The Truncated EM Method of Jump Diffusions with Markovian Switching: A Case Study of Music Signals

Abstract

This paper investigates the strong convergence of jump-diffusion processes with Markovian switching using the truncated Euler–Maruyama (TEM) method. Under the assumption that the drift and diffusion coefficients satisfy a Khasminskii-type condition and the jump coefficient meets a linear growth condition, we derive the convergence rate. Furthermore, we demonstrate that the TEM method effectively preserves both the mean square stability and the asymptotic boundedness of the underlying jump-diffusion process. A case study involving music signals is provided to illustrate the theoretical findings.

1. Introduction

Stochastic hybrid systems are widely employed to model systems subject to frequent and unpredictable structural changes, such as flexible manufacturing systems, air traffic management, power grids, mathematical finance, and risk management. A typical stochastic hybrid system is characterized by a hybrid state space: one component takes continuous values in a Euclidean space, while the other takes discrete values. Due to this structure, such systems have found diverse applications across numerous disciplines. For further details, we refer to [1,2,3,4].

An important class of stochastic hybrid systems is the following jump-diffusion model with Markovian switching:

$$\begin{array}{cc}\hfill X\left(t\right)=& x_0+{\int }_0^{t}b(X\left(s\right),\Lambda \left(s\right))\mathrm{d}s+{\int }_0^t\sigma (X\left(s\right),\Lambda \left(s\right))\mathrm{d}B\left(s\right)\hfill \\ \hfill & +{\int }_0^t{\int }_{{\mathbb{R}}^d\setminus \left\{0\right\}}c(X(s-),\Lambda (s-),u)N(\mathrm{d}s,\mathrm{d}u),\hfill \end{array}$$

(1)

where $\Lambda \left(t\right)$ is a continuous-time Markov chain representing the environmental state or system mode. Such processes are increasingly used to model stochastic systems subject to random impulses and frequent regime switches, typically captured by a Markov chain. A typical application of (1) arises in risk theory within insurance and finance, where $X\left(t\right)$ represents a surplus process in a Markov-modulated market. Applications have also been documented in a broad spectrum of fields, including option pricing [5] and flexible manufacturing systems [6].

Given that many problems involving diffusions with Markovian switching and Lévy jumps cannot be solved explicitly, obtaining approximate solutions—either in explicit form or in a form amenable to numerical computation—is of significant theoretical and practical importance. Although numerical methods for stochastic differential equations (SDEs) have been extensively studied (see, e.g., [7,8,9,10]), considerably less attention has been devoted to numerical schemes for diffusion or jump-diffusion processes with Markovian switching. In this context, Yuan and Mao [11] were the first to investigate numerical solutions for SDEs with Markovian switching. Subsequently, Ye and Li [12] studied numerical methods for a class of jump diffusions with Markovian switching, establishing the convergence of the Euler–Maruyama (EM) numerical solutions to the exact solutions and providing an error estimate. More recently, Yin, Song, and Zhang [13] proposed a numerical algorithm for such systems and proved its convergence via a martingale problem formulation.

In this paper, we propose a jump-adapted numerical scheme for system (1) based on the truncated Euler method. Unlike the methods in [13] and [12], which employ a constant step size, our algorithm incorporates a sequence of jump times generated by a Poisson random measure. Another key distinction concerns the mode of convergence: whereas [13] established weak convergence, we prove strong convergence of the scheme. Moreover, inspired by the techniques in [12], we also investigate the order of strong convergence.

The remainder of the paper is structured as follows: Section 2 briefly reviews the existence and uniqueness of solutions to Equation (1) and describes the modified Euler-type algorithm for numerical solutions. Section 3 establishes the $L^p(p\ge 2)$ convergence of the numerical solution under this algorithm and derives the corresponding error order. Section 4 provides a numerical example to evaluate the performance of the algorithm introduced in Section 3.

2. The Truncated EM Method

Let $(\Omega ,\mathcal{F},P)$ be a probability space. Let ${\left\{{\mathcal{F}}_t\right\}}_{t\ge 0}$ be an increasing family of sub-$\sigma$-algebras of $\mathcal{F}$ and $B\left(t\right)$ an ${\mathcal{F}}_t$-adapted ${\mathbb{R}}^d$-valued Brownian motion and also a martingale with respect to $\left\{{\mathcal{F}}_t\right\}$. Let $\left(X\right(t),\Lambda (t\left)\right)$ be a strong Markov process on ${\mathbb{R}}^d\times \mathbb{S}$ with left-hand limits and right-continuous, where $\mathbb{S}:=\{1,2,\dots ,m\},m<\infty$. The first component $X\left(t\right)$ satisfies the following stochastic differential–integral equation:

$$\begin{array}{cc}\hfill X\left(t\right)=& x_0+{\int }_0^{t}b(X\left(s\right),\Lambda \left(s\right))\mathrm{d}s+{\int }_0^t\sigma (X\left(s\right),\Lambda \left(s\right))\mathrm{d}B\left(s\right)\hfill \\ \hfill & +{\int }_0^t{\int }_{{\mathbb{R}}^d\setminus \left\{0\right\}}c(X(s-),\Lambda (s-),u)N(\mathrm{d}s,\mathrm{d}u),\hfill \end{array}$$

(2)

where $\sigma (x,i)$ is ${\mathbb{R}}^{d\times d}$-matrix valued, $b(x,i)$ and $c(x,i,u)$ are ${\mathbb{R}}^d$-valued for $x\in {\mathbb{R}}^d,u\in {\mathbb{R}}^d\setminus \left\{0\right\}$ and $i\in \mathbb{S}$. Let $N(\mathrm{d}t,\mathrm{d}u)$ (corresponding to a random point function $p\left(t\right)$) be a stationary ${\mathcal{F}}_t$-Poisson point process independent of $B\left(t\right)$, and let $\tilde{N}(\mathrm{d}t,\mathrm{d}u)=N(\mathrm{d}t,\mathrm{d}u)-\Pi \left(\mathrm{d}u\right)\mathrm{d}t$ be the compensated Poisson random measure on $[0,\infty )\times {\mathbb{R}}^d$, where $\Pi (·)$ is a deterministic characteristic measure on the measurable space $({\mathbb{R}}^d\setminus \left\{0\right\},\mathcal{B}({\mathbb{R}}^d\setminus \left\{0\right\}))$, and $\int 1\wedge u^2\Pi \left(\mathrm{d}u\right)<\infty .$ The second component $\Lambda \left(t\right)$, which is ${\mathcal{F}}_t$-adapted and independent of $B\left(t\right)$ and $N(t,A)$, is a right-continuous irreducible Markov chain with finite state space $\mathbb{S}$ and Q-matrix $Q={\left(q_{ij}\right)}_{i,j\in \mathbb{S}}$ such that

$$\begin{array}{c}\hfill P\{\Lambda (t+\Delta )=j\mid \Lambda \left(t\right)=i\}=\left\{\begin{array}{cc}{q}_{ij}\Delta +o(\Delta )\hfill & \mathrm{if}j\ne i,\hfill \\ 1+q_{ii}\Delta +o(\Delta )\hfill & \mathrm{if}j=i,\hfill \end{array}\right.\end{array}$$

(3)

provided that $\Delta \downarrow 0$ (cf. [14]). For $x\in {\mathbb{R}}^d$ and $\sigma =\left({\sigma }_{ij}\right)\in {\mathbb{R}}^{d\times d}$, define $\left|x\right|={\left({\sum }_{k=1}^d{\left|x_k\right|}^2\right)}^{1/2},\left|\sigma \right|={\left({\sum }_{i,j=1}^d{\left|{\sigma }_{ij}\right|}^2\right)}^{1/2}.$ For the existence and uniqueness of the strong Markov process $\left(X\right(t),\Lambda (t\left)\right)$ satisfying (2) and (3), we make the following assumptions.

Assumption 1. 

Assume that the coefficients b and σ satisfy the local Lipschitz condition: for any $R>0$ there exists a constant $L_R>0$ such that

$${|b(x,i)-b(y,i)|}^2{+|\sigma (x,i)-\sigma (y,i)|}^2+{\int }_{{\mathbb{R}}^d\setminus \left\{0\right\}}{|c(x,i,u)-c(y,i,u)|}^2\Pi \left(\mathrm{d}u\right)\le L_R{|x-y|}^2.$$

For all $(i,x,y,u)\in \mathbb{S}\times {\mathbb{R}}^d\times {\mathbb{R}}^d\times {\mathbb{R}}^d\setminus \left\{0\right\}$ with $\left|x\right|\vee \left|y\right|\le R.$

Assumption 2. 

(Khasminskii-type condition) There exist constant $L_2>0,p\ge 2$ such that

$$\begin{array}{c}\hfill x^{T}b(x,i)+{\displaystyle \frac{p-1}{2}}{\left|\sigma (x,i)\right|}^2\le L_2{(1+|x|}^2),\forall x\in {\mathbb{R}}^d.\end{array}$$

(4)

We state a known result (see, e.g., [15]) as a lemma for later use.

Lemma 1. 

