Implicit Numerical Solutions for Solving Stochastic Differential Equations with Jumps

and Applied Analysis 3 Assumption 1. The coefficient functionsf, g, and h satisfy the global Lipschitz condition 󵄨󵄨󵄨󵄨f (x1) − f (x2) 󵄨󵄨󵄨󵄨 + 󵄨󵄨󵄨󵄨g (x1) − g (x2) 󵄨󵄨󵄨󵄨 + 󵄨󵄨󵄨󵄨h (x1) − h (x2) 󵄨󵄨󵄨󵄨 + 󵄨󵄨󵄨󵄨 g (x 1 ) g 󸀠 (x 1 ) − g (x 2 ) g 󸀠 (x 2 ) 󵄨󵄨󵄨󵄨 + 󵄨󵄨󵄨󵄨 g (x 1 ) h 󸀠 (x 1 ) − g (x 2 ) h 󸀠 (x 2 ) 󵄨󵄨󵄨󵄨 ≤ L 󵄨󵄨󵄨󵄨x1 − x2 󵄨󵄨󵄨󵄨 (5) for a positive constant L and any x 1 , x 2 ∈ Rd and the linear growth condition 󵄨󵄨󵄨󵄨f(x) 󵄨󵄨󵄨󵄨 2 + 󵄨󵄨󵄨󵄨g(x) 󵄨󵄨󵄨󵄨 2 + |h(x)| 2 + 󵄨󵄨󵄨󵄨 g(x)g 󸀠 (x) 󵄨󵄨󵄨󵄨 2 + 󵄨󵄨󵄨󵄨 g(x)h 󸀠 (x) 󵄨󵄨󵄨󵄨 2 ≤ L 󸀠 (1 + |x| 2 ) (6) for a positive constant L󸀠 and any x ∈ R. Assumption 2. TheC 0 (⋅) andC 1 (⋅) are bounded d×d-matrixvalued functions. For any real numbers α 0 ∈ [0, α 0 ] and α 1 ∈ [−α 1 , α 1 ] with α 0 ≥ Δ and α 1 ≥ |(ΔW n ) 2 − Δ| for all step-size Δ and x ∈ R, the matrixM(x) = I + α 0 C 0 (x) + α 1 C 1 (x) is reversible and satisfies |(M(x))−1| ≤ B < ∞, where I is a unit matrix and B is a positive constant. In what follows, we will derive the strong convergence orders of the implicit Taylor methods for SDEJs (1). 3.1. Convergence of the θ-Taylor Method. Define x θ (t n+1 ) = x (t n ) + Δ [(1 − θ) f (x (t n )) + θf (x (t n+1 ))] + g (x (t n )) ΔW n


Introduction
Stochastic differential equations (SDEs) have been one of the most important mathematical tools for dealing with many problems in a variety of practical areas.However, SDEs are in general so complex that the analytical solutions can rarely be obtained.Thus, it is a common way to numerically solve SDEs.Since the explicit numerical methods often result in instability and inaccurate approximations to the solutions unless the step-size is very small, it is often necessary to use some implicit methods in numerically solving SDEs.
Generally speaking, there are two kinds of implicit numerical methods.One is the semi-implicit methods in which the drift components are computed implicitly while the diffusion components are computed explicitly.Higham [1,2] studied the stochastic -method for SDEs and SDEs with jumps (SDEJs).When  = 1, the stochastic -method is the backward Euler method.The backward Euler method is discussed in [3][4][5] and the references therein.Hu and Gan [6] proposed a class of drift-implicit one-step methods for neutral stochastic delay differential equations with jump diffusion.Higham and Kloeden [3,7] constructed the splitstep backward Euler method and the compensated splitstep backward Euler method for SDEJs.Ding et al. [8] introduced the split-step -method which is more general than the split-step backward Euler method.Wang and Gan [9] studied split-step one-leg  methods for SDEs.Buckwar and Sickenberger [10] compared the mean-square stability properties of the -Maruyama and -Milstein methods for SDEs.
The other is the fully implicit methods in which both the drift components and the diffusion components are computed implicitly.Since implicit stochastic terms in the implicit methods lead to infinite absolute moments of the numerical solution, extensive research has been done to address this issue [11][12][13][14][15][16][17][18][19][20][21][22][23][24][25][26].For example, Milstein et al. [11] proposed the balanced implicit method for the numerical solutions of SDEs.Burrage and Tian [12] suggested three implicit Taylor methods: the implicit Euler-Taylor method with strong order 0.5, the implicit Milstein-Taylor method with strong order 1.0, and the implicit Taylor method with strong order 1.5.Kahl and Schurz [16] introduced the balanced Milstein method for ordinary SDEs.Wang and Liu [20,21] proposed the semi-implicit Milstein method and the split-step backward balanced Milstein method for stiff stochastic systems.Furthermore, Haghighi and Hosseini [23] developed a class of general split-step balanced numerical methods for SDEs.
Using the idea of the balanced implicit method and combining it with the -Taylor method, we have the following balanced -Taylor method: where with  0 (⋅) and  1 (⋅) called control functions.

