The rate of $L^{p}$-convergence for the Euler-Maruyama method of the stochastic differential equations with Markovian switching ¹¹1This research was supported by NSFC (Grant No.12071101 and No.11671113).

Stochastic differential equations with Markovian switching (SDEwMSs), also known as hybrid stochastic differential equations, play an important role in stochastic theory and have been used in various fields, such as the theory of control and neural networks ([1, 2, 3]). Most of SDEwMSs do not have explicit solutions so it is important to have numerical solutions ([4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15]). [4] is the first research that developed the Euler-Maruyama (EM) scheme for SDEwMSs with the global Lipschitz continuous coefficients and considered the $L^{2}$-convergence rate for EM solutions. [6] designed approximation methods of Milstein type for SDEwMSs and proved the convergence rate is better than the generally adopted EM procedures.

The primary motivation for this work came from the following observation: to our knowledge, there are plenty of papers on the convergence of numerical algorithms for hybrid systems, most of them showed the convergence (without order)(e.g., [10, 11, 12, 13, 14, 15, 16]) or the rate of convergence in the sense of pathwise or mean square (e.g., [4, 5, 6, 7, 8, 9, 17, 18]). However, there are only a few papers revealed the $L^{p}$-convergence order of numerical methods for hybrid systems ([19, 20]). Not only that, the order of $L^{p}$-convergence for Euler type numerical algorithms proved in these papers are no more than $1/p~{}(p\geq 2)$, instead of the well known $1/2$. To be specific, the main result in [19] (Theorem 3) shows

where $x(t)$ is the exact solution of the stochastic delay differential equation with phase semi-Markovian switching and Poisson jumps, $y(t)$ denotes the numerical approximation using the continuous $\theta$ method, $\operatorname{\Delta}$ is the given step-size, $C_{5}$ denotes a generic constant that independent of $\operatorname{\Delta}$, $x_{0}$ is the initial data. By analyzing the details in this paper, we find that the problem first appears in the estimations of

and

(Lemma 4 in [19]), we think these two terms can be seen as the errors in approximating $r(s)$ by $r_{1}(s)$ and $r_{2}(s)$, where $r(s)$ is the given continuous-time Markov chain. Similar estimations also exist in many works aforementioned, for example, Eq.(3.7) in [4], Lemma 3 in [10], Eq.(3.6) in [11], as well as Corollary 3.1 in [20], etc. Based on this fact, the main idea of this work is to use $r(s)$ itself to construct a numerical scheme, rather than its approximation. Therefore, the innovations of this paper are as follows:

• •

  We will use the continuous-time Markov chain itself to develop the numerical scheme, instead of its approximation.

• •

  The order of $L^{p}$-convergence for the EM method given in this work to SDEwMSs can reach $1/2$.

The rest of the paper is arranged as follows. In Section 2, we present some notations and fundamental assumptions, moreover, we further introduce the classical EM method for SDEwMSs which is often used in literatures. Then we develop a different EM scheme in Section 3. The rate of $L^{p}$-convergence for the EM method be proved in Section 4. Finally, we give the conclusion of this paper in Section 5.

In the rest of this work, except as otherwise noted, we let $(\Omega,\mathcal{F},\left\{\mathcal{F}_{t}\right\}_{t\geq 0},\mathbb{P})$ be a complete probability space with a filtration $\left\{\mathcal{F}_{t}\right\}_{t\geq 0}$ which satisfies the general conditions (namely, it is right continuous and $\mathcal{F}_{0}$ involves all $\mathbb{P}$-null sets). The transpose of $A$ is denoted by $A^{\rm T}$ when $A$ is a vector or matrix. $B(t)=(B_{1}(t),\dots,B_{d}(t))^{\rm T}$ represents a $d$-dimensional Brownian motion defined on the $(\Omega,\mathcal{F},\left\{\mathcal{F}_{t}\right\}_{t\geq 0},\mathbb{P})$. If $x$ is a vector, $|x|$ denotes its Euclidean norm. $|A|=\sqrt{\operatorname{trace}(A^{\rm T}A)}$ denotes the trace norm of a matrix $A$. If $u$ and $v$ are two real numbers, let $u\vee v$ and $u\wedge v$ be $\max\left\{u,v\right\}$ and $\min\left\{u,v\right\}$, respectively. Let $\mathcal{L}^{p}([a,b];\mathbb{R}^{n})$ be the family of $\mathbb{R}^{n}$-valued processes $\{f(t)\}_{a\leq t\leq b}$ which satisfies $\mathcal{F}_{t}$-adapted and $\int_{a}^{b}|f(t)|^{p}{\rm{d}}t<\infty$, a.s. $\mathcal{L}^{p}(\mathbb{R}_{+};\mathbb{R}^{n})$ denotes the family of processes $\{f(t)\}_{t\geq 0}$ such that $\{f(t)\}_{0\leq t\leq T}\in\mathcal{L}^{p}([0,T];\mathbb{R}^{n})$ for any $T>0$. In this paper, we use $C$ represents a common positive number independent of $\Delta$, its value may vary with each appearance.