Under Assumptions 1 and 2, SDEs (2) and (3) have a unique global solution $X\left(t\right)$. Moreover, for any $p\ge 2$

$$\underset{0\le t\le T}{sup}\mathbb{E}{\left|X\left(t\right)\right|}^p<\infty ,\forall T>0.$$

In the remainder of this section, we will show that the strong Markov process $\left(X\right(t),\Lambda (t\left)\right)$ can be associated with an appropriate generator. In what follows, let $(·,·)$ and ∇ denote the inner product and the gradient operator in ${\mathbb{R}}^d$, respectively. If A is a vector or matrix, $A^T$ denotes its transpose. For $x\in {\mathbb{R}}^d,i\in \mathbb{S}$, set $a(x,i)=\sigma (x,i)\sigma {(x,i)}^T$. Let $C_c^2({\mathbb{R}}^d\times \mathbb{S})$ denote the family of all functions $V(x,i)$ on ${\mathbb{R}}^d\times \mathbb{S}$ that are twice continuously differentiable in x and have compact support. For $V(x,i)\in C_c^2({\mathbb{R}}^d\times \mathbb{S})$, define an operator $\mathcal{A}$ as follows:

$$\begin{array}{c}\hfill \mathcal{A}V(x,i)=LV(x,i)+\tilde{\Omega }V(x,i)+QV(x,i).\end{array}$$

(5)

Here the operators L, $\tilde{\Omega }$ and Q are further defined by:

$$\begin{array}{c}\hfill LV(x,i)={\displaystyle \frac{1}{2}}\sum _{k,l=1}^d{a}_{kl}(x,i){\displaystyle \frac{{\partial }^2}{\partial x_k\partial x_l}}V(x,i)+\langle b(x,i),{\nabla }_{x}V(x,i)\rangle ,\end{array}$$

$$\begin{array}{c}\hfill \tilde{\Omega }V(x,i)={\int }_{{\mathbb{R}}^d\setminus \left\{0\right\}}[V(x+c(x,i,u),i)-V(x,i)]\Pi \left(\mathrm{d}u\right),\end{array}$$

$$\begin{array}{c}\hfill QV(x,i)=\sum _{j\ne i}{q}_{ij}(V(x,j)-V(x,i)).\end{array}$$

The essence of the truncated method lies in addressing super-linear coefficients. From the standpoint of finite-time convergence, linear coefficients do not pose difficulties for the EM scheme and therefore do not need to be truncated [16]. In our TEM method, we focus exclusively on truncating the super-linear terms, specifically the coefficients b and $\sigma$.

To construct the TEM scheme, we first choose a strictly increasing function $f:{\mathbb{R}}^{+}\to {\mathbb{R}}^{+}$ satisfying $f\left(n\right)\to \infty$, as $n\to \infty$, and

$$\underset{\left|x\right|\le n,i\in \mathbb{S}}{sup}\left(\left|b\right(x,i\left)\right|\vee \left|\sigma \right(x,i\left)\right|\right)\le f\left(n\right),\forall n\ge 1.$$

We denote by $f^{-1}$ the inverse function of f. We observe that $f^{-1}$ is a strictly increasing continuous function mapping from $\left[f\right(0),\infty )$ to ${\mathbb{R}}^{+}$. We also choose a strictly decreasing function $g:(0,1)\to (0,\infty )$ and a number ${\Delta }^{*}\in (0,1]$ such that

$$\begin{array}{c}\hfill g\left({\Delta }^{*}\right)\ge f\left(2\right),\underset{\Delta \to 0}{lim}g(\Delta )=\infty \mathrm{and}{\left(g(\Delta )\right)}^p\le {\Delta }^{-1}\wedge {\Delta }^{-p/4},\forall \Delta \in (0,1],p\ge 2.\end{array}$$

(6)

For a given step size $\Delta \in (0,1)$, we define a mapping ${\pi }_{\Delta }$ from ${\mathbb{R}}^d$ to the closed ball $\{x\in {\mathbb{R}}^d:|x|\le f^{-1}\left(g(\Delta )\right)\}$ by

$${\pi }_{\Delta }\left(x\right)=\left(\left|x\right|\wedge f^{-1}\left(g(\Delta )\right)\right){\displaystyle \frac{x}{\left|x\right|}}.$$

For a given step size $\Delta \in (0,1)$, we define the truncated functions

$$\begin{array}{c}\hfill b_{\Delta }(x,i)=b\left(\left(\right|x|\wedge f^{-1}\left(g(\Delta )\right)){\displaystyle \frac{x}{\left|x\right|}},i\right)\mathrm{and}{\sigma }_{\Delta }(x,i)=\sigma \left(\left(\right|x|\wedge f^{-1}\left(g(\Delta )\right)){\displaystyle \frac{x}{\left|x\right|}},i\right)\end{array}$$

(7)

for $x\in {\mathbb{R}}^d,i\in \mathbb{S}$, where we set ${\displaystyle \frac{x}{\left|x\right|}}=0$ when $x=0$. It is easy to see that

$$\begin{array}{c}\hfill |b_{\Delta }(x,i)|\vee |{\sigma }_{\Delta }(x,i)|\le f\left(f^{-1}\left(g(\Delta )\right)\right)=g(\Delta )\forall x\in {\mathbb{R}}^d,i\in \mathbb{S}.\end{array}$$

(8)

That is, the truncated functions $b_{\Delta }$ and ${\sigma }_{\Delta }$ remain bounded, even when the original coefficients b and $\sigma$ are unbounded. Furthermore, for all $\Delta \in (0,{\Delta }^{*}]$, where ${\Delta }^{*}$ is given in (6), these truncated terms continue to satisfy a Khasminskii-type condition, as stated in the following lemma (see [16]).

Lemma 2. 

Let Assumption 2 hold. Then, for all $\Delta \in (0,{\Delta }^{*}]$, there exist constant $p>2$, $L_4>0$ such that

$$x^T{b}_{\Delta }(x,i)+{\displaystyle \frac{p-1}{2}}|{\sigma }_{\Delta }{(x,i)|}^2\le 2L_4{(1+|x|}^2),\forall x\in {\mathbb{R}}^d,i\in \mathbb{S}.$$

Since $N(\mathrm{d}t,\mathrm{d}u)$ and $\Lambda \left(t\right)$ are independent, for given partition ${\left\{t_k\right\}}_{k\ge 1}$, $\{\Lambda \left(t_k\right),k=1,2,\cdots \}$ is a discrete Markov chain with transition probability matrix ${\left(P(i,j)\right)}_{i,j\in \mathbb{S}}$, where $P(i,j)=P(\Lambda \left(t_{k+1}\right)=j|\Lambda \left(t_k\right)=i)$ is the $(i,j)$-th entry of the matrix $e^{(t_{k+1}-t_k)Q}$. Thus we can simulate the discrete Markov chain ${\{\Lambda \left(t_k\right)\}}_{k\ge 1}$ using the following recursion: Suppose that $\Lambda \left(t_k\right)=i_1$, and generate a random number $\xi$ uniformly distributed in $[0,1]$. Define the next state as:

$$\begin{array}{c}\hfill \Lambda \left(t_{k+1}\right)=\left\{\begin{array}{cc}{i}_2,\hfill & \mathrm{if}{i}_2\in \mathbb{S}-\left\{m\right\}\mathrm{and}\sum _{j=1}^{i_2-1}P(i_1,j)\le \xi <\sum _{j=1}^{i_2}P(i_1,j),\hfill \\ m,\hfill & \mathrm{if}\sum _{j=1}^{m-1}P(i_1,j)\le \xi .\hfill \end{array}\right.\end{array}$$

(9)

By repeating this procedure, a trajectory of ${\{\Lambda \left(t_k\right)\}}_{k\ge 1}$ can be simulated.

We can now form the discrete-time TEM numerical approximations $X_{\Delta }\left(t_k\right)\approx X\left(t_k\right)$ for $t_k=k\Delta$ by setting $X_{\Delta }\left(0\right)=x_0$ and computing

$$\begin{array}{cc}\hfill X_{\Delta }\left(t_{k+1}\right)=& X_{\Delta }\left(t_k\right)+b_{\Delta }(X_{\Delta }\left(t_k\right),{\Lambda }_{\Delta }\left(t_k\right))\Delta +{\sigma }_{\Delta }(X_{\Delta }\left(t_k\right),{\Lambda }_{\Delta }\left(t_k\right))\Delta B_k\hfill \\ \hfill & +\sum _{j\in \mathbb{S}}c(X_{\Delta }\left(t_k\right),{\Lambda }_{\Delta }\left(t_k\right),u_j){\mathbf{I}}_{\{t_{k+1}={\tau }_j\}}\hfill \end{array}$$

(10)

for $k=0,1,\dots$, where $\Delta B_k=B\left(t_{k+1}\right)-B\left(t_k\right)$. Here ${\tau }_j=inf\{t>0:N(t,{\mathbb{R}}^d\setminus \left\{0\right\})\ge j\}$ are the jump times. Denote by $N_t=N(t,T)$ a homogeneous Poisson process with intensity $\lambda =\Pi ({\mathbb{R}}^d\setminus \left\{0\right\})$. Clearly, ${\overline{\tau }}_j={\tau }_j\wedge T$, $j=1,2,\dots$ are also stopping times. Let us now form two versions of the continuous-time TEM solutions. The first one is defined by

$$\begin{array}{c}\hfill {\overline{X}}_{\Delta }\left(t\right)=\sum _{k=0}^{\infty }{X}_{\Delta }\left(t_k\right){\mathbf{I}}_{[t_k,t_{k+1})}\left(t\right),t\ge 0.\end{array}$$

(11)

This is a simple step process, so its sample paths are not continuous. We refer to it as the continuous-time step-process TEM approximation. The other is defined by

$$\begin{array}{cc}\hfill X_{\Delta }\left(t\right)=& x_0+{\int }_0^t{b}_{\Delta }({\overline{X}}_{\Delta }\left(s\right),{\Lambda }_{\Delta }\left(s\right))\mathrm{d}s+{\int }_0^t{\sigma }_{\Delta }({\overline{X}}_{\Delta }\left(s\right),{\Lambda }_{\Delta }\left(s\right))\mathrm{d}B\left(s\right)\hfill \\ \hfill & +{\int }_0^t{\int }_{{\mathbb{R}}^d\setminus \left\{0\right\}}c({\overline{X}}_{\Delta }(s-),{\Lambda }_{\Delta }(s-),u)N(\mathrm{d}s,\mathrm{d}u),\hfill \end{array}$$

(12)

for $t\ge 0$. We refer to this as the continuous-time continuous-sample TEM approximation. Observe that $X_{\Delta }\left(t_k\right)={\overline{X}}_{\Delta }\left(t_k\right)$ for all $k\ge 0$.

3. Convergence of TEM Approximations

In this section, we present the convergence rate between the TEM approximation and the exact solution (for $p\ge 2$). Additionally, we provide some lemmas required to prove the main results, with detailed proofs given separately for $p=2$ and $p>2$.

