of the numerical method is given, and the convergence of the numerical method is obtained. Some earlier results are generalized and improved.
1. Introduction Recently, the study of stochastic pantograph equations (SPEs) has many results [1â3]. SPEs have been extensively applied in many fields such as finance, control, and engineering. However, in general, SPEs have no explicit solutions, and the study of numerical solutions of SPEs has received a great deal of attention. Fan et al. [4] investigate the đźth moment asymptotical stability of the analytic solution and the numerical methods for the stochastic pantograph equation by using the Razumikhin technique. Baker and Buckwar [5] gave strong approximations to the solution obtained by a continuous extension of the đ-Euler scheme and proved that the numerical solution produced by the continuous đ-method converges to the true solution with order 1/2. Fan et al. [6] investigated the existence and uniqueness of the solutions and convergence of semi-implicit Euler methods for stochastic pantograph equations under the local Lipschitz condition and the linear growth condition. Li et al. [7] investigated the convergence of the Euler method of the stochastic pantograph equations with Markovian switching under the weaker conditions. Reference [8] studied convergence and stability of numerical methods of stochastic pantograph differential equations.
In practice, stochastic differential equations with jump and numerical methods are also discussed extensively. In [9â13] strong convergence and mean-square stability properties were analysed in the case of Poisson-driven jumps of deterministic magnitude. References [14, 15] discussed the numerical methods of stochastic differential equations with random jump magnitudes. Motivated by the papers above, in this paper, we focus on stochastic pantograph equations with random jump magnitudes. SPEs with random jump magnitudes may be regarded as an extension of stochastic pantograph equations. Jump models arise in many other application areas and have proved successful at describing unexpected, abrupt changes of state [16â18]. Typically, these models do not admit analytical solutions and hence must be simulated numerically. Similar to stochastic differential equations [19â21], explicit solutions can hardly be obtained for SPEs with random jump magnitudes. Thus, appropriate numerical approximation schemes such as the Euler (or Euler-Maruyama) are needed to apply them in practice or to study their properties. The paper is organised as follows. In Section 2, we introduce the SPEs with random jump magnitudes and define stochastic đ-methods of (1). The main result of the paper is rather technical, so we present several lemmas in Section 3 and then complete the proof in Section 4.
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2. Preliminaries Throughout this paper, we let (Ί, F, {FđĄ }đĄâĽ0 , đ) be a complete probability space with a filtration {FđĄ }đĄâĽ0 satisfying the usual conditions (i.e., it is increasing and right continuous while F0 contains all đ-null sets). Let | â
| be the Euclidean norm in Rđ . Let đĽ0 be F0 -measurable and right-continuous, đ and đ¸|đĽ0 |2 < â. Let đ¤(đĄ) = (đ¤đĄ1 , . . . , đ¤đĄđ ) be a đdimensional Brownian motion defined on the probability space. Consider a class of jump-diffusion stochastic pantograph equations with random magnitude of the form đđĽ (đĄ) = đ (đĽ (đĄâ ) , đĽ (đđĄâ )) đđĄ + đ (đĽ (đĄâ ) , đĽ (đđĄâ )) đđ¤ (đĄ) + â (đĽ (đĄâ ) , đĽ (đđĄâ ) , đžđ(đĄâ )+1 ) đđ (đĄ) , (1) on 0 ⤠đĄ ⤠đ with the initial value đĽ(0â ) = đĽ0 and 0 < đ < 1, where đ(đĄ) is a Poisson process with mean đđĄ; đĽ(đĄâ ) := limđ â đĄâ đĽ(đ ); and đžđ , đ = 1, 2, . . . are independent, identically distributed random variables representing magnitudes for each jump. Throughout, we assume that the jump magnitudes have bounded moments; that is, for some đ ⼠1, there is a constant đľ = đľđ such that óľ¨ óľ¨2đ đ¸ (óľ¨óľ¨óľ¨đžđ óľ¨óľ¨óľ¨ ) ⤠đľ.
(2)
We note for later reference that (1) involves the jump process â
đž (đĄ) := đžđ(đĄâ )+1 = âđžđ+1 1[đđ ,đđ+1 ) (đĄ) , đ=0
(5)
where đ0 = 0 and đđ , đ = 1, 2, . . . are the jump times. One generalisation of stochastic đ-Euler methods [6, 21] to system (1) has the form đŚ0 = đĽ(0) and đŚđ+1 = đŚđ + (1 â đ) đ (đŚđ , đŚ[đđ] ) â + đđ (đŚđ+1 , đŚ[đ(đ+1)] ) â + đ (đŚđ , đŚ[đđ] ) Îđ¤đ + â (đŚđ , đŚ[đđ] , đžđ(đĄđ )+1 ) Îđđ , (6) where đ â [0, 1]. Here â â (0, 1) is a step size, which satisfies đ = đ/â for some positive integer đ, đĄđ = đâ (đ = 0, 1, . . . , đ). đŚđ â đĽ(đĄđ ), Îđ¤đ = đ¤(đĄđ+1 ) â đ¤(đĄđ ), and Îđđ = đ(đĄđ+1 ) â đ(đĄđ ) are the Brownian and Poisson increments, respectively. For đĄ â [đĄđ , đĄđ+1 ], we define đŚ (đĄ) = đŚđ + (1 â đ) đ (đŚđ , đŚ[đđ] ) (đĄ â đĄđ ) + đđ (đŚđ+1 , đŚ[đ(đ+1)] ) (đĄ â đĄđ ) + đ (đŚđ , đŚ[đđ] ) (đ¤ (đĄ) â đ¤ (đĄđ ))
(7)
+ â (đŚđ , đŚ[đđ] , đžđ(đĄđ )+1 ) (đ (đĄ) â đ (đĄđ )) and denote â
đ§1 (đĄ) = âđŚđ 1[đâ,(đ+1)â) (đĄ) ,
We further employ the following assumptions.
đ=0
Assumption 1. The functions đ, đ, and â satisfy the global Lipschitz condition, that is, for each đ = 1, 2, 3, there is a positive constant đž1 such that óľ¨óľ¨ óľ¨2 óľ¨ óľ¨2 óľ¨óľ¨đ (đĽ1 , đĽ2 ) â đ (đŚ1 , đŚ2 )óľ¨óľ¨óľ¨ ⨠óľ¨óľ¨óľ¨đ (đĽ1 , đĽ2 ) â đ (đŚ1 , đŚ2 )óľ¨óľ¨óľ¨ óľ¨2 óľ¨ óľ¨2 óľ¨ â¤ đž1 (óľ¨óľ¨óľ¨đĽ1 â đŚ1 óľ¨óľ¨óľ¨ + óľ¨óľ¨óľ¨đĽ2 â đŚ2 óľ¨óľ¨óľ¨ ) , óľ¨2 óľ¨óľ¨ óľ¨óľ¨â (đĽ1 , đĽ2 , đĽ3 ) â â (đŚ1 , đŚ2 , đŚ3 )óľ¨óľ¨óľ¨
â
đ§2 (đĄ) = âđŚđ+1 1[đâ,(đ+1)â) (đĄ) , đ=0 â
đ§1 (đĄ) = â đŚ[đđ] 1[đâ,(đ+1)â) (đĄ) , đ=0
â
đ§2 (đĄ) = â đŚ[đ(đ+1)] 1[đâ,(đ+1)â) (đĄ) ,
(3)
đ=0 â
óľ¨2 óľ¨ óľ¨2 óľ¨ óľ¨2 óľ¨ â¤ đž1 (óľ¨óľ¨óľ¨đĽ1 â đŚ1 óľ¨óľ¨óľ¨ + óľ¨óľ¨óľ¨đĽ2 â đŚ2 óľ¨óľ¨óľ¨ + óľ¨óľ¨óľ¨đĽ3 â đŚ3 óľ¨óľ¨óľ¨ ) ,
đž (đĄ) = â đž (đĄđ ) 1[đâ,(đ+1)â) (đĄ) . đ=0
where đĽđ , đŚđ â Rđ .
