We are concerned with the stochastic differential delay equations with Poisson jump and Markovian switching (SDDEsPJMSs). Most SDDEsPJMSs cannot be solved explicitly as stochastic differential equations. Therefore, numerical solutions have become an important issue in the study of SDDEsPJMSs. The key contribution of this paper is to investigate the strong convergence between the true solutions and the numerical solutions to SDDEsPJMSs when the drift and diffusion coefficients are Taylor approximations.

1. Introduction

Recently there has been an increasing interest in the study of stochastic differential delay equations with Poisson jump and Markovian switching (SDDEsPJMSs). Svīshchuk and Kazmerchuk [1] investigated stability of stochastic differential delay equations with linear Poisson jumps and Markovian switchings. While Luo [2] discussed the comparison principle and several kinds of stability of Itô stochastic differential delay equations with Poisson jump and Markovian switching. Besides, Li and Chang [3] discussed the convergence of the numerical solutions of stochastic differential delay equations with Poisson jump and Markovian switching. In the present paper we will further research this topic and our focus is on the convergence of numerical solution to stochastic differential delay equation with Poisson jump and Markovian switching when the coefficients are Taylor approximations.

Stochastic differential delay equation with Poisson jump and Markovian switching may be considered as extension of stochastic differential delay equation with Poisson jump. Of course, it may also be regarded as an generalization of stochastic differential delay equation with Markovian switching. Similar to stochastic differential delay equations with Poisson jump, explicit solutions can hardly be obtained for the stochastic differential delay equations with Poisson jump and Markovian switching. Thus appropriate numerical approximation schemes such as the Euler (or Euler-Maruyama) are needed if we apply them in practice or to study their properties. There is an extensive literature concerning the approximate schemes for either stochastic differential delay equations with Poisson jump or stochastic differential delay equations with Markovian switching [4–12].

However, the rate of convergence to the true solution by the numerical solution is different for different numerical schemes [13]. Recently, Janković and Ilić [14] have investigated the rate of convergence between the true solution and numerical solution of the stochastic differential equations in the sense of the L^p-norm when the drift and diffusion coefficients are Taylor approximations, up to arbitrary fixed derivatives. Moreover, Jiang et al. [15] generalized the Taylor method to stochastic differential delay equations with Poisson jump. Since the rate of convergence for such a numerical method is faster than the result obtained in [13], in this paper, we intend to generalize the above method to the SDDEsPJMSs case and consider the strong convergence between the true solution and numerical; solution to SDDEsPJMSs if the drift and diffusion coefficients are Taylor approximations, up to arbitrary fixed derivatives. To the best of our knowledge, so far there seem to be no existing results. Therefore, the aim of this paper is to close this gap.

In Section 2, we introduce necessary notations and approximation scheme. Then comes our main result that the Taylor approximate solutions will converge to the true solutions of SDDEsPJMSs. The proof of this main result is rather technical so we present several lemmas in Section 3 and then complete the proof in Section 4.

2. Approximation Scheme and Hypotheses

Throughout this paper, we let (Ω, ℱ, {ℱ_t} _t≥0, P) be a complete probability space with a filtration {ℱ_t} _t≥0 satisfying the usual conditions (i.e., it is increasing and right-continuous while ℱ₀ contains all P-null sets). Let C([−τ, 0]; R) be the family of continuous function ξ from [−τ, 0] to R with the norm ∥ξ∥ = sup _{−τ≤t≤0} | ξ(t)|. We also denote by the family of all bounded, ℱ₀-measurable, C([−τ, 0]; R)-valued random variables and denote by the family of all ℱ_t-measurable, C([−τ, 0]; R)-valued random variables ξ = {ξ(t):−τ ≤ t ≤ 0} satisfying sup _{−τ≤t≤0}E | ξ(t)|^p < κ < ∞, where κ is a positive constant.

Let w(t), t ≥ 0, be a one-dimensional Brownian motion defined on the probability space and let N(t), t ≥ 0, be a scalar Poisson process with intensity λ which is independent of w(t). Let r(t), t ≥ 0, be a right-continuous Markov chain on the probability space taking values in a finite state space S = {1,2, …, N} with the generator Γ = (γ_ij) _N×N given by

()

where δ > 0. Here γ_ij ≥ 0 is the transition rate from i to j if i ≠ j while γ_ii = −∑_i≠jγ_ij. We assume that the Markov chain r(·) is independent of the Brownian motion w(·) and Poisson process N(t). It is well known that almost every sample path of r(·) is a right-continuous step function with finite number of simple jumps in any finite subinterval of ℜ₊ = [0, ∞).