3.1. Convergence in $L^2$

First, we state the following assumptions.

Assumption 3. 

For all $(x,i)\in {\mathbb{R}}^d\times \mathbb{S}$, there exists a constant $L_2>0$ such that

$${\int }_{{\mathbb{R}}^d\setminus \left\{0\right\}}{\left|c(x,i,u)\right|}^2\Pi \left(\mathrm{d}u\right)\le L_2{(1+|x|}^2).$$

By (8), it is obvious that

$$\underset{0\le t\le T}{sup}\mathbb{E}{\left|X_{\Delta }\left(t\right)\right|}^2<\infty ,\forall T>0.$$

However, it is not obvious that

$$\underset{0<\Delta \le {\Delta }^{*}}{sup}\underset{0\le t\le T}{sup}\mathbb{E}{\left|X_{\Delta }\left(t\right)\right|}^2<\infty ,\forall T>0,$$

and this is what we are going to establish in this subsection. Let us first present a lemma which shows that $X_{\Delta }\left(t\right)$ and ${\overline{X}}_{\Delta }\left(t\right)$ are close to each other in the $L^2$ sense.

Lemma 3. 

For any ${\Delta }^{*}$ given in (6) and $\Delta \in (0,{\Delta }^{*}]$, we have

$$\begin{array}{c}\hfill \mathbb{E}|X_{\Delta }\left(t\right)-{\overline{X}}_{\Delta }{\left(t\right)|}^2\le C\Delta {\left(g(\Delta )\right)}^2,\forall t\ge 0.\end{array}$$

(13)

Consequently,

$$\begin{array}{c}\hfill \underset{\Delta \to 0}{lim}\mathbb{E}{|X_{\Delta }\left(t\right)-{\overline{X}}_{\Delta }\left(t\right)|}^2=0,\forall t\ge 0.\end{array}$$

(14)

Proof. 

Fix any $\Delta \in (0,{\Delta }^{*}]$ and $t\ge 0$. There is a unique integer $k\ge 0$ such that $t_k\le t\le t_{k+1}$. Let $V(x,i)={\left|x\right|}^2\in C^2({\mathbb{R}}^d\times \mathbb{S})$. For each $i\in \mathbb{S}$, $V(·,i)$ is continuous and nonnegative and tends to infinity as $\left|x\right|\to \infty$. Using (8) and the properties of the Itô integral (see, e.g., [17]), we derive from (12) that

$$\begin{array}{cc}\hfill & \mathbb{E}|X_{\Delta }\left(t\right)-{\overline{X}}_{\Delta }{\left(t\right)|}^2=\mathbb{E}{|X_{\Delta }\left(t\right)-X_{\Delta }\left(t_k\right)|}^2\hfill \\ \hfill \le & |x_0{|}^2+\mathbb{E}{\int }_{t_k}^t\left(2|X_{\Delta }\left(s\right)-{\overline{X}}_{\Delta }\left(s\right)\left|\right|b_{\Delta }({\overline{X}}_{\Delta }\left(s\right),{\Lambda }_{\Delta }\left(s\right))|+|{\sigma }_{\Delta }({\overline{X}}_{\Delta }\left(s\right),{\Lambda }_{\Delta }\left(s\right)){|}^2\right)\mathrm{d}s\hfill \\ \hfill & +\mathbb{E}{\int }_{t_k}^t{\int }_{{\mathbb{R}}^d\setminus \left\{0\right\}}[{\left(X_{\Delta }\left(s\right)+c({\overline{X}}_{\Delta }(s-),{\Lambda }_{\Delta }(s-),u)\right)}^2-|X_{\Delta }\left(s\right){|}^2]\Pi \left(\mathrm{d}u\right)\mathrm{d}s.\hfill \end{array}$$

(15)

Denote the two integrals on the right-hand side by $I_1$ and $I_2$ respectively. Using (8), we bound $I_1$

$$\begin{array}{cc}\hfill I_1=& \mathbb{E}{\int }_{t_k}^t\left(2|X_{\Delta }\left(s\right)-{\overline{X}}_{\Delta }\left(s\right)\left|\right|b_{\Delta }({\overline{X}}_{\Delta }\left(s\right),{\Lambda }_{\Delta }\left(s\right))|+|{\sigma }_{\Delta }({\overline{X}}_{\Delta }\left(s\right),{\Lambda }_{\Delta }\left(s\right)){|}^2\right)\mathrm{d}s\hfill \\ \hfill \le & \mathbb{E}{\int }_{t_k}^{t}2|X_{\Delta }\left(s\right)-{\overline{X}}_{\Delta }\left(s\right)|(g(\Delta )+g^2(\Delta ))\mathrm{d}s\hfill \\ \hfill \le & C{g}^2(\Delta )\mathbb{E}{\int }_{t_k}^{t}2|X_{\Delta }\left(s\right)-{\overline{X}}_{\Delta }\left(s\right)|\mathrm{d}s.\hfill \end{array}$$

(16)

For $I_2$, using Assumption 3 and the fact that ${\overline{X}}_{\Delta }\left(s\right)$ and $X_{\Delta }\left(s\right)$ coincide at the grid points, we obtain

$$\begin{array}{cc}\hfill I_2=& \mathbb{E}{\int }_{t_k}^t{\int }_{{\mathbb{R}}^d\setminus \left\{0\right\}}[{\left(X_{\Delta }\left(s\right)+c({\overline{X}}_{\Delta }(s-),{\Lambda }_{\Delta }(s-),u)\right)}^2-|X_{\Delta }\left(s\right){|}^2]\Pi \left(\mathrm{d}u\right)\mathrm{d}s\hfill \\ \hfill \le & \mathbb{E}{\int }_{t_k}^t{\int }_{{\mathbb{R}}^d\setminus \left\{0\right\}}|c({\overline{X}}_{\Delta }(s-),{\Lambda }_{\Delta }(s-),u){|}^2+2|X_{\Delta }\left(s\right)\left|\right|c({\overline{X}}_{\Delta }(s-),{\Lambda }_{\Delta }(s-),u)|\Pi \left(\mathrm{d}u\right)\mathrm{d}s\hfill \\ \hfill \le & \mathbb{E}{\int }_{t_k}^t(1+|{\overline{X}}_{\Delta }{\left(s\right)|}^2\mathrm{d}s\hfill \\ \hfill \le & C\Delta .\hfill \end{array}$$

(17)

Combining the estimates for $I_1$ and $I_2$ and applying Gronwall’s inequality, we obtain

$$\begin{array}{c}\hfill \mathbb{E}|X_{\Delta }\left(t\right)-{\overline{X}}_{\Delta }{\left(t\right)|}^2\le C\Delta {\left(g(\Delta )\right)}^2,\forall t\ge 0,\end{array}$$

which completes the proof. □

Lemma 4. 

Let Assumptions 1 and 3 hold. Then

$$\begin{array}{c}\hfill \underset{0<\Delta \le {\Delta }^{*}}{sup}\underset{0\le t\le T}{sup}\mathbb{E}{\left|X_{\Delta }\left(t\right)\right|}^2\le C,\forall T>0,\end{array}$$

(18)

where ${\Delta }^{*}$ satisfies (6). Here and in the sequel, C denotes a generic positive constant dependent on T and $x_0$ but independent of Δ, and its value may change from line to line.

Proof. 

Fix any $\Delta \in (0,{\Delta }^{*}]$ and $T\ge 0$. Apply Itô’s formula to $V(x,i)={\left|x\right|}^2$. For $0\le t\le T$,

$$\begin{array}{cc}\hfill & \mathbb{E}|X_{\Delta }{\left(t\right)|}^2\hfill \\ \hfill \le & |x_0{|}^2+\mathbb{E}{\int }_0^t\left(2X_{\Delta }^T\left(s\right)b_{\Delta }({\overline{X}}_{\Delta }\left(s\right),{\Lambda }_{\Delta }\left(s\right))+{\left|{\sigma }_{\Delta }({\overline{X}}_{\Delta }\left(s\right),{\Lambda }_{\Delta }\left(s\right))\right|}^2\right)\mathrm{d}s\hfill \\ \hfill & +\mathbb{E}{\int }_{t_k}^t{\int }_{{\mathbb{R}}^d\setminus \left\{0\right\}}\left({\left(X_{\Delta }\left(s\right)+c({\overline{X}}_{\Delta }(s-),{\Lambda }_{\Delta }(s-),u)\right)}^2-{\left|X_{\Delta }\left(s\right)\right|}^2\right)\Pi \left(\mathrm{d}u\right)\mathrm{d}s\hfill \\ \hfill =& C|x_0{|}^2+{\overline{I}}_1+{\overline{I}}_2.\hfill \end{array}$$

(19)

Using (8) and Lemma 2, ${\overline{I}}_1$ can be bounded by $C{\int }_0^t(1+\mathbb{E}|X_{\Delta }\left(s\right)\left|\mathrm{d}s\right)$. For ${\overline{I}}_2$, using Assumption 3 as in (16) gives ${\overline{I}}_2\le C\Delta .$ Consequently,

$$\begin{array}{c}\hfill \mathbb{E}|X_{\Delta }{\left(t\right)|}^2\le C_1+C_2{\int }_0^t\left(\mathbb{E}|X_{\Delta }{\left(s\right)|}^2\right)\mathrm{d}s.\end{array}$$

Taking the supremum over $t\in [0,T]$ and applying Gronwall’s inequality yields

$$\underset{0\le t\le T}{sup}\mathbb{E}{\left|X_{\Delta }\left(t\right)\right|}^2\le C.$$

Since the constant C is independent of $\Delta$, the required assertion (36) follows. □

Next, we present the first main result of this paper, namely, the convergence in $L^2$ between the TEM approximation $X_{\Delta }\left(T\right)$ and the exact solution $X\left(T\right)$.

Theorem 1. 