Then, we define the continuous-time approximation
Assumption 2 (linear growth condition). There is a positive constant đž2 such that for all đĄ â [0, đ] óľ¨óľ¨ óľ¨ óľ¨ óľ¨ óľ¨óľ¨đ (đĽ1 , đĽ2 )óľ¨óľ¨ ⨠óľ¨óľ¨óľ¨đ (đĽ1 , đĽ2 )óľ¨óľ¨ ⤠đž2 (1 + óľ¨óľ¨óľ¨đĽ1 óľ¨óľ¨ + óľ¨óľ¨óľ¨đĽ2 óľ¨óľ¨ ) , óľ¨óľ¨2
(8)
óľ¨óľ¨2
óľ¨óľ¨2
đĄ
= đŚ0 + ⍠[(1 â đ) đ (đ§1 (đ ) , đ§1 (đ )) + đđ (đ§2 (đ ) , đ§2 (đ ))] đđ
óľ¨óľ¨2
óľ¨2 óľ¨ óľ¨2 óľ¨ óľ¨2 óľ¨ óľ¨2 óľ¨óľ¨ óľ¨óľ¨â (đĽ1 , đĽ2 , đĽ3 )óľ¨óľ¨óľ¨ ⤠đž2 (1 + óľ¨óľ¨óľ¨đĽ1 óľ¨óľ¨óľ¨ + óľ¨óľ¨óľ¨đĽ2 óľ¨óľ¨óľ¨ + óľ¨óľ¨óľ¨đĽ3 óľ¨óľ¨óľ¨ ) ,
đŚ (đĄ) 0
(4)
for all đĽ1 , đĽ2 , đĽ3 â Rđ . In fact, the global Lipschitz condition (3) implies the linear growth condition (4). Under these conditions, it can be shown similarly as in [20] that (1) has a unique solution with all moments bounded.
đĄ
+ ⍠đ (đ§1 (đ ) , đ§1 (đ )) đđ¤ (đ ) 0 đĄ
+ ⍠â (đ§1 (đ ) , đ§1 (đ ) , đž (đ )) đđ (đ ) , 0
(9)
which interpolates the discrete numerical approximation (6). So a convergence result for đŚ(đĄ) immediately provides a result for đŚđ .
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3. Lemmas
Combining (13), (14), and (15) with (12) yields
Throughout our analysis, đśđ , đˇđ , đ = 1, 2, . . . denote generic constants, independent of â. The main theorem of the paper is rather technical. We will present a number of useful lemmas in the section and then complete the proof in Section 4. â
Lemma 3. Under Assumption 2, there exists 0 < â < 1 such that for all 0 < â ⤠ââ , óľ¨ óľ¨2 đ¸óľ¨óľ¨óľ¨đŚđ óľ¨óľ¨óľ¨ ⤠đś1 ,
đđđ đ = 1, . . . , đ.
(10)
Proof. From (6), we have
óľ¨2 óľ¨2 óľ¨ óľ¨ + đ´ 4 (â) đ¸óľ¨óľ¨óľ¨óľ¨đŚ[đđ] óľ¨óľ¨óľ¨óľ¨ + đ´ 5 (â) đ¸óľ¨óľ¨óľ¨óľ¨đŚ[đ(đ+1)] óľ¨óľ¨óľ¨óľ¨ óľ¨2 óľ¨ + 4đž2 đâ (1 + đâ) đ¸óľ¨óľ¨óľ¨óľ¨đžđ(đĄđ )+1 óľ¨óľ¨óľ¨óľ¨
(15)
óľ¨2 óľ¨ â¤ đ´ 1 (â) + đ´ 3 (â) đ¸óľ¨óľ¨óľ¨đŚ(đ+1) óľ¨óľ¨óľ¨ óľ¨2 óľ¨ + 4đž2 đâ (1 + đâ) đ¸óľ¨óľ¨óľ¨óľ¨đžđ(đĄđ )+1 óľ¨óľ¨óľ¨óľ¨ óľ¨ óľ¨2 + (đ´ 2 (â) + đ´ 4 (â) + đ´ 5 (â)) max đ¸óľ¨óľ¨óľ¨đŚđ óľ¨óľ¨óľ¨ ,
óľ¨2 óľ¨ óľ¨ óľ¨2 đ¸óľ¨óľ¨óľ¨đŚđ+1 óľ¨óľ¨óľ¨ ⤠4đ¸óľ¨óľ¨óľ¨đŚđ óľ¨óľ¨óľ¨
[đđ]â¤đâ¤đ
óľ¨ + 4đ¸ óľ¨óľ¨óľ¨óľ¨(1 â đ) đ (đŚđ , đŚ[đđ] ) â óľ¨2 +đđ (đŚđ+1 , đŚ[đ(đ+1)] ) âóľ¨óľ¨óľ¨óľ¨
(11)
óľ¨2 óľ¨ + 4đ¸óľ¨óľ¨óľ¨óľ¨đ (đŚđ , đŚ[đđ] ) Îđ¤đ óľ¨óľ¨óľ¨óľ¨ óľ¨2 óľ¨ + 4đ¸óľ¨óľ¨óľ¨óľ¨â (đŚđ , đŚ[đđ] , đžđ(đĄđ )+1 ) Îđđ óľ¨óľ¨óľ¨óľ¨ . Note that 2đźđ˝ ⤠|đź|2 + |đ˝|2 . Now, using Assumption 2, óľ¨ óľ¨2 đ¸óľ¨óľ¨óľ¨óľ¨(1 â đ) đ (đŚđ , đŚ[đđ] ) â + đđ (đŚđ+1 , đŚ[đ(đ+1)] ) âóľ¨óľ¨óľ¨óľ¨ óľ¨ óľ¨ óľ¨2 óľ¨2 ⤠(1 â đ)2 â2 đ¸óľ¨óľ¨óľ¨óľ¨đ (đŚđ , đŚ[đđ] )óľ¨óľ¨óľ¨óľ¨ + đ2 â2 óľ¨óľ¨óľ¨óľ¨đ (đŚđ+1 , đŚ[đ(đ+1)] )óľ¨óľ¨óľ¨óľ¨ óľ¨óľ¨2
óľ¨ óľ¨ + (1 â đ) đâ2 đ¸ [óľ¨óľ¨óľ¨óľ¨đ (đŚđ , đŚ[đđ] )óľ¨óľ¨óľ¨ + óľ¨óľ¨óľ¨óľ¨đ (đŚđ+1 , đŚ[đ(đ+1)] )óľ¨óľ¨óľ¨ ]
óľ¨óľ¨2
⤠đž2 [(1 â đ)2 â2 + đ2 â2 + (1 â đ) đâ2 ] óľ¨ óľ¨2 + đž2 [(1 â đ)2 â2 + (1 â đ) đâ2 ] đ¸óľ¨óľ¨óľ¨đŚđ óľ¨óľ¨óľ¨ 2
The result then follows from an application of the discrete Gronwall inequality. The proof is complete. Lemma 4. Under Assumption 2, there exists ââ > 0 such that, for all 0 < â ⤠ââ , óľ¨2 óľ¨ đ¸óľ¨óľ¨óľ¨đŚ (đĄ) â đ§1 (đĄ)óľ¨óľ¨óľ¨ ⤠đś3 â,
đđđ đĄ â [0, đ] ,
(17)
óľ¨2 óľ¨ đ¸óľ¨óľ¨óľ¨đŚ (đĄ) â đ§2 (đĄ)óľ¨óľ¨óľ¨ ⤠đś4 â,
đđđ đĄ â [0, đ] .