Consider stochastic differential delay equations with Poisson jump and Markovian switching of the form

()

on the time interval [0, T] with initial data

being independent of w(t) and N(t), and r(0) = i₀ ∈ S, where

()

In the paper, , and denote, respectively, jth-order partial derivatives of f, g and h with respect to x. Further, to ensure the existence and uniqueness of the solution to (2.2), we impose the following hypotheses.

(H₁) f, g, and h satisfy Lipschitz and linear growth condition; that is, there exists a positive constant L > 0 such that

()

for x₁, x₂, y₁, y₂ ∈ R and i ∈ S.

(H₂) There exist constants K₁ > 0 and γ ∈ (0,1] such that, for all −τ ≤ s < t ≤ 0 and ρ ≥ 2,

()

(H₃) f, g, and h have Taylor approximations in the second argument, up to m₁th, m₂th, and m₃th derivatives, respectively.

(H₄) Partial derivatives of the order m₁ + 1, m₂ + 1, and m₃ + 1 of the functions f, g, and h,

, and

, are uniformly bounded; that is, there exist positive constants L₁, L₂, and L₃ obeying

()

For some sufficiently large integer M, we define the time step by h = τ/M, where 0 < h ≪ 1. Then, the approximation solution to (2.2) is computed by y(t) = ξ(t) on −τ ≤ t ≤ 0, and for any t ≥ 0,

()

where, for t_k = kh with integer k ≥ 0,

()

In order to show the strong convergence of the numerical solutions and the exact solutions to (2.2), we will need the following assumption.

(H₅) There exists a positive constant K₂ such that

()

for any p > 0, where m = max {m₁, m₂, m₃}.

Remark 2.1. (H₁) shows that the exact solutions to (2.2) admit finite moments; see [15]. If m₁ = m₂ = m₃ = 0, then (H₁) shows also that the numerical solutions and the exact solutions admit finite moments (see, [1, 3]).

We can now state our main result of this paper.

Theorem 2.2. Under assumptions(H₁)–(H₅), for any p ≥ 2, then

()

The proof of this theorem is rather technical. We will present a number of useful lemmas in Section 3 and then complete the proof in Section 4.

3. Lemmas

Throughout our analysis, C_i, i = 1,2, … denote generic constants, independent of h. In order to prove the main theorem, the following lemmas are useful.

Lemma 3.1. If assumptions(H₁), (H₃), (H₄), and (H₅)hold, then, for2 ≤ ρ ≤ (m + 1)p,

()

where C is a positive constant independent of h.

Proof. For notation simplicity reason, let us denote that

()

Obviously, for any t ≥ 0, there exists an integer k ≥ 0 such that t ∈ [t_k, t_k+1]. Then, by (2.7) and (2.8) we obtain

()

Hence, we have

()

By the Hölder inequality, we have

()

By the Burkholder-Davis-Gundy inequality and the Hölder inequality, we yield, for some positive constant C_ρ,

()

For the jump integral, we convert to the compensated Poisson process

, which is a martingale with

()

see, for example, [1, 4, 12, 16]. By the Burkholder-Davis-Gundy inequality and the Hölder inequality, for some positive constant C_λ,ρ, we then obtain

()

Now, by the mean value theorem there is a θ ∈ (0,1) such that

()

This, together with (H₁), (H₃)–(H₅), yields that

()

Similarly, we can also show that there are two positive constant C₂ and C₃ for which

()

Next, since the boundedness of J₁(t), J₂(t), and J₃(t), we then have some positive constant C, independent of h, such that

()

The desired assertion is complete.

Lemma 3.2. If assumptions (H₁)–(H₅) hold, then, for 2 ≤ ρ ≤ (m + 1)p,

()

where γ ∈ (0,1] and

is a positive constant independent of h.

Proof. Clearly, for any t ≥ 0, there exists an integer k ≥ 0 such that t ∈ [t_k, t_k+1). In what follows, we split the following three cases to complete the proof.