Let Assumptions 1–3 hold. Then,

$$\underset{\Delta \to 0}{lim}\mathbb{E}|X_{\Delta }{\left(T\right)-X\left(T\right)|}^2=0and\underset{\Delta \to 0}{lim}\mathbb{E}{|{\overline{X}}_{\Delta }\left(T\right)-X\left(T\right)|}^2=0.$$

Proof. 

For every $0\le T^{\prime }\le T$, by (2) and (12), we have

$$\begin{array}{cc}\hfill & \mathbb{E}\left(\underset{0\le t\le T^{\prime }}{sup}{|X\left(t\right)-X_{\Delta }\left(t\right)|}^2\right)\hfill \\ \hfill \le & 3T\mathbb{E}{\int }_0^{T^{\prime }}{|b_{\Delta }({\overline{X}}_{\Delta }\left(s\right),{\Lambda }_{\Delta }\left(s\right))-b(X\left(s\right),\Lambda \left(s\right))|}^2\mathrm{d}s\hfill \\ \hfill & +12\mathbb{E}{\int }_0^{T^{\prime }}{|{\sigma }_{\Delta }({\overline{X}}_{\Delta }\left(s\right),{\Lambda }_{\Delta }\left(s\right))-\sigma (X\left(s\right),\Lambda \left(s\right))|}^2\mathrm{d}s\hfill \\ \hfill & +C(T^2+1)\mathbb{E}\left({\left|{\int }_0^{T^{\prime }}{\int }_{{\mathbb{R}}^d\setminus \left\{0\right\}}\left(c({\overline{X}}_{\Delta }(s-),{\Lambda }_{\Delta }(s-),u)-c(X(s-),\Lambda \left(s\right),u)\right)N(\mathrm{d}s,\mathrm{d}u)\right|}^2\right).\hfill \end{array}$$

(20)

We focus on the third term. Observing that $\tilde{N}(\mathrm{d}t,\mathrm{d}u)=N(\mathrm{d}t,\mathrm{d}u)-\Pi \left(\mathrm{d}u\right)\mathrm{d}t$ is the compensated Poisson random measure and the process

$${\int }_0^t{\int }_{{\mathbb{R}}^d\setminus \left\{0\right\}}c(X_{\Delta }(s-),{\Lambda }_{\Delta }(s-),u)\tilde{N}(\mathrm{d}s,\mathrm{d}u)$$

is a martingale (see [18]), we have

$$\begin{array}{cc}\hfill & \underset{0\le t\le T}{sup}\mathbb{E}\left({\left|{\int }_0^t{\int }_{{\mathbb{R}}^d\setminus \left\{0\right\}}\left(c({\overline{X}}_{\Delta }(s-),{\Lambda }_{\Delta }(s-),u)-c(X(s-),\Lambda (s-),u)\right)N(\mathrm{d}s,\mathrm{d}u)\right|}^2\right)\hfill \\ \hfill \le & \mathbb{E}\left(\underset{0\le t\le T}{sup}{\left|{\int }_0^t{\int }_{{\mathbb{R}}^d\setminus \left\{0\right\}}\left(c({\overline{X}}_{\Delta }(s-),{\Lambda }_{\Delta }(s-),u)-c(X(s-),\Lambda (s-),u)\right)N(\mathrm{d}s,\mathrm{d}u)\right|}^2\right)\hfill \\ \hfill =& \mathbb{E}\left(\underset{0\le t\le T}{sup}\right|{\int }_0^t{\int }_{{\mathbb{R}}^d\setminus \left\{0\right\}}\left(c({\overline{X}}_{\Delta }(s-),{\Lambda }_{\Delta }(s-),u)-c(X(s-),\Lambda (s-),u)\right)\tilde{N}(\mathrm{d}s,\mathrm{d}u)\hfill \\ \hfill & +{\int }_0^t{\int }_{{\mathbb{R}}^d\setminus \left\{0\right\}}\left(c({\overline{X}}_{\Delta }(s-),{\Lambda }_{\Delta }(s-),u)-c(X(s-),\Lambda (s-),u)\right)\Pi \left(\mathrm{d}u\right)\mathrm{d}s{|}^2)\hfill \\ \hfill \le & 2\mathbb{E}\left(\underset{0\le t\le T}{sup}{\left|{\int }_0^t{\int }_{{\mathbb{R}}^d\setminus \left\{0\right\}}\left(c({\overline{X}}_{\Delta }(s-),{\Lambda }_{\Delta }(s-),u)-c(X(s-),\Lambda (s-),u)\right)\tilde{N}(\mathrm{d}s,\mathrm{d}u)\right|}^2\right)\hfill \\ \hfill & +2\mathbb{E}\left(\underset{0\le t\le T}{sup}{\left|{\int }_0^t{\int }_{{\mathbb{R}}^d\setminus \left\{0\right\}}\left(c({\overline{X}}_{\Delta }(s-),{\Lambda }_{\Delta }(s-),u)-c(X(s-),\Lambda (s-),u)\right)\Pi \left(\mathrm{d}u\right)\mathrm{d}s\right|}^2\right).\hfill \end{array}$$

(21)

By applying the Doob martingale inequality to the first term and using Assumptions 1 and 3 to handle the second term, we obtain

$$\begin{array}{cc}\hfill & \mathbb{E}\left(\underset{0\le t\le T}{sup}{\left|{\int }_0^t{\int }_{{\mathbb{R}}^d\setminus \left\{0\right\}}\left(c({\overline{X}}_{\Delta }(s-),{\Lambda }_{\Delta }(s-),u)-c(X(s-),\Lambda (s-),u)\right)\tilde{N}(\mathrm{d}s,\mathrm{d}u)\right|}^2\right)\hfill \\ \hfill \le & c_p\mathbb{E}{\left|{\int }_0^T{\int }_{{\mathbb{R}}^d\setminus \left\{0\right\}}\left(c({\overline{X}}_{\Delta }(s-),{\Lambda }_{\Delta }(s-),u)-c(X(s-),\Lambda (s-),u)\right)\tilde{N}(\mathrm{d}s,\mathrm{d}u)\right|}^2\hfill \\ \hfill =& C\mathbb{E}{\int }_0^T{\int }_{{\mathbb{R}}^d\setminus \left\{0\right\}}{|c({\overline{X}}_{\Delta }(s-),{\Lambda }_{\Delta }(s-),u)-c(X(s-),\Lambda (s-),u)|}^2\Pi \left(\mathrm{d}u\right)\mathrm{d}s.\hfill \end{array}$$

(22)

On the other hand, we can get

$$\begin{array}{cc}\hfill & \mathbb{E}\left(\underset{0\le t\le T}{sup}{\left|{\int }_0^t{\int }_{{\mathbb{R}}^d\setminus \left\{0\right\}}[c({\overline{X}}_{\Delta }(s-),{\Lambda }_{\Delta }(s-),u)-c(X(s-),\Lambda (s-),u)]\Pi \left(\mathrm{d}u\right)\mathrm{d}s\right|}^2\right)\hfill \\ \hfill \le & \mathbb{E}\left(\underset{0\le t\le T}{sup}{\int }_0^t{\int }_{{\mathbb{R}}^d\setminus \left\{0\right\}}{|c({\overline{X}}_{\Delta }(s-),{\Lambda }_{\Delta }(s-),u)-c(X(s-),\Lambda (s-),u)|}^2\Pi \left(\mathrm{d}u\right)\mathrm{d}s\right)·\sqrt{\Pi ({\mathbb{R}}^d\setminus \left\{0\right\})T}\hfill \\ \hfill \le & C\mathbb{E}{\int }_0^T{\int }_{{\mathbb{R}}^d\setminus \left\{0\right\}}{|c({\overline{X}}_{\Delta }(s-),{\Lambda }_{\Delta }(s-),u)-c(X(s-),\Lambda (s-),u)|}^2\Pi \left(\mathrm{d}u\right)\mathrm{d}s\hfill \\ \hfill \le & C\mathbb{E}{\int }_0^{T^{\prime }}{\int }_{{\mathbb{R}}^d\setminus \left\{0\right\}}{|c({\overline{X}}_{\Delta }(s-),\Lambda (s-),u)-c(X(s-),\Lambda (s-),u)|}^2\Pi \left(\mathrm{d}u\right)\mathrm{d}s\hfill \\ \hfill & +C\mathbb{E}{\int }_0^{T^{\prime }}{\int }_{{\mathbb{R}}^d\setminus \left\{0\right\}}{|c({\overline{X}}_{\Delta }(s-),{\Lambda }_{\Delta }(s-),u)-c({\overline{X}}_{\Delta }(s-),\Lambda (s-),u)|}^2\Pi \left(\mathrm{d}u\right)\mathrm{d}s.\hfill \end{array}$$

(23)

By Assumption 1, we have

$$\begin{array}{cc}\hfill & \mathbb{E}{\int }_0^{T^{\prime }}{\int }_{{\mathbb{R}}^d\setminus \left\{0\right\}}{|c({\overline{X}}_{\Delta }(s-),\Lambda (s-),u)-c(X(s-),\Lambda (s-),u)|}^2\Pi \left(\mathrm{d}u\right)\mathrm{d}s\hfill \\ \hfill \le & C\mathbb{E}{\int }_0^{T^{\prime }}{|\overline{X}(s-)-X(s-)|}^2\mathrm{d}s.\hfill \end{array}$$

(24)