(18)
đŚ (đĄ) â đ§1 (đĄ) = đŚ (đĄ) â đŚđ
óľ¨2 óľ¨ + đž2 [(1 â đ)2 â2 + (1 â đ) đâ2 ] đ¸óľ¨óľ¨óľ¨óľ¨đŚ[đđ] óľ¨óľ¨óľ¨óľ¨ óľ¨2 óľ¨ + đž2 [đ2 â2 + (1 â đ) đâ2 ] đ¸óľ¨óľ¨óľ¨óľ¨đŚ[đ(đ+1)] óľ¨óľ¨óľ¨óľ¨ .
where đ´ đ (â), đ = 1, . . . , 5 is a constant dependent on â and đ´ 3 (â) = 4đž2 â2 (đ2 + (1 â đ)đ). Now choosing â sufficiently small such that 1 â đ´ 3 (â) ⼠1/2 and noting that (2) implies that each đ¸|đž(đĄđ )|2 ⤠đľ1 , we obtain óľ¨ óľ¨2 đ¸óľ¨óľ¨óľ¨đŚđ+1 óľ¨óľ¨óľ¨ ⤠2đ´ 1 (â) + 8đž2 đâ (1 + đâ) đľ1 óľ¨ óľ¨2 + 2 (đ´ 2 (â) + đ´ 4 (â) + đ´ 5 (â)) max đ¸óľ¨óľ¨óľ¨đŚđ óľ¨óľ¨óľ¨ . đđ â¤đâ¤đ [ ] (16)
Proof. Consider đĄ â [đâ, (đ + 1)â] â [0, đ]. In this interval we have
óľ¨2 óľ¨ + đž2 [đ â + (1 â đ) đâ ] đ¸óľ¨óľ¨óľ¨đŚđ+1 óľ¨óľ¨óľ¨ 2 2
óľ¨2 óľ¨2 óľ¨ óľ¨2 óľ¨ óľ¨ đ¸óľ¨óľ¨óľ¨đŚđ+1 óľ¨óľ¨óľ¨ ⤠đ´ 1 (â) + đ´ 2 (â) đ¸óľ¨óľ¨óľ¨đŚđ óľ¨óľ¨óľ¨ + đ´ 3 (â) đ¸óľ¨óľ¨óľ¨đŚ(đ+1) óľ¨óľ¨óľ¨
= (1 â đ) đ (đŚđ , đŚ[đđ] ) (đĄ â đĄđ ) (12)
Using đ¸|Îđ¤đ |2 = đâ and Assumption 2, we have óľ¨2 óľ¨2 óľ¨ óľ¨ óľ¨ óľ¨2 đ¸óľ¨óľ¨óľ¨óľ¨đ (đŚđ , đŚ[đđ] ) Îđ¤đ óľ¨óľ¨óľ¨óľ¨ = đâđž2 (1 + đ¸óľ¨óľ¨óľ¨đŚđ óľ¨óľ¨óľ¨ + đ¸óľ¨óľ¨óľ¨óľ¨đŚ[đđ] óľ¨óľ¨óľ¨óľ¨ ) . (13) For the jump, we convert to the compensated Poisson Ěđ ) = 0 and đ¸(Îđ Ěđ )2 = Ěđ := Îđđ âđâ with đ¸(Îđ increment Îđ đâ and Assumption 2. We then obtain óľ¨2 óľ¨ đ¸óľ¨óľ¨óľ¨óľ¨â (đŚđ , đŚ[đđ] , đžđ(đĄđ )+1 ) Îđ (đ )óľ¨óľ¨óľ¨óľ¨ óľ¨ Ě (đ ) + đâ)óľ¨óľ¨óľ¨óľ¨2 = đ¸óľ¨óľ¨óľ¨óľ¨â (đŚđ , đŚ[đđ] , đžđ(đĄđ )+1 ) (Îđ óľ¨ 2 óľ¨ óľ¨2 óľ¨ óľ¨ óľ¨ óľ¨2 = đâ (1 + đâ) đž2 (1 + đ¸óľ¨óľ¨óľ¨đŚđ óľ¨óľ¨óľ¨ + đ¸óľ¨óľ¨óľ¨óľ¨đŚ[đđ] óľ¨óľ¨óľ¨óľ¨ + đ¸óľ¨óľ¨óľ¨óľ¨đžđ(đĄđ )+1 óľ¨óľ¨óľ¨óľ¨ ) . (14)
+ đđ (đŚđ+1 , đŚ[đ(đ+1)] ) (đĄ â đĄđ ) + đ (đŚđ , đŚ[đđ] ) (đ¤ (đĄ) â đ¤ (đĄđ )) + â (đŚđ , đŚ[đđ] , đžđ(đĄđ )+1 ) (đ (đĄ) â đ (đĄđ )) . (19) Thus, óľ¨2 óľ¨ đ¸óľ¨óľ¨óľ¨đŚ (đĄ) â đ§1 (đĄ)óľ¨óľ¨óľ¨ óľ¨ â¤ 3đ¸ óľ¨óľ¨óľ¨óľ¨(1 â đ) đ (đŚđ , đŚ[đđ] ) (đĄ â đĄđ ) óľ¨2 +đđ (đŚđ+1 , đŚ[đ(đ+1)] ) (đĄ â đĄđ )óľ¨óľ¨óľ¨óľ¨ óľ¨ óľ¨2 + 3đ¸óľ¨óľ¨óľ¨óľ¨đ (đŚđ , đŚ[đđ] ) (đ¤ (đĄ) â đ¤ (đĄđ ))óľ¨óľ¨óľ¨óľ¨ óľ¨ óľ¨2 + 3đ¸óľ¨óľ¨óľ¨óľ¨â (đŚđ , đŚ[đđ] , đžđ(đĄđ )+1 ) (đ (đĄ) â đ (đĄđ ))óľ¨óľ¨óľ¨óľ¨ .
