Marcus Stochastic Differential Equations: Representation of Probability Density

Abstract

Marcus stochastic delay differential equations are often used to model stochastic dynamical systems with memory in science and engineering. It is challenging to study the existence, uniqueness, and probability density of Marcus stochastic delay differential equations, due to the fact that the delays cause very complicated correction terms. In this paper, we identify Marcus stochastic delay differential equations with some Marcus stochastic differential equations without delays but subject to extra constraints. This helps us to obtain the following two main results: (i) we establish a sufficient condition for the existence and uniqueness of the solution to the Marcus delay differential equations; and (ii) we establish a representation formula for the probability density of the Marcus stochastic delay differential equations. In the representation formula, the probability density for Marcus stochastic differential equations with memory is analytically expressed in terms of probability density for the corresponding Marcus stochastic differential equations without memory.

1. Introduction

Stochastic differential equations (SDEs) driven by Lévy processes are widely used to model stochastic dynamical systems perturbed by non-Gaussian white noises. While Itô SDEs driven by Lévy processes are widely applied in biology and finance [1,2], Marcus SDEs [3,4,5], also known as SDEs with DiPaola–Falsone correction, are more appropriate models in physics and engineering [6,7,8]. To study the propagation and evolution of uncertainty in stochastic dynamical systems, it is essential to study the dynamics of the probability density, which contains all the statistical information about the system. For SDEs without delays, the dynamical behavior of the probability density is governed by Fokker–Planck equations. Fokker–Planck equations corresponding to Marcus SDEs driven by Lévy processes are derived in [7]. Other forms of Fokker–Planck equations, such as fractional Fokker–Planck equations, have been extensively studied in recent years; see [9,10], among others.

Stochastic delay differential equations (SDDEs) are appropriate models for some stochastic dynamical systems with memory; i.e., the future states of the system depend not only on the current but also the past states. Itô SDDEs have been well studied. Sufficient conditions for the existence and uniqueness of the solutions to Itô SDDEs can be found in [11,12,13]. Probability densities for Itô SDDEs have been investigated by many authors; see [14,15,16], among others. It is much more challenging to study Marcus SDDEs than their Itô counterparts. The main difficulty lies in the fact that the time delays, especially those appearing in diffusion terms, cause very complicated correction terms in Marcus SDDEs, which make conventional techniques for Itô SDDEs no longer effective for Marcus SDDEs. Existing studies on Marcus SDDEs mostly focus on cases where the time delay appears only in drift terms. The existence and uniqueness of the solution to Marcus SDDEs without delays in diffusion terms are established in Theorem 6.2 in [17].

The objective of this paper is twofold: (i) to establish the existence and uniqueness of the solutions to general Marcus SDDEs driven by Lévy processes; and (ii) to derive a representation formula for the probability density associated with these Marcus SDDEs. This paper is organized as follows. In Section 2, we present some preliminary results that will be used in later sections. We deal with the above objectives (i) and (ii) in Section 3 and Section 4, respectively. Section 5 is the conclusion. We only consider Marcus SDDEs in this paper. Note that there are other models, such as stochastic integral equations with delay [18,19,20] that have important applications in random control but will not be considered here.

2. Some Preliminary Results

First, we introduce some notations for vectors. Throughout this paper, each element in ${\mathbb{R}}^d$ is represented as a column vector. Given $x_1,x_2,\cdots ,x_k\in {\mathbb{R}}^d$, then ${\left(x_1^T,x_2^T,\cdots ,x_k^T\right)}^T$ is a column vector in ${\mathbb{R}}^{kd}$. Here ${\{·\}}^T$ represents the transpose of $\{·\}$.

Now, we review the definition of Marcus SDEs driven by Lévy processes and without delays. Consider

$$\begin{array}{c}\hfill \begin{array}{c}\hfill \left\{\begin{array}{c}\mathrm{d}Z\left(t\right)=\alpha \left(Z\right(t),t)\mathrm{d}t+\beta \left(Z\right(t-),t)\diamond \mathrm{d}L\left(t\right),\mathrm{for}t>0,\hfill \\ Z\left(0\right)=z_0\in {\mathbb{R}}^d,\hfill \end{array}\right.\end{array}\end{array}$$

(1)

where $Z\left(t\right)\in {\mathbb{R}}^d$, $\alpha :{\mathbb{R}}^d\times {\mathbb{R}}^{+}\to {\mathbb{R}}^d,(x,t)\to \alpha (x,t)$, $\beta :{\mathbb{R}}^d\times {\mathbb{R}}^{+}\to {\mathcal{M}}^{d\times n},(x,t)\to \beta (x,t)={\left({\beta }_{ij}\right)}_{d\times n}$, and $L\left(t\right)\in {\mathbb{R}}^n$. Here, ${\mathcal{M}}^{d\times n}$ is the set of all d-by-n real matrix.