By Assumption 3, we can compute that

$$\begin{array}{cc}\hfill & \mathbb{E}{\int }_0^{T^{\prime }}{\int }_{{\mathbb{R}}^d\setminus \left\{0\right\}}{|c({\overline{X}}_{\Delta }(s-),{\Lambda }_{\Delta }(s-),u)-c({\overline{X}}_{\Delta }(s-),\Lambda (s-),u)|}^2\Pi \left(\mathrm{d}u\right)\mathrm{d}s\hfill \\ \hfill =& \mathbb{E}\left[\mathbb{E}\left[{\int }_0^T{\int }_{{\mathbb{R}}^d\setminus \left\{0\right\}}{|c({\overline{X}}_{\Delta }(s-),{\Lambda }_{\Delta }(s-),u)-c({\overline{X}}_{\Delta }(s-),\Lambda (s-),u)|}^2\Pi \left(\mathrm{d}u\right)\mathrm{d}s\mid N_T,{\tau }_1,\dots ,{\tau }_{N_T}\right]\right]\hfill \\ \hfill =& \mathbb{E}\left[\sum _{k\ge 0}\mathbb{E}\left[{\int }_{t_k}^{t_{k+1}}{\int }_{{\mathbb{R}}^d\setminus \left\{0\right\}}{|c({\overline{X}}_{\Delta }(s-),{\Lambda }_{\Delta }(s-),u)-c({\overline{X}}_{\Delta }(s-),\Lambda (s-),u)|}^2\Pi \left(\mathrm{d}u\right)\mathrm{d}s\mid N_T,{\tau }_1,\dots ,{\tau }_{N_T}\right]\right]\hfill \\ \hfill =& \mathbb{E}\left[\sum _{k\ge 0}\mathbb{E}\right[{\int }_{t_k}^{t_{k+1}}{\int }_{{\mathbb{R}}^d\setminus \left\{0\right\}}{|c({\overline{X}}_{\Delta }(s-),{\Lambda }_{\Delta }(s-),u)-c({\overline{X}}_{\Delta }(s-),\Lambda (s-),u)|}^2\hfill \\ \hfill & ·{\mathbf{I}}_{\{\Lambda (s-)\ne \Lambda \left(t_k\right)\}}\Pi \left(\mathrm{d}u\right)\mathrm{d}s\mid N_T,{\tau }_1,\dots ,{\tau }_{N_T}\left]\right]\hfill \\ \hfill \le & 2\mathbb{E}\left[\sum _{k\ge 0}\mathbb{E}\right[{\int }_{t_k}^{t_{k+1}}{\int }_{{\mathbb{R}}^d\setminus \left\{0\right\}}\left(\right|c({\overline{X}}_{\Delta }(s-),{\Lambda }_{\Delta }(s-),u){|}^2+|c({\overline{X}}_{\Delta }(s-),\Lambda (s-),u){|}^2)\hfill \\ \hfill & ·{\mathbf{I}}_{\{\Lambda (s-)\ne \Lambda \left(t_k\right)\}}\Pi \left(\mathrm{d}u\right)\mathrm{d}s\mid N_T,{\tau }_1,\dots ,{\tau }_{N_T}\left]\right]\hfill \\ \hfill \le & C\mathbb{E}\left[\sum _{k\ge 0}\mathbb{E}\left[{\int }_{t_k}^{t_{k+1}}[1+|{\overline{X}}_{\Delta }\left(t_k\right){|}^2]{\mathbf{I}}_{\{\Lambda \left(s\right)\ne \Lambda \left(t_k\right)\}}\mathrm{d}s\mid N_T,{\tau }_1,\dots ,{\tau }_{N_T}\right]\right].\hfill \end{array}$$

(25)

Note that given $\{N_T,{\tau }_1,\dots ,{\tau }_{N_T}\}$, the terms $1+|{\overline{X}}_{\Delta }\left(t_k\right){|}^2$ and ${\mathbf{I}}_{\{\Lambda \left(s\right)\ne \Lambda \left(t_k\right)\}}$ are conditionally independent with respect to $\Lambda \left(t_k\right)$; thus

$$\begin{array}{cc}\hfill & \mathbb{E}{\int }_{t_k}^{t_{k+1}}[1+|{\overline{X}}_{\Delta }\left(t_k\right){|}^2]{\mathbf{I}}_{\{\Lambda \left(s\right)\ne \Lambda \left(t_k\right)\}}\mathrm{d}s\hfill \\ \hfill =& {\int }_{t_k}^{t_{k+1}}\mathbb{E}\left[\mathbb{E}\left[(1+|{\overline{X}}_{\Delta }\left(t_k\right){|}^2){\mathbf{I}}_{\{\Lambda \left(s\right)\ne \Lambda \left(t_k\right)\}}\mid \Lambda \left(t_k\right)\right]\right]\mathrm{d}s\hfill \\ \hfill =& {\int }_{t_k}^{t_{k+1}}\mathbb{E}\left[\mathbb{E}\left[1+|{\overline{X}}_{\Delta }\left(t_k\right){|}^2\mid \Lambda \left(t_k\right)\right]\mathbb{E}\left[{\mathbf{I}}_{\{\Lambda \left(s\right)\ne \Lambda \left(t_k\right)\}}\mid \Lambda \left(t_k\right)\right]\right]\mathrm{d}s.\hfill \end{array}$$

(26)

On the other hand, by the Markov property, we get

$$\begin{array}{cc}\hfill \mathbb{E}\left[{\mathbf{I}}_{\{\Lambda (s-)\ne \Lambda \left(t_k\right)\}}\mid \Lambda \left(t_k\right)\right]& =\sum _{i\in \mathbb{S}}{\mathbf{I}}_{\{\Lambda \left(t_k\right)=i\}}P(\Lambda \left(s\right)\ne i\mid \Lambda \left(t_k\right)=i)\hfill \\ \hfill & =\sum _{i\in \mathbb{S}}{\mathbf{I}}_{\{\Lambda \left(t_k\right)=i\}}(-q_{ii}(s-t_k)+o(s-t_k))\hfill \\ \hfill & \le \sum _{i\in \mathbb{S}}{\mathbf{I}}_{\{\Lambda \left(t_k\right)=i\}}(max(-q_{ii})\Delta +o(\Delta ))\hfill \\ \hfill & \le C\Delta +o(\Delta ).\hfill \end{array}$$

(27)

Substituting (27) into (26) gives

$$\begin{array}{cc}\hfill & \mathbb{E}{\int }_{t_k}^{t_{k+1}}[1+|{\overline{X}}_{\Delta }\left(t_k\right){|}^2]{\mathbf{I}}_{\{\Lambda \left(s\right)\ne \Lambda \left(t_k\right)\}}\mathrm{d}s\hfill \\ \hfill \le & {\int }_{t_k}^{t_{k+1}}\mathbb{E}\left[\mathbb{E}\left[(1+|{\overline{X}}_{\Delta }\left(t_k\right){|}^2)\mid \Lambda \left(t_k\right)\right](C\Delta +o(\Delta ))\right]\mathrm{d}s\hfill \\ \hfill \le & (C\Delta +o(\Delta )){\int }_{t_k}^{t_{k+1}}\mathbb{E}\left[\mathbb{E}\left[1+|{\overline{X}}_{\Delta }\left(t_k\right){|}^2\mid \Lambda \left(t_k\right)\right]\right]\mathrm{d}s\hfill \\ \hfill =& (C\Delta +o(\Delta )){\int }_{t_k}^{t_{k+1}}\mathbb{E}\left[1+|{\overline{X}}_{\Delta }\left(t_k\right){|}^2\right]\mathrm{d}s.\hfill \end{array}$$

By Lemma 1, there exists a positive constant C such that

$$\mathbb{E}[1+|{\overline{X}}_{\Delta }\left(t_k\right){|}^2]\le 1+\mathbb{E}(\underset{0\le t\le T}{sup}|X\left(t\right){|}^2)\le C,$$

then

$$\begin{array}{c}\hfill \mathbb{E}{\int }_{t_k}^{t_{k+1}}[1+|{\overline{X}}_{\Delta }\left(t_k\right){|}^2]{\mathbf{I}}_{\{\Lambda \left(s\right)\ne \Lambda \left(t_k\right)\}}\mathrm{d}s\le (t_{k+1}-t_k)(C\Delta +o(\Delta )).\end{array}$$

(28)

Substitute (28) into (25)

$$\begin{array}{cc}\hfill & \mathbb{E}{\int }_0^{T^{\prime }}{\int }_{{\mathbb{R}}^d\setminus \left\{0\right\}}{|c({\overline{X}}_{\Delta }(s-),\Lambda (s-),u)-c(X(s-),\Lambda (s-),u)|}^2\Pi \left(\mathrm{d}u\right)\mathrm{d}s\hfill \\ \hfill \le & C\mathbb{E}\left[\sum _{k\ge 0}\mathbb{E}\left[(t_{k+1}-t_k)(C\Delta +o(\Delta ))\mid N_T,{\tau }_1,\dots ,{\tau }_{N_T}\right]\right]\hfill \\ \hfill =& C\mathbb{E}\left[\sum _{k\ge 0}(t_{k+1}-t_k)(C\Delta +o(\Delta ))\right]\hfill \\ \hfill \le & C\Delta +o(\Delta ).\hfill \end{array}$$

(29)

Here we use the fact that $(t_{k+1}-t_k)(C\Delta +o(\Delta ))$ are measurable with respect to the $\sigma$-algebra generated by $\{N_T,{\tau }_1,\dots ,{\tau }_{N_T}\}$.