(20)
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The Scientific World Journal Therefore, in view of the H¨older inequality and đđĄ â [đđ]â ⤠2â, we have
Thus, by virtue of (12)â(14) and Lemma 3, we have óľ¨ óľ¨2 đ¸óľ¨óľ¨óľ¨đŚ (đĄ) â đ§1 (đĄ)óľ¨óľ¨óľ¨ ⤠3đâ2 đž2 (1 + 2đś1 ) + 3đâ (1 + đâ) Ă đž2 (1 + 2đś1 + đľ1 ) + 15đž2 â2
(21)
óľ¨2 óľ¨ đ¸óľ¨óľ¨óľ¨đŚ (đđĄ) â đ§1 (đĄ)óľ¨óľ¨óľ¨ đđĄ
óľ¨2 óľ¨ óľ¨2 óľ¨ â¤ 12âđ¸ ⍠[óľ¨óľ¨óľ¨đ (đ§1 (đ ) , đ§1 (đ ))óľ¨óľ¨óľ¨ + óľ¨óľ¨óľ¨đ (đ§2 (đ ) , đ§2 (đ ))óľ¨óľ¨óľ¨ ] đđ [đđ]â
⤠đś3 â, where đś3 = 3đđž2 (1+2đś1 )+3đ(1+đ)đž2 (1+2đś1 +đľ1 )+15đž2 . In a similar way we obtain (18). The proof is complete. Lemma 5. Under Assumption 2, there exists ââ > 0 such that, for all 0 < â ⤠ââ , óľ¨2 óľ¨ đ¸óľ¨óľ¨óľ¨đŚ (đđĄ) â đ§1 (đĄ)óľ¨óľ¨óľ¨ ⤠đś5 â, đđđ đĄ â [0, đ] , (22) óľ¨2 óľ¨ đ¸óľ¨óľ¨óľ¨đŚ (đđĄ) â đ§2 (đĄ)óľ¨óľ¨óľ¨ ⤠đś6 â đđđ đĄ â [0, đ] . Proof. Consider đĄ â [đâ, (đ + 1)â] â [0, đ]. By (9), we have
óľ¨óľ¨ đđĄ óľ¨óľ¨2 óľ¨ óľ¨ đ (đ§1 (đ ) , đ§1 (đ )) đđ¤ (đ )óľ¨óľ¨óľ¨óľ¨ + 12đ¸óľ¨óľ¨óľ¨óľ¨âŤ óľ¨óľ¨ [đđ]â óľ¨óľ¨ óľ¨óľ¨ đđĄ óľ¨óľ¨2 óľ¨ Ě (đ )óľ¨óľ¨óľ¨óľ¨ + 24đ¸óľ¨óľ¨óľ¨óľ¨âŤ â (đ§1 (đ ) , đ§1 (đ ) , đž (đ )) đđ óľ¨óľ¨ óľ¨óľ¨ [đđ]â óľ¨ + 12âđ2 đ¸ âŤ
đđĄ
[đđ]â
óľ¨óľ¨ óľ¨2 óľ¨óľ¨â (đ§1 (đ ) , đ§1 (đ ) , đž (đ ))óľ¨óľ¨óľ¨ đđ . (25)
đŚ (đđĄ) â đ§1 (đĄ)
Then, applying ItËo and martingale isometries and Assumption 2, we have
= đŚ (đđĄ) â đŚ[đđ] = đŚ (đđĄ) â đŚ ([đđ] â) đđĄ
[(1 â đ) đ (đ§1 (đ ) , đ§1 (đ )) + đđ (đ§2 (đ ) , đ§2 (đ ))] đđ =⍠[đđ]â đđĄ
+⍠đ (đ§1 (đ ) , đ§1 (đ )) đđ¤ (đ ) [đđ]â +âŤ
đđĄ
[đđ]â
óľ¨2 óľ¨ đ¸óľ¨óľ¨óľ¨đŚ (đđĄ) â đ§1 (đĄ)óľ¨óľ¨óľ¨ đđĄ
óľ¨2 óľ¨2 óľ¨ óľ¨ â¤ 12âđž2 ⍠[2 + đ¸óľ¨óľ¨óľ¨đ§1 (đ )óľ¨óľ¨óľ¨ + đ¸óľ¨óľ¨óľ¨đ§1 (đ )óľ¨óľ¨óľ¨ [đđ]â óľ¨2 óľ¨2 óľ¨ óľ¨ +đ¸óľ¨óľ¨óľ¨đ§2 (đ )óľ¨óľ¨óľ¨ + đ¸óľ¨óľ¨óľ¨đ§2 (đ )óľ¨óľ¨óľ¨ ] đđ
â (đ§1 (đ ) , đ§1 (đ ) , đž (đ )) đđ (đ ) .
đđĄ
(23) Thus, óľ¨2 óľ¨óľ¨ óľ¨óľ¨đŚ (đđĄ) â đ§1 (đĄ)óľ¨óľ¨óľ¨ óľ¨óľ¨2 óľ¨óľ¨ đđĄ óľ¨ óľ¨ â¤ 3óľ¨óľ¨óľ¨óľ¨âŤ [(1 â đ) đ (đ§1 (đ ) , đ§1 (đ )) + đđ (đ§2 (đ ) , đ§2 (đ ))] đđ óľ¨óľ¨óľ¨óľ¨ óľ¨óľ¨ óľ¨óľ¨ [đđ]â óľ¨óľ¨ đđĄ óľ¨óľ¨2 óľ¨ óľ¨ + 3óľ¨óľ¨óľ¨óľ¨âŤ đ (đ§1 (đ ) , đ§1 (đ )) đđ¤ (đ )óľ¨óľ¨óľ¨óľ¨ óľ¨óľ¨ [đđ]â óľ¨óľ¨ óľ¨óľ¨ đđĄ óľ¨óľ¨2 óľ¨ óľ¨ + 3óľ¨óľ¨óľ¨óľ¨âŤ â (đ§1 (đ ) , đ§1 (đ ) , đž (đ )) đđ (đ )óľ¨óľ¨óľ¨óľ¨ óľ¨óľ¨ [đđ]â óľ¨óľ¨ óľ¨óľ¨ đđĄ óľ¨óľ¨2 óľ¨óľ¨ óľ¨ óľ¨ â¤ 3óľ¨óľ¨âŤ [(1 â đ) đ (đ§1 (đ ) , đ§1 (đ )) + đđ (đ§2 (đ ) , đ§2 (đ ))] đđ óľ¨óľ¨óľ¨óľ¨ óľ¨óľ¨ [đđ]â óľ¨óľ¨ óľ¨óľ¨óľ¨ đđĄ óľ¨óľ¨óľ¨2 + 3óľ¨óľ¨óľ¨óľ¨âŤ đ (đ§1 (đ ) , đ§1 (đ )) đđ¤ (đ )óľ¨óľ¨óľ¨óľ¨ óľ¨óľ¨ [đđ]â óľ¨óľ¨ óľ¨óľ¨óľ¨ đđĄ óľ¨óľ¨2 Ě (đ )óľ¨óľ¨óľ¨óľ¨ + 6óľ¨óľ¨óľ¨óľ¨âŤ â (đ§1 (đ ) , đ§1 (đ ) , đž (đ )) đđ óľ¨óľ¨ óľ¨óľ¨ [đđ]â óľ¨ óľ¨óľ¨2 óľ¨óľ¨ đđĄ óľ¨ óľ¨ + 6đ2 óľ¨óľ¨óľ¨óľ¨âŤ â (đ§1 (đ ) , đ§1 (đ ) , đž (đ )) đđ óľ¨óľ¨óľ¨óľ¨ . óľ¨óľ¨ óľ¨óľ¨ [đđ]â (24)
óľ¨2 óľ¨2 óľ¨ óľ¨ + 12đž2 ⍠(1 + đ¸óľ¨óľ¨óľ¨đ§1 (đ )óľ¨óľ¨óľ¨ + đ¸óľ¨óľ¨óľ¨đ§1 (đ )óľ¨óľ¨óľ¨ ) đđ [đđ]â đđĄ
óľ¨2 óľ¨2 óľ¨2 óľ¨ óľ¨ óľ¨ (1 + đ¸óľ¨óľ¨óľ¨đ§1 (đ )óľ¨óľ¨óľ¨ + đ¸óľ¨óľ¨óľ¨đ§1 (đ )óľ¨óľ¨óľ¨ + đ¸óľ¨óľ¨óľ¨đž (đ )óľ¨óľ¨óľ¨ ) đđ + 24đđž2 ⍠[đđ]â + 12âđ2 đž2 đđĄ
óľ¨2 óľ¨2 óľ¨2 óľ¨ óľ¨ óľ¨ Ă⍠(1 + đ¸óľ¨óľ¨óľ¨đ§1 (đ )óľ¨óľ¨óľ¨ + đ¸óľ¨óľ¨óľ¨đ§1 (đ )óľ¨óľ¨óľ¨ + đ¸óľ¨óľ¨óľ¨đž (đ )óľ¨óľ¨óľ¨ ) đđ . [đđ]â
(26)
Now, note that (2) implies that each đ¸|đž(đĄđ )|2 ⤠đľ1 , on [đâ, (đ + 1)â], đ§1 ⥠đŚđ , đ§2 ⥠đŚđ+1 , đ§1 ⥠đŚđâđ , đ§2 ⥠đŚđâđ+1 , and đž ⥠đžđ . Hence, applying Lemma 3, we obtain óľ¨2 óľ¨ đ¸óľ¨óľ¨óľ¨đŚ (đđĄ) â đ§1 (đĄ)óľ¨óľ¨óľ¨ ⤠48đž2 â2 (1 + 2đś1 ) + 24đž2 â (1 + 2đś1 ) + 48đž2 đâ (1 + 3đś1 ) + 24đž2 đ2 â2 (1 + 3đś1 ) ⤠đś5 â, (27)
The Scientific World Journal
5
where đś5 = 72đž2 (1 + 2đś1 ) + 24đž2 đ(2 + đ)(1 + 3đś1 ). In the following we consider đ¸|đŚ(đđĄ) â đ§2 (đĄ)|2 : óľ¨2 óľ¨ đ¸óľ¨óľ¨óľ¨đŚ (đđĄ) â đ§2 (đĄ)óľ¨óľ¨óľ¨ óľ¨2 óľ¨ = đ¸óľ¨óľ¨óľ¨óľ¨đŚ (đđĄ) â đŚ[đ(đ+1)] óľ¨óľ¨óľ¨óľ¨ óľ¨2 óľ¨2 óľ¨ óľ¨ â¤ 2đ¸óľ¨óľ¨óľ¨óľ¨đŚ (đđĄ) â đŚ[đđ] óľ¨óľ¨óľ¨óľ¨ + 2đ¸óľ¨óľ¨óľ¨óľ¨đŚ[đđ] â đŚ[đ(đ+1)] óľ¨óľ¨óľ¨óľ¨ ⤠4đś5 â.