Case 1. If −τ ≤ t_k − τ ≤ t − τ ≤ 0, we then have, from (H₂),

()

Case 2. If 0 ≤ t_k − τ ≤ t − τ, then, by Lemma 3.1, we have

()

Case 3. If −τ ≤ t_k − τ ≤ 0 ≤ t − τ, note that

()

Then, by the above two cases, it follows easily that

()

Now, combining the three cases altogether, for 2 ≤ ρ ≤ (m + 1)P and γ ∈ (0,1],

()

where

is a positive constant independent of h. The proof is complete.

Lemma 3.3. If assumptions (H₁) and (H₅) hold, then, for p ≥ 2,

()

where

, and

are positive constants dependent on max _0≤i≤N (−γ_ii), but independent of h.

Proof. Let n = [T/h] be the integer part of T/h. Then

()

with t_n+1 being T. By (H₁) and (H₅), we derive

()

Now, by the Markov property [6], we have

()

So, (3.19) is complete. Similarly, we can show (3.20) and (3.21).

4. Proof of Theorem 2.2

Let us now begin to prove our main result Theorem 2.2. Clearly,

()

So, for any t₁ ≤ T, by the Hölder inequality,

()

By the Burkholder-Davis-Gundy inequality, for some positive constant C_p, we have

()

By Doob’s martingale inequality and the compensated Poisson integral, for some positive constant C_λ,p,T, we yield

()

Now, by virtue of the assumption (H₁), we obtain that

()

Moreover, by (H₁) and Lemma 3.2, we have

()

and by Lemma 3.1 and (H₄), there exists a θ₁ ∈ (0,1) such that

()

Hence, by (4.6) and (4.7), together with Lemma 3.3, we yield

()

Similarly, we have

()

Now we use K to denote a generic positive constant that may change between occurrences. Hence, by (4.2)–(4.4) and (4.8)–(4.10), we yield that

()

The continuous Gronwall inequality then gives

()

This proof is therefore complete.

Remark 4.1. If m₁ = m₂ = m₃ = 0, then the approximate scheme (2.7) is reduced to Euler-Maruyama method for stochastic differential delay equations with Poisson Jump and Markovian switching, which has been discussed in [3].

Remark 4.2. As is well known, Taylor approximation is effectively applicable in engineering if the equations can be solved explicitly. If not, since polynomials are very useful analytic functions, the approximation in the paper can be useful in other applications of stochastic Taylor expansion, especially in the construction of various time discrete approximations of Itô processes by using Itô-Taylor expansion such as Euler-Maruyama approximation and Milstein approximation, which has order 1/2 and 1, respectively [3, 8, 13]. All these show that numerical methods based on Taylor expansions of higher degrees could be improved by combining them with analytic approximations presented in this paper.

Acknowledgments

The project reported here was supported by the Foundation of Wuhan Polytechnic University and Zhongnan University of Economics and Law for Zhenxing.