Now substitute (24) and (29) into (23); we obtain that

$$\begin{array}{cc}\hfill & \mathbb{E}{\int }_0^{T^{\prime }}{\int }_{{\mathbb{R}}^d\setminus \left\{0\right\}}{|c({\overline{X}}_{\Delta }(s-),{\Lambda }_{\Delta }(s-),u)-c(X(s-),\Lambda (s-),u)|}^2\Pi \left(\mathrm{d}u\right)\mathrm{d}s\hfill \\ \hfill \le & 2C{\int }_0^T\mathbb{E}{|{\overline{X}}_{\Delta }(s-)-X(s-)|}^2\mathrm{d}s+C\Delta +o(\Delta ).\hfill \end{array}$$

(30)

Similar to (30), we have

$$\begin{array}{cc}\hfill \mathbb{E}{\int }_0^{T^{\prime }}{|b_{\Delta }({\overline{X}}_{\Delta }\left(s\right),{\Lambda }_{\Delta }\left(s\right))-b(X\left(s\right),\Lambda \left(s\right))|}^2\mathrm{d}s& \le 2C{\int }_0^{T^{\prime }}\mathbb{E}{|{\overline{X}}_{\Delta }\left(s\right)-X\left(s\right)|}^2\mathrm{d}s+C\Delta +o(\Delta ),\hfill \end{array}$$

(31)

and

$$\begin{array}{cc}\hfill \mathbb{E}{\int }_0^{T^{\prime }}{|{\sigma }_{\Delta }({\overline{X}}_{\Delta }\left(s\right),{\Lambda }_{\Delta }\left(s\right))-\sigma (X\left(s\right),\Lambda \left(s\right))|}^2\mathrm{d}s& \le 2C{\int }_0^{T^{\prime }}\mathbb{E}{|{\overline{X}}_{\Delta }\left(s\right)-X\left(s\right)|}^2\mathrm{d}s+C\Delta +o(\Delta ).\hfill \end{array}$$

(32)

Substituting (30)–(32) into (20) yields

$$\begin{array}{cc}\hfill \mathbb{E}\left(\underset{0\le t\le T^{\prime }}{sup}{|X_{\Delta }\left(t\right)-X\left(t\right)|}^2\right)\le & C{\int }_0^{T^{\prime }}\mathbb{E}{|{\overline{X}}_{\Delta }\left(s\right)-X\left(s\right)|}^2\mathrm{d}s\hfill \\ \hfill & +C{\int }_0^{T^{\prime }}\mathbb{E}{|{\overline{X}}_{\Delta }(s-)-X(s-)|}^2\mathrm{d}s+C\Delta +o(\Delta ).\hfill \end{array}$$

(33)

Note that

$$\begin{array}{cc}\hfill \mathbb{E}|{\overline{X}}_{\Delta }{\left(s\right)-X\left(s\right)|}^2& \le 2\mathbb{E}|{\overline{X}}_{\Delta }\left(s\right)-X_{\Delta }{\left(s\right)|}^2+2\mathbb{E}{|X_{\Delta }\left(s\right)-X\left(s\right)|}^2.\hfill \end{array}$$

(34)

$$\begin{array}{cc}\hfill \mathbb{E}|{\overline{X}}_{\Delta }{(s-)-X(s-)|}^2& \le 2\mathbb{E}|{\overline{X}}_{\Delta }(s-)-X_{\Delta }{(s-)|}^2+2\mathbb{E}{|X_{\Delta }(s-)-X(s-)|}^2.\hfill \end{array}$$

(35)

By Lemma 3 that $\mathbb{E}|{\overline{X}}_{\Delta }\left(s\right)-X_{\Delta }{\left(s\right)|}^2\le C\Delta {\left(g(\Delta )\right)}^2$. Similarly, $\mathbb{E}|{\overline{X}}_{\Delta }(s-)-X_{\Delta }{(s-)|}^2\le C\Delta {\left(g(\Delta )\right)}^2.$ Substituting these and (34)–(35) into (33) gives

$$\begin{array}{cc}\hfill \mathbb{E}\left(\underset{0\le t\le T^{\prime }}{sup}{|X_{\Delta }\left(t\right)-X\left(t\right)|}^2\right)& \le C{\int }_0^{T^{\prime }}\left(\mathbb{E}\right|X_{\Delta }{\left(t\right)-X\left(t\right)|}^2+\mathbb{E}|X_{\Delta }(t-)-X(t-){|}^2)\mathrm{d}t+C\Delta +o(\Delta )\hfill \\ \hfill & \le C{\int }_0^{T^{\prime }}\mathbb{E}\left(\underset{0\le s\le t}{sup}{|X_{\Delta }\left(s\right)-X\left(s\right)|}^2\right)\mathrm{d}t+C\Delta +o(\Delta ).\hfill \end{array}$$

An application of Gronwall’s inequality then yields

$$\mathbb{E}\left(\underset{0\le t\le T}{sup}{|X_{\Delta }\left(t\right)-X\left(t\right)|}^2\right)\le C\Delta +o(\Delta ).$$

The proof is complete. □

3.2. Strong Convergence in $L^p,p>2$

In this subsection we will show that

$$\underset{\Delta \to 0}{lim}\mathbb{E}\left(\underset{0\le t\le T}{sup}{|X\left(t\right)-X_{\Delta }\left(t\right)|}^q\right)=0\mathrm{and}\underset{\Delta \to 0}{lim}\mathbb{E}{|{\overline{X}}_{\Delta }\left(t\right)-X\left(t\right)|}^q=0,$$

for any $T>0$ and $2<q\le p$. In the remainder of this subsection, we fix an arbitrary $T>0$. First, we state an additional assumption.

Assumption 4. 

For all $(x,i)\in {\mathbb{R}}^d\times \mathbb{S}$,

$${\int }_{{\mathbb{R}}^d\setminus \left\{0\right\}}\left({\displaystyle \frac{{|x+c(x,i,u)|}^p}{{\left|x\right|}^p}}-1\right)\Pi \left(\mathrm{d}u\right)\le \widehat{c}\left(i\right)<\infty .$$

This section begins by establishing some lemmas, which will then lead to the statement of the main result. In what follows, we use $c_p$ to denote generic positive constants that may depend only on p and whose values may change from occurrence to occurrence.

Lemma 5. 

Let Assumption 1 hold. For any real number $R > |x_0|$, define the stopping time

$${\rho }_R=inf\{t\ge 0:|X\left(t\right)|\ge R\}.$$

Then

$$\begin{array}{c}\hfill P({\rho }_R\le T)\le {\displaystyle \frac{C}{R^2}}.\end{array}$$

Proof. 

A detailed proof can be found in [19]. □

Lemma 6. 

Let Assumptions 1 and 4 hold. Then for $p>2$,

$$\begin{array}{c}\hfill \underset{0<\Delta \le {\Delta }^{*}}{sup}\underset{0\le t\le T}{sup}\mathbb{E}{\left|X_{\Delta }\left(t\right)\right|}^p\le C,\forall T>0,\end{array}$$

(36)

where C denotes a generic positive constant that may depend on T, p and $x_0$ but is independent of Δ, and its values may change between occurrences.

Proof. 

Apply Itô’s formula to $V(x,i)={\left|x\right|}^p.$ Fix any $\Delta \in (0,{\Delta }^{*}]$ and $T\ge 0$. For $0\le t\le T$,

$$\begin{array}{cc}\hfill & \mathbb{E}|X_{\Delta }{\left(t\right)|}^p\hfill \\ \hfill \le & c_p|x_0{|}^p+\mathbb{E}{\int }_0^{t}p{\left|X_{\Delta }\left(s\right)\right|}^{p-2}\left(X_{\Delta }^T\left(s\right)b_{\Delta }({\overline{X}}_{\Delta }\left(s\right),{\Lambda }_{\Delta }\left(s\right))+{\displaystyle \frac{p-1}{2}}{\left|{\sigma }_{\Delta }({\overline{X}}_{\Delta }\left(s\right),{\Lambda }_{\Delta }\left(s\right))\right|}^2\right)\mathrm{d}s\hfill \\ \hfill & +\mathbb{E}{\int }_0^{t}p{\left|X_{\Delta }\left(s\right)\right|}^{p-2}{(X_{\Delta }\left(s\right)-{\overline{X}}_{\Delta }^T\left(s\right))}^T{b}_{\Delta }({\overline{X}}_{\Delta }\left(s\right),{\Lambda }_{\Delta }\left(s\right))\mathrm{d}s\hfill \\ \hfill & +\mathbb{E}{\int }_{t_k}^t{\int }_{{\mathbb{R}}^d\setminus \left\{0\right\}}\left({\left(X_{\Delta }\left(s\right)+c({\overline{X}}_{\Delta }(s-),{\Lambda }_{\Delta }(s-),u)\right)}^p-{\left|X_{\Delta }\left(s\right)\right|}^p\right)\Pi \left(\mathrm{d}u\right)\mathrm{d}s\hfill \\ \hfill =& c_p{\left|x_0\right|}^p+{\tilde{I}}_1+{\tilde{I}}_2+{\tilde{I}}_3.\hfill \end{array}$$

(37)

By Lemma 2 and the Young inequality

$$a^{p-2}b\le {\displaystyle \frac{p-2}{p}}{a}^p+{\displaystyle \frac{2}{p}}{b}^{p/2},\forall a,b\ge 0,p>2,$$

we then have

$$\begin{array}{cc}\hfill {\tilde{I}}_1+{\tilde{I}}_2\le & \mathbb{E}{\int }_0^{t}p|X_{\Delta }{\left(s\right)|}^{p-2}(L_3(1+|{\overline{X}}_{\Delta }\left(s\right){|}^2)\mathrm{d}s\hfill \\ \hfill & +\mathbb{E}{\int }_0^t(p-2)|X_{\Delta }{\left(s\right)|}^p\mathrm{d}s+\mathbb{E}{\int }_0^t|X_{\Delta }\left(s\right)-{\overline{X}}_{\Delta }{\left(s\right)|}^{p-2}{\left|b_{\Delta }({\overline{X}}_{\Delta }\left(s\right),{\Lambda }_{\Delta }\left(s\right))\right|}^{p/2}\mathrm{d}s\hfill \\ \hfill \le & C+C{\int }_0^T\left(\mathbb{E}\right|X_{\Delta }{\left(s\right)|}^p+\mathbb{E}|{\overline{X}}_{\Delta }\left(s\right){|}^p)\mathrm{d}s\hfill \\ \hfill & +2{\int }_0^t\mathbb{E}|X_{\Delta }\left(s\right)-{\overline{X}}_{\Delta }{\left(s\right)|}^{p/2}{\left|b_{\Delta }({\overline{X}}_{\Delta }\left(s\right),{\Lambda }_{\Delta }\left(s\right))\right|}^{p/2}\mathrm{d}s.\hfill \end{array}$$

(38)