(28)
óľ¨óľ¨ đĄ óľ¨ + 4đ¸ ( sup óľ¨óľ¨óľ¨âŤ [đ (đ§1 (đ ) , đ§1 (đ )) óľ¨ đĄâ[0,đĄ1 ] óľ¨ 0 óľ¨óľ¨2 óľ¨ âđ (đĽ (đ â ) , đĽ (đđ â ))] đđ¤ (đ )óľ¨óľ¨óľ¨ ) óľ¨óľ¨ óľ¨óľ¨ đĄ óľ¨ + 4đ¸ ( sup óľ¨óľ¨óľ¨âŤ [â (đ§1 (đ ) , đ§1 (đ ) , đž (đ )) óľ¨ đĄâ[0,đĄ1 ] óľ¨ 0 óľ¨óľ¨2 óľ¨ ââ (đ§1 (đ ) , đ§1 (đ ) , đž (đ â ))] đđ (đ )óľ¨óľ¨óľ¨ ) óľ¨óľ¨
Let đś6 = 4đś5 ; the proof is complete.
4. Main Results We can now state and prove our main result of this paper.
óľ¨óľ¨ đĄ óľ¨ + 4đ¸ ( sup óľ¨óľ¨óľ¨âŤ [â (đ§1 (đ ) , đ§1 (đ ) , đž (đ â )) óľ¨ đĄâ[0,đĄ1 ] óľ¨ 0 óľ¨óľ¨2 óľ¨ ââ (đĽ (đ â ), đĽ (đđ â ), đž (đ â ))] đđ (đ )óľ¨óľ¨óľ¨ ) . óľ¨óľ¨
Theorem 6. Under Assumption 1 for some đ > 1 and Assumptions 1â2, there exists ââ > 0 and đś = đś(đ) such that, for all 0 < â < ââ ,
(31)
óľ¨2 óľ¨ đ¸ [ sup óľ¨óľ¨óľ¨đŚ (đĄ) â đĽ (đĄ)óľ¨óľ¨óľ¨ ] ⤠đśâ1â(1/đ) .
(29)
đĄâ[0,đ]
Proof. The analysis uses ideas from [15], where analogous results are derived in the stochastic differential equations. By construction, we have
By Assumption 1 and H¨older inequality, we have óľ¨óľ¨ đĄ óľ¨ đ¸ ( sup óľ¨óľ¨óľ¨âŤ ((1 â đ) [đ (đ§1 (đ ) , đ§1 (đ ))âđ (đĽ (đ â ) , đĽ (đđ â ))] óľ¨ đĄâ[0,đĄ1 ]óľ¨ 0 + đ [đ (đ§2 (đ ) , đ§2 (đ ))
đŚ (đĄ) â đĽ (đĄ) đĄ
â
óľ¨óľ¨2 óľ¨ âđ (đĽ (đ â ) , đĽ (đđ â ))]) đđ óľ¨óľ¨óľ¨ ) óľ¨óľ¨
â
= ⍠(1 â đ) [đ (đ§1 (đ ) , đ§1 (đ )) â đ (đĽ (đ ) , đĽ (đđ ))] đđ 0
đĄ
đĄ
+ ⍠đ [đ (đ§2 (đ ) , đ§2 (đ )) â đ (đĽ (đ â ) , đĽ (đđ â ))] đđ 0 đĄ
+ ⍠[đ (đ§1 (đ ) , đ§1 (đ )) â đ (đĽ (đ â ) , đĽ (đđ â ))] đđ¤ (đ )
óľ¨2 âđ (đĽ (đ â ) , đĽ (đđ â ))óľ¨óľ¨óľ¨ óľ¨ + óľ¨óľ¨óľ¨đ (đ§2 (đ ) , đ§2 (đ ))
0 đĄ
+ ⍠[â (đ§1 (đ ) , đ§1 (đ ) , đž (đ )) 0
óľ¨2 âđ(đĽ (đ â ) , đĽ(đđ â ))óľ¨óľ¨óľ¨ ) đđ )
ââ (đ§1 (đ ) , đ§1 (đ ) , đž (đ â ))] đđ (đ )
đĄ
+ ⍠[â (đ§1 (đ ) , đ§1 (đ ) , đž (đ â )) 0
đĄ
1 óľ¨ â¤ 2đ¸ ( sup ⍠12 đđ ⍠(óľ¨óľ¨óľ¨đ (đ§1 (đ ) , đ§1 (đ )) 0 đĄâ[0,đĄ1 ] 0
đĄ
1 óľ¨ óľ¨2 óľ¨ óľ¨2 ⤠2đđž1 ⍠(đ¸óľ¨óľ¨óľ¨đ§1 (đ ) â đĽ (đ â )óľ¨óľ¨óľ¨ + đ¸óľ¨óľ¨óľ¨đ§1 (đ ) â đĽ (đđ â )óľ¨óľ¨óľ¨ 0
ââ (đĽ (đ â ) , đĽ (đđ â ) , đž (đ â ))] đđ (đ ) .