Convergence of the Implicit Taylor Methods
where the constants  1 and  2 are independent of Δ.
Proof.To obtain the convergence rate of the -Taylor method, we firstly introduce the local Taylor numerical approximation   +1 which is defined by Then, there exists some constant  1 > 0 such that Since we obtain On the other hand, since we have Therefore, the result ( 9) is obtained.
Theorem 4.Under Assumption 1, the -Taylor method (3) is convergent with order 1 in the mean square.That is, the global error satisfies where  3 is independent of Δ.
Proof.From the definitions of   and   , we have where Since   is F   -measurable, we have from Theorem 3 that where ⟨⋅, ⋅⟩ indicates the scalar product.

Convergence of the
by replacing the numerical approximations with the exact solution values on the right-hand side of (4).Then, the local error of method ( 4) is   ( +1 ) = ( +1 ) −   ( +1 ) and the global error of method ( 4) is   = (  ) −   .
Theorem 5.Under Assumptions 1 and 2, the balanced -Taylor method (4) is consistent with order 2 in the mean and with order 1.5 in the mean square.That is, the local mean error and mean-square error of the balanced -Taylor method (4) satisfy where the constants  4 and  5 are independent of Δ.
Proof.From Theorem 3, we have From the definitions of   +1 and   +1 in ( 7) and ( 26), we can write Abstract and Applied Analysis 7 Therefore, On the other hand, since we have ≤  (Δ 3 ) .
(33) Theorem 6.Under Assumptions 1 and 2, the balanced -Taylor method (4) is convergent with order 1 in the mean square.That is, the global error satisfies where  6 is independent of Δ.
Proof.From the definitions of   and   , we have where Thus, there exists a constant  5 such that where  9 =  8 ((  7  − 1)/ 7 ).

Stability of the Implicit Taylor Methods
In this section, we will discuss the stability properties of the numerical methods introduced in Section 2. Consider a scalar linear test equation, d () =  () d +  () d () +  () d () , where , , and  are real constants.The solution of (43) is where Let  = Δ,  =  √ Δ, and  = Δ.Then the MS-stability function of the Taylor method is Thus, the strong Taylor method (2) for the linear test equation ( 43) is MS-stable if  1 (, , , ) < 1.
Applying the -Taylor method (3) to the test equation (43), we obtain where Then the MS-stability function of the -Taylor method is

Numerical Examples
In this section, we conduct some simulation to demonstrate the convergence of the proposed implicit Taylor numerical solutions (3) and (4) for the equation system (43) with the coefficients  = −4,  = 1,  = −0.5 and the jump intensity  = 2.We compare the explicit solutions with the numerical approximations for the step-sizes Δ = 2 −1 , 2 −2 , . . ., 2 −8 .
To measure the accuracy and convergence property of the proposed methods, we compute mean of the absolute errors as In Table 1, we report the simulated errors of the -Taylor method and the balanced -Taylor method with  0 = 1 and  1 = 1 for different values of  and Δ.Note that the Taylor method is a special case of the -Taylor method with  = 0. From Table 1, we know that the accuracy of the -Taylor method with  = 1/2 and the balanced -Taylor method with  = 0 is higher than that of the Taylor method.The accuracy of the balanced -Taylor method with  = 0 is the highest for Δ ≤ 2 −2 .When  ≥ 1/2, the accuracy of the -Taylor method is higher than that of the balanced -Taylor method with  0 = 1 and  1 = 1.

Conclusions
In this paper, we introduce two kinds of the implicit methods, the -Taylor method and the balanced -Taylor method, for solving stochastic differential equations with Poisson jumps.It is proved that the proposed numerical methods have a strong convergence order of 1.0.Moreover, the MS-stable regions of the proposed numerical methods are derived for a linear scalar test equation and it is demonstrated that the -Taylor method and the balanced -Taylor method have better stable properties than the Taylor method.As has been confirmed by the theoretical and the numerical results, the proposed numerical methods perform satisfactorily in solving SDEJs.

Figure 1 :
Figure 1: MS-stable regions of the -Taylor methods and the balanced -Taylor method with  0 = − and  1 = 0.

Figure 2 :
Figure 2: MS-stable regions of the -Taylor methods and the balanced -Taylor method with  0 = − and  1 =  2 .

Table 1 :
Mean of the absolute errors for different values of Δ and different methods.