Combining (21), (22) and Assumption 4, we obtain that

$$\begin{array}{cc}\hfill {\tilde{I}}_3=& \mathbb{E}{\int }_{t_k}^t{\int }_{{\mathbb{R}}^d\setminus \left\{0\right\}}[{\left(X_{\Delta }\left(s\right)+c({\overline{X}}_{\Delta }(s-),{\Lambda }_{\Delta }(s-),u)\right)}^p-|X_{\Delta }\left(s\right){|}^p]\Pi \left(\mathrm{d}u\right)\mathrm{d}s\hfill \\ \hfill \le & \mathbb{E}{\int }_{t_k}^t{\int }_{{\mathbb{R}}^d\setminus \left\{0\right\}}{\left|X_{\Delta }\left(s\right)\right|}^p\left[{\displaystyle \frac{{(X_{\Delta }\left(s\right)+c({\overline{X}}_{\Delta }(s-),{\Lambda }_{\Delta }(s-),u))}^p}{|X_{\Delta }{\left(s\right)|}^p}}-1\right]\Pi \left(\mathrm{d}u\right)\mathrm{d}s\hfill \\ \hfill \le & \mathbb{E}{\int }_{t_k}^t|X_{\Delta }{\left(s\right)|}^p\left|\widehat{c}\left({\Lambda }_{\Delta }(s-)\right)\right|\mathrm{d}s\hfill \\ \hfill \le & \underset{i\in \mathbb{S}}{max}|\widehat{c}\left({\Lambda }_{\Delta }(s-)\right)|\mathbb{E}{\int }_{t_k}^t{\left|X_{\Delta }\left(s\right)\right|}^p\mathrm{d}s\hfill \\ \hfill \le & C\mathbb{E}{\int }_{t_k}^t{\left|X_{\Delta }\left(s\right)\right|}^p\mathrm{d}s\hfill \end{array}$$

By Lemma 3 and inequalities (8) and (6), we have

$$\begin{array}{cc}\hfill \mathbb{E}{\int }_0^T|X_{\Delta }\left(s\right)-{\overline{X}}_{\Delta }{\left(s\right)|}^{p/2}{\left|b_{\Delta }({\overline{X}}_{\Delta }\left(s\right),{\Lambda }_{\Delta }\left(s\right))\right|}^{p/2}\mathrm{d}s& \le {\left(g(\Delta )\right)}^{p/2}{\int }_0^T\mathbb{E}\left(\right|X_{\Delta }\left(s\right)-{\overline{X}}_{\Delta }\left(s\right){|}^{p/2})\mathrm{d}s\hfill \\ \hfill & \le {\left(g(\Delta )\right)}^{p/2}{\int }_0^T\left(\mathbb{E}\right|X_{\Delta }\left(s\right)-{\overline{X}}_{\Delta }\left(s\right){{|}^p)}^{1/2}\mathrm{d}s\hfill \\ \hfill & \le c_{p}T{\left(g(\Delta )\right)}^p{\Delta }^{p/4}\le c_{p}T.\hfill \end{array}$$

We therefore have

$$\begin{array}{cc}\hfill \mathbb{E}|X_{\Delta }{\left(t\right)|}^p& \le C+C{\int }_0^t\left(\mathbb{E}|X_{\Delta }{\left(s\right)|}^p+\mathbb{E}{\left|{\overline{X}}_{\Delta }\left(s\right)\right|}^p\right)\mathrm{d}s\hfill \\ \hfill & \le C+C{\int }_0^t\left(\underset{0\le v\le s}{sup}\mathbb{E}{\left|X_{\Delta }\left(v\right)\right|}^p\right)\mathrm{d}s.\hfill \end{array}$$

As this holds for any $t\in [0,T]$, while the right-hand side is non-decreasing in t, we then see

$$\underset{0\le v\le t}{sup}\mathbb{E}{\left|X_{\Delta }\left(v\right)\right|}^p\le C+C{\int }_0^t\left(\underset{0\le v\le s}{sup}\mathbb{E}{\left|X_{\Delta }\left(v\right)\right|}^p\right)\mathrm{d}s.$$

Applying Gronwall’s inequality yields

$$\underset{0\le t\le T}{sup}\mathbb{E}{\left|X_{\Delta }\left(t\right)\right|}^p\le C.$$

As this holds for any $\Delta \in (0,{\Delta }^{*}]$ while C is independent of $\Delta$, we see the required assertion (36). □

Lemma 7. 

Let Assumptions 1 and 2 hold. For any real number $R > |x_0|$ and $\Delta \in (0,{\Delta }^{*})$, define the stopping time

$${\rho }_{\Delta ,R}=inf\{t\ge 0: |X_{\Delta }\left(t\right)|\ge R\}.$$

Then, there exists a constant C which is independent of Δ and R, such that

$$\begin{array}{c}\hfill P({\rho }_{\Delta ,R}\le T)\le {\displaystyle \frac{C}{R^2}}.\end{array}$$

(39)

Proof. 

We simply write ${\rho }_{\Delta ,R}=\rho$. By Itô’s formula, we have that for $0\le t\le T$,

$$\begin{array}{cc}\hfill \mathbb{E}|X_{\Delta }{(t\wedge \rho )|}^2\le & |x_0{|}^2+\mathbb{E}{\int }_0^{t\wedge \rho }\left(2X_{\Delta }^T\left(s\right)b_{\Delta }({\overline{X}}_{\Delta }\left(s\right),{\Lambda }_{\Delta }\left(s\right))+{\left|{\sigma }_{\Delta }({\overline{X}}_{\Delta }\left(s\right),{\Lambda }_{\Delta }\left(s\right))\right|}^2\right)\mathrm{d}s\hfill \\ \hfill & +\mathbb{E}{\int }_0^{t\wedge \rho }{\int }_{{\mathbb{R}}^d\setminus \left\{0\right\}}\left({\left(X_{\Delta }\left(s\right)+c({\overline{X}}_{\Delta }(s-),{\Lambda }_{\Delta }(s-),u)\right)}^2-{\left|X_{\Delta }\left(s\right)\right|}^2\right)\Pi \left(\mathrm{d}u\right)\mathrm{d}s\hfill \\ \hfill =& |x_0{|}^2+\mathbb{E}{\int }_0^{t\wedge \rho }\left(2{\overline{X}}_{\Delta }^T\left(s\right)b_{\Delta }({\overline{X}}_{\Delta }\left(s\right),{\Lambda }_{\Delta }\left(s\right))+{\left|{\sigma }_{\Delta }({\overline{X}}_{\Delta }\left(s\right),{\Lambda }_{\Delta }\left(s\right))\right|}^2\right)\mathrm{d}s\hfill \\ \hfill & +\mathbb{E}{\int }_0^{t\wedge \rho }2{(X_{\Delta }\left(s\right)-{\overline{X}}_{\Delta }\left(s\right))}^T{b}_{\Delta }({\overline{X}}_{\Delta }\left(s\right),{\Lambda }_{\Delta }\left(s\right))\mathrm{d}s\hfill \\ \hfill & +\mathbb{E}{\int }_0^{t\wedge \rho }{\int }_{{\mathbb{R}}^d\setminus \left\{0\right\}}\left(|c({\overline{X}}_{\Delta }(s-),{\Lambda }_{\Delta }(s-),u){|}^2+2|X_{\Delta }\left(s\right)\left|\right|c({\overline{X}}_{\Delta }(s-),{\Lambda }_{\Delta }(s-),u)|\right)\Pi \left(\mathrm{d}u\right)\mathrm{d}s.\hfill \end{array}$$

By Lemma 2, we then derive that

$$\begin{array}{cc}\hfill \mathbb{E}|X_{\Delta }{(t\wedge \rho )|}^2& \le |x_0{|}^2+\mathbb{E}{\int }_0^{t\wedge \rho }2C(1+|{\overline{X}}_{\Delta }{\left(s\right)|}^2)\mathrm{d}s\mathbb{E}{\int }_0^{t\wedge \rho }2|X_{\Delta }\left(s\right)-{\overline{X}}_{\Delta }\left(s\right)\left|\right|b_{\Delta }({\overline{X}}_{\Delta }\left(s\right),{\Lambda }_{\Delta }\left(s\right))|\mathrm{d}s\hfill \\ \hfill & \le |x_0{|}^2+2CT+4C{\int }_0^t\mathbb{E}|X_{\Delta }{(s\wedge \rho )|}^2\mathrm{d}s+4C{\int }_0^T\mathbb{E}{|X_{\Delta }\left(s\right)-{\overline{X}}_{\Delta }\left(s\right)|}^2\mathrm{d}s\hfill \\ \hfill & +2\mathbb{E}{\int }_0^T|X_{\Delta }\left(s\right)-{\overline{X}}_{\Delta }\left(s\right)\left|\right|b_{\Delta }({\overline{X}}_{\Delta }\left(s\right),{\Lambda }_{\Delta }\left(s\right))|\mathrm{d}s.\hfill \end{array}$$

But, by Lemma 3, we have

$${\int }_0^T\mathbb{E}{|X_{\Delta }\left(s\right)-{\overline{X}}_{\Delta }\left(s\right)|}^2\mathrm{d}s\le C,$$

while by Lemma 3 and inequalities (6) and (8), we derive

$$\begin{array}{cc}\hfill & \mathbb{E}{\int }_0^T|X_{\Delta }\left(s\right)-{\overline{X}}_{\Delta }\left(s\right)\left|\right|b_{\Delta }({\overline{X}}_{\Delta }\left(s\right),{\Lambda }_{\Delta }\left(s\right))|\mathrm{d}s\hfill \\ \hfill & \le g(\Delta )\mathbb{E}{\int }_0^T{\left(\mathbb{E}|X_{\Delta }\left(s\right)-{\overline{X}}_{\Delta }{\left(s\right)|}^2\right)}^{1/2}\mathrm{d}s\hfill \\ \hfill & \le Tg(\Delta ){\left(c_p\Delta {\left(g(\Delta )\right)}^2\right)}^{1/2}\hfill \\ \hfill & \le C{\left(g(\Delta ){\Delta }^{1/4}\right)}^2\le C.\hfill \end{array}$$

We hence have

$$\mathbb{E}|X_{\Delta }{(t\wedge \rho )|}^2\le C+4C{\int }_0^t\mathbb{E}{\left|X_{\Delta }(s\wedge \rho )\right|}^2\mathrm{d}s.$$

Gronwall’s inequality shows that

$$\mathbb{E}|X_{\Delta }{(T\wedge \rho )|}^2\le C.$$

This implies the required assertion (39) easily. □

We are now ready to state the main convergence result for $p\ge 2$.