óľ¨ óľ¨2 óľ¨ óľ¨2 +đ¸óľ¨óľ¨óľ¨đ§2 (đ ) â đĽ (đ â )óľ¨óľ¨óľ¨ + đ¸óľ¨óľ¨óľ¨đ§2 (đ ) â đĽ (đđ â )óľ¨óľ¨óľ¨ ) đđ . (32)
(30) Now for any 0 ⤠đĄ1 ⤠đ we have
By Assumption 1, the Cauchy-Schwarz inequality, and the Doob inequality in the two martingale terms and the martingale isometry,
óľ¨2 óľ¨ đ¸ ( sup óľ¨óľ¨óľ¨đŚ (đĄ) â đĽ (đĄ)óľ¨óľ¨óľ¨ ) đĄâ[0,đĄ1 ] óľ¨óľ¨ đĄ óľ¨ = 4đ¸ ( sup óľ¨óľ¨óľ¨âŤ ((1 â đ) [đ (đ§1 (đ ) , đ§1 (đ )) óľ¨ đĄâ[0,đĄ1 ] óľ¨ 0 âđ (đĽ (đ â ) , đĽ (đđ â ))] + đ [đ (đ§2 (đ ) , đ§2 (đ )) óľ¨óľ¨2 óľ¨ âđ (đĽ (đ â ) , đĽ (đđ â ))]) đđ óľ¨óľ¨óľ¨ ) óľ¨óľ¨
óľ¨óľ¨ đĄ óľ¨óľ¨óľ¨2 óľ¨ đ¸ ( sup óľ¨óľ¨óľ¨âŤ [đ (đ§1 (đ ) , đ§1 (đ ))âđ (đĽ (đ â ) , đĽ (đđ â ))] đđ¤ (đ )óľ¨óľ¨óľ¨ ) óľ¨ óľ¨óľ¨ đĄâ[0,đĄ1 ]óľ¨ 0 đĄ
1 óľ¨ óľ¨2 ⤠4đ¸ ⍠óľ¨óľ¨óľ¨đ (đ§1 (đ ) , đ§1 (đ )) â đ (đĽ (đ â ) , đĽ (đđ â ))óľ¨óľ¨óľ¨ đđ 0
đĄ
1 óľ¨ óľ¨2 óľ¨ óľ¨2 ⤠4đž1 ⍠[đ¸óľ¨óľ¨óľ¨đ§1 (đ ) â đĽ (đ â )óľ¨óľ¨óľ¨ + đ¸óľ¨óľ¨óľ¨đ§1 (đ ) â đĽ (đđ â )óľ¨óľ¨óľ¨ ] đđ ,
0
(33)
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óľ¨óľ¨ đĄ óľ¨ đ¸ ( sup óľ¨óľ¨óľ¨âŤ [â (đ§1 (đ ) , đ§1 (đ ) , đž (đ â )) óľ¨ đĄâ[0,đĄ1 ] óľ¨ 0 óľ¨óľ¨2 óľ¨ ââ (đĽ (đ â ) , đĽ (đđ â ) , đž (đ â ))] đđ (đ )óľ¨óľ¨óľ¨ ) óľ¨óľ¨ óľ¨óľ¨ đĄ óľ¨ = đ¸ ( sup óľ¨óľ¨óľ¨âŤ [â (đ§1 (đ ) , đ§1 (đ ) , đž (đ â )) óľ¨ đĄâ[0,đĄ1 ] óľ¨ 0 Ě (đ ) ââ (đĽ (đ â ) , đĽ (đđ â ) , đž (đ â ))] đđ đĄ
We also have óľ¨óľ¨ đĄ óľ¨ đ¸ ( sup óľ¨óľ¨óľ¨âŤ [â (đ§1 (đ ) , đ§1 (đ ) , đž (đ )) óľ¨ đĄâ[0,đĄ1 ] óľ¨ 0 óľ¨óľ¨2 óľ¨ ââ (đ§1 (đ ) , đ§1 (đ ) , đž (đ â ))] đđ (đ ) óľ¨óľ¨óľ¨ ) óľ¨óľ¨ óľ¨óľ¨ đĄ óľ¨ = đ¸ ( sup óľ¨óľ¨óľ¨âŤ [â (đ§1 (đ ) , đ§1 (đ ) , đž (đ )) óľ¨ đĄâ[0,đĄ1 ] óľ¨ 0
+ đ ⍠[â (đ§1 (đ ) , đ§1 (đ ) , đž (đ â ))
Ě (đ ) ââ (đ§1 (đ ) , đ§1 (đ ) , đž (đ â ))] đđ
0
óľ¨óľ¨2 óľ¨ ââ (đĽ (đ â ) , đĽ (đđ â ) , đž (đ â ))] đđ óľ¨óľ¨óľ¨ ) óľ¨óľ¨
đĄ
+ đ ⍠[â (đ§1 (đ ) , đ§1 (đ ) , đž (đ )) 0
óľ¨óľ¨2 óľ¨ ââ (đ§1 (đ ) , đ§1 (đ ) , đž (đ â ))] đđ óľ¨óľ¨óľ¨ ) óľ¨óľ¨
óľ¨óľ¨ đĄ óľ¨ â¤ 2đ¸ ( sup óľ¨óľ¨óľ¨âŤ [â (đ§1 (đ ) , đ§1 (đ ) , đž (đ â )) óľ¨ đĄâ[0,đĄ1 ] óľ¨ 0 óľ¨óľ¨2 Ě (đ )óľ¨óľ¨óľ¨ ) ââ (đĽ (đ â ) , đĽ (đđ â ) , đž (đ â ))] đđ óľ¨óľ¨ óľ¨
óľ¨óľ¨ đĄ óľ¨ â¤ 2đ¸ ( sup óľ¨óľ¨óľ¨âŤ [â (đ§1 (đ ) , đ§1 (đ ) , đž (đ )) óľ¨ đĄâ[0,đĄ1 ] óľ¨ 0 óľ¨óľ¨2 Ě (đ ) óľ¨óľ¨óľ¨ ) ââ (đ§1 (đ ) , đ§1 (đ ) , đž (đ â ))] đđ óľ¨óľ¨ óľ¨
óľ¨óľ¨ đĄ óľ¨ + 2đ2 đ¸ ( sup óľ¨óľ¨óľ¨âŤ [â (đ§1 (đ ) , đ§1 (đ ) , đž (đ â )) óľ¨ đĄâ[0,đĄ1 ] óľ¨ 0 óľ¨óľ¨2 óľ¨ ââ (đĽ (đ â ) , đĽ (đđ â ) , đž (đ â ))] đđ óľ¨óľ¨óľ¨ ) óľ¨óľ¨
óľ¨óľ¨ đĄ óľ¨ + 2đ2 đ¸ ( sup óľ¨óľ¨óľ¨âŤ [â (đ§1 (đ ) , đ§1 (đ ) , đž (đ )) óľ¨ đĄâ[0,đĄ1 ] óľ¨ 0
óľ¨óľ¨ đĄ1 óľ¨ â¤ 8đ¸ óľ¨óľ¨óľ¨âŤ [â (đ§1 (đ ) , đ§1 (đ ) , đž (đ â )) óľ¨óľ¨ 0 óľ¨óľ¨2 Ě (đ )óľ¨óľ¨óľ¨ ââ (đĽ (đ â ) , đĽ (đđ â ) , đž (đ â ))] đđ óľ¨óľ¨ óľ¨ đĄ1 óľ¨óľ¨ 2 â + 2đ đđ¸ (⍠óľ¨óľ¨â (đ§1 (đ ) , đ§1 (đ ) , đž (đ )) 0 óľ¨2 ââ (đĽ (đ â ) , đĽ (đđ â ) , đž (đ â ))óľ¨óľ¨óľ¨ đđ ) đĄ1
óľ¨ â¤ 8đđ¸ (⍠óľ¨óľ¨óľ¨â (đ§1 (đ ) , đ§1 (đ ) , đž (đ â )) 0
óľ¨óľ¨2 óľ¨ ââ (đ§1 (đ ) , đ§1 (đ ) , đž (đ â ))] đđ óľ¨óľ¨óľ¨ ) óľ¨óľ¨ óľ¨óľ¨ đĄ1 óľ¨ â¤ 8đ¸ óľ¨óľ¨óľ¨âŤ [â (đ§1 (đ ) , đ§1 (đ ) , đž (đ )) óľ¨óľ¨ 0 óľ¨óľ¨2 Ě (đ ) óľ¨óľ¨óľ¨ ââ (đ§1 (đ ) , đ§1 (đ ) , đž (đ â ))] đđ óľ¨óľ¨ óľ¨ đĄ
1 óľ¨ + 2đ2 đđ¸ (⍠óľ¨óľ¨óľ¨â (đ§1 (đ ) , đ§1 (đ ) , đž (đ )) 0
óľ¨2 ââ (đ§1 (đ ) , đ§1 (đ ) , đž (đ â ))óľ¨óľ¨óľ¨ đđ )