References

1 Svīshchuk A. V. and Kazmerchuk Y. I., Stability of stochastic Itô equations with delays, with jumps and Markov switchings, and their application in finance, Theory of Probability and Mathematical Statistics. (2002) no. 64, 141–151, 1922963.
Google Scholar
2 Luo J., Comparison principle and stability of Ito stochastic differential delay equations with Poisson jump and Markovian switching, Nonlinear Analysis: Theory, Methods & Applications. (2006) 64, no. 2, 253–262, https://doi.org/10.1016/j.na.2005.06.048, 2188457, ZBL1082.60054.
10.1016/j.na.2005.06.048
Web of Science® Google Scholar
3 Li R. and Chang Z., Convergence of numerical solution to stochastic delay differential equation with Poisson jump and Markovian switching, Applied Mathematics and Computation. (2007) 184, no. 2, 451–463, https://doi.org/10.1016/j.amc.2006.06.112, 2294860, ZBL1120.65003.
10.1016/j.amc.2006.06.112
Web of Science® Google Scholar
4 Higham D. J. and Kloeden P. E., Strong convergence rates for backward Euler on a class of nonlinear jump-diffusion problems, Journal of Computational and Applied Mathematics. (2007) 205, no. 2, 949–956, https://doi.org/10.1016/j.cam.2006.03.039, 2329668, ZBL1128.65007.
10.1016/j.cam.2006.03.039
Web of Science® Google Scholar
5 Bao J. and Hou Z., An analytic approximation of solutions of stochastic differential delay equations with Markovian switching, Mathematical and Computer Modelling. (2009) 50, no. 9-10, 1379–1384, https://doi.org/10.1016/j.mcm.2009.07.006, 2583427, ZBL1185.65009.
10.1016/j.mcm.2009.07.006
Google Scholar
6 Mao X. and Yuan C., Stochastic Differential Equations with Markovian Switching, 2006, Imperial College Press, London, UK, 2256095.
10.1142/p473
Google Scholar
7 Mao X., Yuan C., and Yin G., Approximations of Euler-Maruyama type for stochastic differential equations with Markovian switching, under non-Lipschitz conditions, Journal of Computational and Applied Mathematics. (2007) 205, no. 2, 936–948, https://doi.org/10.1016/j.cam.2006.01.052, 2329667, ZBL1121.65011.
10.1016/j.cam.2006.01.052
Web of Science® Google Scholar
8 Saito Y. and Mitsui T., Stability analysis of numerical schemes for stochastic differential equations, SIAM Journal on Numerical Analysis. (1996) 33, no. 6, 2254–2267, https://doi.org/10.1137/S0036142992228409, 1427462, ZBL0869.60052.
10.1137/S0036142992228409
Web of Science® Google Scholar
9 Li B. and Hou Y., Convergence and stability of numerical solutions to SDDEs with Markovian switching, Applied Mathematics and Computation. (2006) 175, no. 2, 1080–1091, https://doi.org/10.1016/j.amc.2005.08.026, 2220315, ZBL1095.65005.
10.1016/j.amc.2005.08.026
Web of Science® Google Scholar
10 Wang L.-S., Mei C., and Xue H., The semi-implicit Euler method for stochastic differential delay equations with jumps, Applied Mathematics and Computation. (2007) 192, no. 2, 567–578, https://doi.org/10.1016/j.amc.2007.03.027, 2385622.
10.1016/j.amc.2007.03.027
Web of Science® Google Scholar
11 Yuan C. and Glover W., Approximate solutions of stochastic differential delay equations with Markovian switching, Journal of Computational and Applied Mathematics. (2006) 194, no. 2, 207–226, https://doi.org/10.1016/j.cam.2005.07.004, 2239069, ZBL1098.65005.
10.1016/j.cam.2005.07.004
Web of Science® Google Scholar
12 Jiang F., Shen Y., and Hu J., Stability of the split-step backward Euler scheme for stochastic delay integro-differential equations with Markovian switching, Communications in Nonlinear Science and Numerical Simulation. (2011) 16, no. 2, 814–821, https://doi.org/10.1016/j.cnsns.2010.04.034, 2725805.
10.1016/j.cnsns.2010.04.034
Web of Science® Google Scholar
13 Kloeden P. E. and Platen E., Numerical Solution of Stochastic Differential Equations, 1992, 23, Springer, Berlin, Germany, Applications of Mathematics, 1214374.
10.1007/978-3-662-12616-5
Google Scholar
14 Janković S. and Ilić D., An analytic approximation of solutions of stochastic differential equations, Computers & Mathematics with Applications. (2004) 47, no. 6-7, 903–912, https://doi.org/10.1016/S0898%2D1221(04)90074%2D0, 2060324, ZBL1075.60067.
10.1016/S0898-1221(04)90074-0
Web of Science® Google Scholar
15 Jiang F., Shen Y., and Liu L., Taylor approximation of the solutions of stochastic differential delay equations with Poisson jump, Communications in Nonlinear Science and Numerical Simulation. (2011) 16, no. 2, 798–804, https://doi.org/10.1016/j.cnsns.2010.04.032, 2725803.
10.1016/j.cnsns.2010.04.032
Web of Science® Google Scholar
16 Higham D. J. and Kloeden P. E., Numerical methods for nonlinear stochastic differential equations with jumps, Numerische Mathematik. (2005) 101, no. 1, 101–119, https://doi.org/10.1007/s00211%2D005%2D0611%2D8, 2194720, ZBL1186.65010.
10.1007/s00211-005-0611-8
Web of Science® Google Scholar

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Analytic Approximation of the Solutions of Stochastic Differential Delay Equations with Poisson Jump and Markovian Switching

Abstract

1. Introduction

2. Approximation Scheme and Hypotheses

3. Lemmas

4. Proof of Theorem 2.2

Acknowledgments

References

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Analytic Approximation of the Solutions of Stochastic Differential Delay Equations with Poisson Jump and Markovian Switching

Abstract

1. Introduction

2. Approximation Scheme and Hypotheses

3. Lemmas

4. Proof of Theorem 2.2

Acknowledgments

References

Citing Literature

References

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