Theorem 2. 

Let Assumptions 1 and 2 hold. Then, for any $2<q\le p,$

$$\underset{\Delta \to 0}{lim}\mathbb{E}|X_{\Delta }{\left(T\right)-X\left(T\right)|}^q=0and\underset{\Delta \to 0}{lim}\mathbb{E}{|{\overline{X}}_{\Delta }\left(T\right)-X\left(T\right)|}^q=0.$$

Proof. 

The proof of this theorem is similar to that of Theorem 3.5 in Mao (2015) [16]; therefore, redundant details are omitted. □

4. Numerical Example

In this section, we discuss a numerical example integrated with music elements to illustrate the theory established in the previous sections. By incorporating music-related parameters and scenario settings, we demonstrate the feasibility and efficiency of computer simulations based on the TEM method for jump-diffusion processes with Markovian switching while highlighting the method’s adaptability in interdisciplinary scenarios.

Example 1. 

Take the dimension $d=1$, and consider the following jump-diffusion model with Markovian switching integrated with music elements:

$$\mathrm{d}X\left(t\right)=b(\Lambda \left(t\right))X\left(t\right)\mathrm{d}t+\sigma ·M\left(t\right)X\left(t\right)\mathrm{d}B\left(t\right)+{\int }_{\mathbb{R}\setminus \left\{0\right\}}\beta (\Lambda \left(t\right))·\gamma \left(u\right)X\left(t^{-}\right)N(\mathrm{d}t,\mathrm{d}u),$$

where $\sigma >0$ is a constant diffusion intensity coefficient; $B\left(t\right)$ is a one-dimensional Brownian motion, describing the continuous random fluctuations of the system; $N(\mathrm{d}t,\mathrm{d}u)$ is a stationary Poisson point process independent of $B\left(t\right)$, and its compensated Poisson random measure is $\tilde{N}(\mathrm{d}t,\mathrm{d}u)=N(\mathrm{d}t,\mathrm{d}u)-\Pi \left(\mathrm{d}u\right)\mathrm{d}t$; and $\Pi (·)$ is a deterministic characteristic measure defined on the measurable space $(\Gamma ,\mathcal{B}(\Gamma \left)\right)$, where $\Gamma$ is a compact set excluding the origin and $\mathcal{B}(\Gamma )$ is the Borel $\sigma$-algebra on $\Gamma$. Since the jump coefficient takes the linear form $c(x,i,u)=\beta \left(i\right)\gamma \left(u\right)x$, Assumption 4 reduces to ${\int \left(\right|1+\beta \left(i\right)\gamma \left(u\right)|}^p-1)\Pi \left(\mathrm{d}u\right)\le \widehat{c}\left(i\right).$ If we set $\gamma \left(u\right)=u$ and $\Pi$ is a Gamma process measure satisfying $\int u^p\Pi \left(\mathrm{d}u\right)<\infty$, then $\widehat{c}\left(i\right)$ can be explicitly constructed via convex inequalities.

$\Lambda \left(t\right)$ is a two-state Markov chain independent of $B\left(t\right)$ and $N(t,A)$ with state space $\mathbb{S}=\{1,2\}$, corresponding to two core states of music: State 1 represents the “melody-dominated mode”, and State 2 represents the “rhythm-dominated mode”. Its Q-matrix is defined as

$$\left(\begin{array}{cc}-q_{12}& q_{12}\\ q_{21}& -q_{21}\end{array}\right),$$

where $q_{12}=0.3$ (switching rate from melody-dominated to rhythm-dominated mode) and $q_{21}=0.2$ (switching rate from rhythm-dominated to melody-dominated mode), which is consistent with the natural transition characteristics of the two modes in music. The drift coefficient is bound to the state: $b\left(1\right)=1$ (in the melody-dominated state, the system mean shows an increasing trend, simulating the driving force of melody progression) and $b\left(2\right)=-0.5$ (in the rhythm-dominated state, the system mean shows a mild decreasing trend, simulating the restrictive effect of rhythmic stability).

$M\left(t\right)$ is the music dynamic intensity factor, whose value depends on the musical segment characteristics corresponding to time t. Let $M\left(t\right)=sin(2\pi ft+\varphi )$, where $f=0.01$ (simulating the basic beat frequency of music) and $\varphi =0$ (initial phase). The value range of $M\left(t\right)$ is $[-1,1]$, which is used to adjust the intensity of the diffusion term and reflect the periodic fluctuation in music intensity over time. $\beta \left(1\right)=0.015$ and $\beta \left(2\right)=0.008$ are the basic jump amplitude coefficients in the two states, respectively. The jump amplitude coefficient in State 1 (melody-dominated) is larger, simulating the sudden pitch changes in the melody, and the jump amplitude coefficient in State 2 (rhythm-dominated) is smaller, simulating the regular pulses of rhythm. $\gamma \left(u\right)$ is the correlation function between jumps and music elements. Let $\Gamma =\{1,2,3\}$ correspond to three typical jump events in music (such as strong beats, weak beats, and grace notes). Define $\gamma \left(1\right)=0.8$ (strong beat jumps with the largest impact), $\gamma \left(2\right)=0.4$ (weak beat jumps with medium impact) and $\gamma \left(3\right)=0.1$ (grace note jumps, with the smallest impact amplitude); the characteristic measure satisfies

$$\Pi \left(\right\{1\left\}\right)=4\times 0.25,\Pi \left(\right\{2\left\}\right)=4\times 0.5,\Pi \left(\right\{3\left\}\right)=4\times 0.25,\mathrm{and}\Pi (\mathbb{R}\setminus \{1,2,3\left\}\right)=0,$$

which is consistent with the occurrence probability distribution of strong beats, weak beats, and grace notes in music.

Select the truncation-related function $g(\Delta )={\Delta }^{-1/4}$. At this time, all conditions in Equation (6) hold for any ${\Delta }^{*}\in (0,1]$. Therefore, the truncation threshold is defined as $f^{-1}\left(g(\Delta )\right)={\Delta }^{-1/20}$, and the truncation functions are:

$$b_{\Delta }(x,i)=\left\{\begin{array}{cc}b\left(i\right)x,\hfill & \left|x\right|\le {\Delta }^{-1/20},\hfill \\ b\left(i\right){\Delta }^{-1/20}·{\displaystyle \frac{x}{\left|x\right|}},\hfill & \left|x\right|>{\Delta }^{-1/20},\hfill \end{array}\right.$$

$${\sigma }_{\Delta }(x,i)=\left\{\begin{array}{cc}\sigma ·M\left(t\right)x,\hfill & \left|x\right|\le {\Delta }^{-1/20},\hfill \\ \sigma ·M\left(t\right){\Delta }^{-1/20}·{\displaystyle \frac{x}{\left|x\right|}},\hfill & \left|x\right|>{\Delta }^{-1/20}.\hfill \end{array}\right.$$

The initial value is set to $x_0=1$. For different time step sizes $\Delta$ and total duration $T=100$ (simulating the duration of a complete piece of music), the numerical solution $X_{k+1}$ is calculated by the following recursive formula:

$$X_{\Delta }\left(t_{k+1}\right)=X_{\Delta }\left(t_k\right)+b_{\Delta }(X_{\Delta }\left(t_k\right),\Lambda \left(t_k\right))\Delta +{\sigma }_{\Delta }(X_{\Delta }\left(t_k\right),\Lambda \left(t_k\right))\Delta B_k+\sum _{j:N(\Delta t_k,\Delta t_{k+1}]\times \left\{u\right\}\ne \varnothing }\beta (\Lambda \left(t_k\right))\gamma \left(u\right)X_{\Delta }\left(t_k^{-}\right),$$

where $\Delta t_k=t_{k+1}-t_k=\Delta$ and $\Delta B_k=B\left(t_{k+1}\right)-B\left(t_k\right)$.

Based on the above settings, numerical simulations are performed using the TEM method, and the following results are obtained.

1. Figure 1 ($\Delta =0.001$) shows the trajectory of the numerical solution when the step size is extremely small. Since the step size is small enough, although the truncation threshold ${\Delta }^{-1/20}$ is large, the truncation operation is rarely triggered, and the numerical solution is almost equivalent to the result of the standard EM method. In the trajectory, the combination of the periodic fluctuation brought by $M\left(t\right)$ and the Markov state switching (melody/rhythm dominance) makes the trajectory show the characteristics of increasing fluctuation in the melody segment and stable fluctuation in the rhythm segment. It can accurately capture the time nodes and impact amplitudes of jump events (strong beats/weak beats/grace notes), which is highly consistent with the system dynamics driven by real music, and the trajectory is smooth and consistent with the continuous characteristics of the theoretical solution.

2. Figure 2 ($\Delta =0.01$) shows the trajectory of the numerical solution when the step size is relatively large. Due to the increased step size, the capture accuracy of Markov state switching decreases, the periodic characteristics of $M\left(t\right)$ are not fully reflected, and the time positioning of jump events is deviated, resulting in more intense fluctuations and obvious deviations from the theoretical solution.

5. Conclusions and Discussion

The core idea of the TEM method is to control the explosive growth of numerical solutions through reasonable truncation, which is still applicable in models integrated with music elements.

When the step size $\Delta$ is small enough, the truncation operation is rarely triggered. The method can accurately restore the impact of music elements (beat fluctuation, state switching, and jump events) on the system, and the numerical solution converges to the true solution of the stochastic differential equation. When the step size increases, the fluctuation and deviation of the numerical solution intensify, which verifies the importance of step size selection for the numerical simulation of jump-diffusion processes containing dynamic music elements.

This example shows that the TEM method is not only applicable to traditional jump-diffusion processes with Markovian switching but can also effectively adapt to extended models integrated with interdisciplinary elements such as music, providing a reliable numerical tool for fields such as music signal processing and music-driven dynamic system analysis.