óľ¨2 ââ (đĽ (đ â ) , đĽ (đđ â ) , đž (đ â ))óľ¨óľ¨óľ¨ đđ ) đĄ
1 óľ¨ + 2đ2 đđ¸ (⍠óľ¨óľ¨óľ¨â (đ§1 (đ ) , đ§1 (đ ) , đž (đ â )) 0
óľ¨2 ââ (đĽ (đ â ) , đĽ (đđ â ) , đž (đ â ))óľ¨óľ¨óľ¨ đđ ) đĄ
1 óľ¨ óľ¨2 óľ¨ óľ¨2 ⤠8đđž1 ⍠(đ¸óľ¨óľ¨óľ¨đ§1 (đ ) â đĽ (đ â )óľ¨óľ¨óľ¨ + đ¸óľ¨óľ¨óľ¨đ§1 (đ ) â đĽ (đđ â )óľ¨óľ¨óľ¨ ) đđ
0
đĄ
1 óľ¨ óľ¨2 óľ¨ óľ¨2 + 2đ2 đđž1 ⍠(đ¸óľ¨óľ¨óľ¨đ§1 (đ )âđĽ (đ â )óľ¨óľ¨óľ¨ +đ¸óľ¨óľ¨óľ¨đ§1 (đ )âđĽ (đđ â )óľ¨óľ¨óľ¨ ) đđ . 0 (34)
đĄ
1 óľ¨ â¤ 8đđ¸ (⍠óľ¨óľ¨óľ¨â (đ§1 (đ ) , đ§1 (đ ) , đž (đ )) 0
óľ¨2 ââ (đ§1 (đ ) , đ§1 (đ ) , đž (đ â ))óľ¨óľ¨óľ¨ đđ ) đĄ
1 óľ¨ + 2đ2 đđ¸ (⍠óľ¨óľ¨óľ¨â (đ§1 (đ ) , đ§1 (đ ) , đž (đ )) 0
óľ¨2 ââ (đ§1 (đ ) , đ§1 (đ ) , đž (đ â ))óľ¨óľ¨óľ¨ đđ )
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đĄ
1 óľ¨2 óľ¨ â¤ 2đ (4 + đđ) đž1 đ¸ (⍠óľ¨óľ¨óľ¨đž (đ ) â đž (đ â )óľ¨óľ¨óľ¨ đđ ) 0
đó¸ â1
đĄđ+1
⤠2đ (4 + đđ) đž1 ( â đ¸ (âŤ
đĄđ
đ=0
â¤
óľ¨óľ¨ â óľ¨2 óľ¨óľ¨đž (đ ) â đž (đ )óľ¨óľ¨óľ¨ đđ )) , â¤
where đó¸ is the smallest integer such that đó¸ â ⼠đĄ1 . Now the number of nonzero terms in the summation in (34) is a random variable that is not independent of the summands. To obtain a useful bound, we recall the following Youngâs inequality: đ â 1 1/(đâ1) đ/(đâ1) 1 đ đ + đ, đ đ đđ
đ¸ (âŤ
đĄđ
= đ¸ (1{Îđđ âĽ1} âŤ
đĄđ
â¤
âđâ1 2đâ1 đĄđ+1 óľ¨óľ¨ óľ¨2đ óľ¨ óľ¨2đ 2 ⍠(đ¸óľ¨óľ¨đž (đ )óľ¨óľ¨óľ¨ + đ¸óľ¨óľ¨óľ¨đž (đ â )óľ¨óľ¨óľ¨ ) đđ đđ đĄđ
+ â¤
đ â 1 1/(đâ1) âđ đľ 2đ đâ + đ 2 . đ đđ (40) 2
Choosing đ = â(đâ1) /đ , we have đ¸ (âŤ
đĄđ+1
đĄđ
óľ¨óľ¨ â óľ¨2 óľ¨óľ¨đž (đ ) â đž (đ )óľ¨óľ¨óľ¨ đđ ) đĄđ+1
đ â 1 1/(đâ1) đâ đ đ
(36)
where đ, đ, đ > 0 and 1 < đ < â. Hence, đĄđ+1
âđâ1 đĄđ+1 óľ¨óľ¨ óľ¨2đ ⍠đ¸óľ¨óľ¨đž (đ ) â đž (đ â )óľ¨óľ¨óľ¨ đđ đđ đĄđ
+ (35)
đđ â¤
đ â 1 1/(đâ1) đ (Îđđ ⼠1) đ đ
1 óľ¨óľ¨ â óľ¨2 2đ 2â1/đ . óľ¨óľ¨đž (đ ) â đž (đ )óľ¨óľ¨óľ¨ đđ ) ⤠((đ â 1) đ + 2 đľ) â đ (41)
Substituting (40) into (34) yields óľ¨óľ¨ đĄ óľ¨ đ¸ ( sup óľ¨óľ¨óľ¨âŤ [â (đ§1 (đ ) , đ§1 (đ ) , đž (đ )) óľ¨ đĄâ[0,đĄ1 ] óľ¨ 0
óľ¨óľ¨ â óľ¨2 óľ¨óľ¨đž (đ ) â đž (đ )óľ¨óľ¨óľ¨ đđ ) (37)
đ â 1 1/(đâ1) đ¸ (1{Îđđ âĽ1} ) đ đ
óľ¨óľ¨2 óľ¨ ââ (đ§1 (đ ) , đ§1 (đ ) , đž (đ â ))] đđ (đ )óľ¨óľ¨óľ¨ ) óľ¨óľ¨
đ
+
đĄđ+1 1 óľ¨ óľ¨2 đ¸(⍠óľ¨óľ¨óľ¨đž (đ ) â đž (đ â )óľ¨óľ¨óľ¨ đđ ) . đđ đĄđ
đâ1
1 ((đ â 1) đ + 22đ đľ) â2â1/đ đ đ=0
⤠2đ (4 + đđ) đž1 â
Now we can apply the H¨older inequality as follows: đ¸(âŤ
đĄđ+1
đĄđ
đ
óľ¨óľ¨ â óľ¨2 óľ¨óľ¨đž (đ ) â đž (đ )óľ¨óľ¨óľ¨ đđ )
⤠đ¸ [(âŤ
đĄđ+1
đĄđ
= âđâ1 đ¸ [âŤ
đâ1
đđ )
đĄđ+1
đĄđ
đĄđ+1
âŤ
đĄđ
óľ¨óľ¨ â óľ¨2đ óľ¨óľ¨đž (đ ) â đž (đ )óľ¨óľ¨óľ¨ đđ ]
(38)
đĄđ+1
đĄđ
â¤
óľ¨óľ¨ â óľ¨2 óľ¨óľ¨đž (đ ) â đž (đ )óľ¨óľ¨óľ¨ đđ ) đ â 1 1/(đâ1) đ¸ (1{Îđđ âĽ1} ) đ đ +
đâ1
đĄđ+1
â đ¸ (⍠đđ đĄđ
óľ¨óľ¨ â óľ¨2đ óľ¨óľ¨đž (đ ) â đž (đ )óľ¨óľ¨óľ¨ đđ )
2đ (4 + đđ) đđž1 ((đ â 1) đ + 22đ đľ) â1â1/đ đ
óľ¨2 óľ¨ đ¸ ( sup óľ¨óľ¨óľ¨đŚ (đĄ) â đĽ (đĄ)óľ¨óľ¨óľ¨ ) đĄâ[0,đĄ1 ] (39)
which follows from the H¨older inequality and (2); we yield đ¸ (âŤ
â¤
as đâ ⤠đ. Now, substituting (31), (32), (33), and (41) into (30) yields
Using (37) in (35), we have
óľ¨2đ óľ¨ óľ¨ óľ¨2đ ⤠22đâ1 (đ¸ [óľ¨óľ¨óľ¨đž (đ )óľ¨óľ¨óľ¨ ] + đ¸ [óľ¨óľ¨óľ¨đž (đ â )óľ¨óľ¨óľ¨ ]) ,
2đ (4 + đđ) đž1 ((đ â 1) đ + 22đ đľ) đâ2â1/đ đ
(42)
óľ¨óľ¨ â óľ¨2đ óľ¨óľ¨đž (đ ) â đž (đ )óľ¨óľ¨óľ¨ đđ ] .