Suppose $\alpha(t),t\geq 0,$ is a right-continuous Markov chain taking values in $S=\left\{1,2,\dots,N\right\}$ with generator $\Gamma=(\gamma_{ij})_{N\times N}$, $\gamma_{ij}\geq 0$ denotes the transition rate from $i$ to $j$ when $i\neq j$, and

Assuming that $\alpha(\cdot)$ is independent of the Brownian motion $B(\cdot)$.

Let $T>0$, consider the following SDEwMS

on $t\in[0,T]$, where

We impose the following conditions:

In the following, we will first introduce the classical methods for simulating discrete-time Markov chain that have been used in many papers, and further present the well known EM method. In the next section, we will introduce the method used to simulate Markov chain in this paper, and further construct another type of EM method for SDEwMS which is different from the one given in the Ref.[4].

For the generation of the Markov chain $\alpha(t)$, we cite the methodology of formulating the Markov chain from Section 2.4 in [22]. In order to get the sample paths of $\alpha(t)$, we need to determine the time of residence in each state and the succeeding actions. The chain remains at any given state $i_{0}(i_{0}\in S)$ for a random length of time, $\tau_{1}$, which follows an exponential distribution with parameter $-\gamma_{i_{0}i_{0}}$, hence $\tau_{1}$ can be obtained by

where $\zeta_{1}$ is a random variable uniformly distributed in $(0,1)$. Then, the process will enter another state. In addition, the probability that state $j$ (with $j\in S,j\neq i_{0}$) becomes the next residence of the chain is $\gamma_{i_{0}j}/(-\gamma_{i_{0}i_{0}})$. The position after the jump is determined by a discrete random variable $i_{1}(i_{1}\in S\setminus\{i_{0}\})$, namely $\alpha(\tau_{1})=i_{1}$. The value of $i_{1}$ is given by

where $\xi_{1}$ is a random variable uniformly distributed in $(0,1)$.

The chain remains at state $i_{1}$ for a random length of time, $\tau_{2}$ , which follows an exponential distribution with parameter $-\gamma_{i_{1}i_{1}}$, thus

where $\zeta_{2}$ is also a random variable uniformly distributed in $(0,1)$. Then, the process will enter another state. The post-jump location is identified by a discrete random variable $i_{2}(i_{2}\in S\setminus\{i_{1}\})$, which implies $\alpha(\tau_{1}+\tau_{2})=i_{2}$. The value of $i_{2}$ is determined by

where $\xi_{2}$ is a random variable uniformly distributed in $(0,1)$. Therefore, repeating the procedure above, the sampling path of $\alpha(t),t\geq 0$ is composed of exponential random variables and $U(0,1)$ random variables alternately.

Recall that nearly all sample paths of $\alpha(\cdot)$ are right-continuous piecewise constant function with finite sample jumps in $[0,T]$. Thus, there are stopping times $0=\bar{\tau}_{0}<\bar{\tau}_{1}<\bar{\tau}_{2}<\cdots<\bar{\tau}_{\bar{N}}=T~{}(\bar{N}\in\mathbb{N}_{+})$, where $\bar{\tau}_{k}=\sum_{j=1}^{k}\tau_{j},k=1,2,\dots,\bar{N}-1$, such that

Now we are in a position to define the EM method to SDEwMS (2). Given a step size $\Delta>0$, let $t_{k}=k\Delta$ $(k\in\mathbb{N})$ be the gridpoints.

Define

According to the definition of $J_{i}$, it is easy to know that

where $\#\{J_{i}\}$ denotes the number of elements in set $\{J_{i}\}$. Then we define the EM method to (2) of the following type by setting $Z_{0}=z(0)=z_{0}$,

for $k\in\mathbb{N}$, where $\operatorname{\Delta}J_{i}=J_{i+1}-J_{i},\Delta B_{J_{i}}=B(J_{i+1})-B(J_{i})$. $Z_{k}$ is the approximate value of $z(t_{k})$. Let

the continuous EM method is given by

It can be verified that $Z(t_{k})=\bar{Z}(t_{k})=Z_{k},~{}\forall k\geq 0$.

Similar to Lemma 4.1 in [23], we can easily obtain the following conclusion.

The proof is omitted because it is similar to that for Lemma 4.1 in [23].

In this paper, we develop the EM scheme, which is different from the one given in the references (such as [4, 10]), to generate the approximate solutions to a class of SDEwMSs, and analyze the order of the errors in the $L^{p}$-sense. It has been proved that the $L^{p}$-convergence rate of the EM scheme given in this paper for SDEwMSs can reach $1/2$. We further point out that the techniques used in this paper to construct the EM method can also be used to construct other schemes for hybrid systems, such as stochastic theta method, tamed EM method, and Milstein method, etc. We believe that this approach will also contribute to the $L^{p}$-convergence order of these numerical methods for hybrid systems.