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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.

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 [

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 [

However, the rate of convergence to the true solution by the numerical solution is different for different numerical schemes [

In Section

Throughout this paper, we let

Let

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

In the paper,

(H_{1})

for

(H_{2}) There exist constants

(H_{3})

(H_{4}) Partial derivatives of the order

For some sufficiently large integer

In order to show the strong convergence of the numerical solutions and the exact solutions to (

(H_{5}) There exists a positive constant

We can now state our main result of this paper.

Under assumptions

The proof of this theorem is rather technical. We will present a number of useful lemmas in Section

Throughout our analysis,

If assumptions

For notation simplicity reason, let us denote that

If assumptions

Clearly, for any

If

If

If

If assumptions

Let

Let us now begin to prove our main result Theorem

Moreover, by

The continuous Gronwall inequality then gives

This proof is therefore complete.

If

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 [

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