óľ¨2đ óľ¨ đ¸ [óľ¨óľ¨óľ¨đž (đ ) â đž (đ â )óľ¨óľ¨óľ¨ đđ ]
â¤
â¤
8đđđž1 (4 + đđ) ((đ â 1) đ + 22đ đľ) â1â1/đ đ đĄ
1 óľ¨ óľ¨2 + 8đž1 (đ + 2 + 4đ + đ2 đ) ⍠đ¸óľ¨óľ¨óľ¨đ§1 (đ ) â đĽ (đ â )óľ¨óľ¨óľ¨ đđ 0
đĄ
1 óľ¨ óľ¨2 + 8đž1 (đ + 2 + 4đ + đ2 đ) ⍠đ¸óľ¨óľ¨óľ¨đ§1 (đ ) â đĽ (đđ â )óľ¨óľ¨óľ¨ đđ 0
đĄ
1 óľ¨ óľ¨2 + 8đđž1 ⍠đ¸óľ¨óľ¨óľ¨đ§2 (đ ) â đĽ (đ â )óľ¨óľ¨óľ¨ đđ
0
đĄ
1 óľ¨ óľ¨2 + 8đđž1 ⍠đ¸óľ¨óľ¨óľ¨đ§2 (đ ) â đĽ (đđ â )óľ¨óľ¨óľ¨ đđ
0
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8đđđž1 (4 + đđ) ((đ â 1) đ + 22đ đľ) â1â1/đ đ
This problem class is now widely used in mathematical finance. By Theorem 6, we obtain the following corollaries.
+ 16đž1 (đ + 2 + 4đ + đ2 đ)
Corollary 8. Under Assumption 1,
đĄ1
óľ¨2 óľ¨ óľ¨ óľ¨2 à ⍠(đ¸óľ¨óľ¨óľ¨đ§1 (đ ) â đŚ (đ )óľ¨óľ¨óľ¨ + đ¸óľ¨óľ¨óľ¨đŚ (đ ) â đĽ (đ â )óľ¨óľ¨óľ¨ ) đđ
óľ¨2 óľ¨ lim đ¸ [ sup óľ¨óľ¨óľ¨đŚ (đĄ) â đĽ (đĄ)óľ¨óľ¨óľ¨ ] = 0.
0
ââ0
+ 16đž1 (đ + 2 + 4đ + đ2 đ) đĄ1
óľ¨2 óľ¨ óľ¨ óľ¨2 à ⍠(đ¸óľ¨óľ¨óľ¨đ§1 (đ ) â đŚ (đ â đ)óľ¨óľ¨óľ¨ + đ¸óľ¨óľ¨óľ¨đŚ (đ â đ) â đĽ (đđ â )óľ¨óľ¨óľ¨ ) đđ 0 đĄ
1 óľ¨2 óľ¨ óľ¨ óľ¨2 + 16đđž1 ⍠(đ¸óľ¨óľ¨óľ¨đ§2 (đ ) â đŚ (đ )óľ¨óľ¨óľ¨ + đ¸óľ¨óľ¨óľ¨đŚ (đ ) â đĽ (đ â )óľ¨óľ¨óľ¨ ) đđ 0
Corollary 9. Under the local Lipschitz condition and Assumption 2,
đĄ1
óľ¨2 óľ¨ lim đ¸ [ sup óľ¨óľ¨óľ¨đŚ (đĄ) â đĽ (đĄ)óľ¨óľ¨óľ¨ ] = 0.
ââ0
óľ¨ óľ¨2 +đ¸óľ¨óľ¨óľ¨đŚ (đ â đ) â đĽ (đđ â )óľ¨óľ¨óľ¨ ) đđ . (43) From Lemmas 4 and 5, we have óľ¨2 óľ¨ đ¸ ( sup óľ¨óľ¨óľ¨đŚ (đĄ) â đĽ (đĄ)óľ¨óľ¨óľ¨ ) đĄâ[0,đĄ1 ] 8đđđž1 (4 + đđ) ((đ â 1) đ + 22đ đľ) â1â1/đ đ
(46)
The convergent result can be extended to the case of nonlinear coefficients that are local Lipschitz [6, 7, 12] based on the style of analysis in [22].
óľ¨2 óľ¨ + 16đđž1 ⍠(đ¸óľ¨óľ¨óľ¨đ§2 (đ ) â đŚ (đ â đ)óľ¨óľ¨óľ¨ 0
â¤
đĄâ[0,đ]
đĄâ[0,đ]
(47)
Remark 10. Corollary 9 shows that the numerical solution converges to the true solution. However, the order of the convergence of the numerical method is not given under the local Lipschitz condition. If we remove jump and discuss the system without time lag, our results are reduced to the results derived in [6, 14]. In other words, our results are the generalization of paper [6, 14].
Conflict of Interests
+ 16đž1 đś3 (đ + 2 + 4đ + đ2 đ) â
The authors declare that there is no conflict of interests regarding the publication of this paper.
+ 16đž1 đś5 (đ + 2 + 4đ + đ2 đ) â
Acknowledgments
+ 16đđž1 đś4 â + 16đđž1 đś6 â + 32đž1 (2đ + 2 + 4đ + đ2 đ)
The work is supported by the Fundamental Research Funds for the Central Universities, the National Natural Science Foundation of China under Grant 61304067, 11271146 and 61304175 the Natural Science Foundation of Hubei Province of China under Grant 2013CFB443.
đĄ
1 óľ¨ óľ¨2 à ⍠đ¸ sup óľ¨óľ¨óľ¨đŚ (đĄ) â đĽ (đĄâ )óľ¨óľ¨óľ¨ đđ
0
đĄâ[0,đ ]
⤠32đž1 (2đ + 2 + 4đ + đ2 đ)
References
đĄ1
óľ¨ óľ¨2 à ⍠đ¸ sup óľ¨óľ¨óľ¨đŚ (đĄ) â đĽ (đĄâ )óľ¨óľ¨óľ¨ đđ + đˇ1 â1â1/đ , 0 đĄâ[0,đ ]
(44) where đˇ1 := (8đđđž1 /đ)(4+đđ)((đâ1)đ+22đ đľ)+16đž1 ((đś3 + đś5 )(đ + 2 + 4đ + đ2 đ) + đđś4 + đđś6 ). By the Gronwall inequality, we have 2 óľ¨2 óľ¨ đ¸ ( sup óľ¨óľ¨óľ¨đŚ (đĄ) â đĽ (đĄ)óľ¨óľ¨óľ¨ ) ⤠đˇ1 â1â1/đ đ32đž1 (2đ+2+4đ+đ đ) .
đĄâ[0,đ]
(45) The proof is complete. Remark 7. Theorem 6 shows that the order of convergence in mean square is close to 1. Moreover, stochastic đ-methods give strong convergence rate arbitrarily close to order 1/2 under appropriate moment bounds on the jump magnitude.
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