To understand the Marcus canonical integral $“\diamond ”$, we need the following prerequisite knowledge.

By Lévy–It$\widehat{\mathrm{o}}$ decomposition [1], the Lévy process $L\left(t\right)$ can be expressed as

$$\begin{array}{c}\hfill L\left(t\right)=bt+B_A\left(t\right)+{\int }_{\parallel y\parallel <1}y\tilde{N}(t,\mathrm{d}y)+{\int }_{\parallel y\parallel \ge 1}yN(t,\mathrm{d}y),\end{array}$$

(2)

where $b\in {\mathbb{R}}^n$ is a drift vector, $B_A\left(t\right)$ is the n-dimensional Brownian motion with the covariance matrix A, $\parallel y\parallel =\sqrt{y_1^2+y_2^2+\cdots +y_d^2}$ is the usual Euclidean norm of y, and $N(t,\mathrm{d}y)$ is the Poisson random measure defined as

$$\begin{array}{c}\hfill N(t,S)\left(\omega \right)=\#\left\{s|0\le s\le t;\Delta L(s\left)\right(\omega )\in S\right\},\end{array}$$

(3)

with $\#\{·\}$ representing the number of elements in the set $“·”$. S is the Borel set in $\mathcal{B}\left({\mathbb{R}}^d\setminus \left\{0\right\}\right)$, $\Delta L\left(s\right)$ is the jump of $L\left(s\right)$ at time s defined as $\Delta L\left(s\right)=L\left(s\right)-L(s-)$, and $\tilde{N}(\mathrm{d}t,\mathrm{d}y)$ is the compensated Poisson measure defined as $\tilde{N}(\mathrm{d}t,\mathrm{d}y)=N(\mathrm{d}t,\mathrm{d}y)-\mathrm{d}t\nu \left(\mathrm{d}y\right)$.

Note that it is convenient to write the n-dimensional Brownian motion $B_A\left(t\right)$ in the form of $B_A\left(t\right)=\sigma B\left(t\right)$ [1], where $B\left(t\right)$ is a standard n-dimensional Brownian motion and $\sigma$ is an $n\times n$ nonzero matrix for which $A=\sigma {\sigma }^T$, and $B_A^i\left(t\right)={\sum }_{j=1}^n{\sigma }_{ij}{B}_j\left(t\right)$ for $i=1,2,\cdots ,n$.

To proceed, the following definition is needed.

Definition 1.

For $u={(u_1,u_2,\cdots ,u_d)}^T\in {\mathbb{R}}^d$, $v={(v_1,v_2,\cdots ,v_n)}^T\in {\mathbb{R}}^n$, and $t\in {\mathbb{R}}^{+}$, the mapping H is defined by

$$\begin{array}{c}\hfill H:{\mathbb{R}}^d\times {\mathbb{R}}^n\times {\mathbb{R}}^{+}\to {\mathbb{R}}^d,(u,v,t)\mapsto H(u,v,t)={\mathsf{\Psi }}_t\left(1\right),\end{array}$$

(4)

where ${\mathsf{\Psi }}_t:\mathbb{R}\to {\mathbb{R}}^d,r\mapsto {\mathsf{\Psi }}_t\left(r\right)$ is the solution of the following ordinary differential equation (ODE) with parametert,

$$\begin{array}{c}\hfill {\displaystyle \frac{\mathrm{d}{\mathsf{\Psi }}_t\left(r\right)}{\mathrm{d}r}}=\beta \left({\mathsf{\Psi }}_t\left(r\right),t\right)v,{\mathsf{\Psi }}_t\left(0\right)=u.\end{array}$$

(5)

The ${\mathbb{R}}^d$-valued strong solution $Z\left(t\right)$ of Marcus SDE (1) is defined as

$$\begin{array}{c}\hfill Z\left(t\right)=z_0+{\int }_0^t\alpha \left(Z\left(s\right),s\right)\mathrm{d}s+{\int }_0^t\beta \left(Z(s-),s\right)\diamond \mathrm{d}L\left(s\right),\end{array}$$

(6)

where the left limit $Z(s-)=\underset{u<s,u\to s}{\lim }Z\left(u\right)$ and $“\diamond ”$ indicates the Marcus canonical integral defined below. For each $t\ge 0$,

$$\begin{array}{c}\hfill \begin{array}{cc}\hfill {\int }_0^t\beta \left(Z(s-),s\right)\diamond \mathrm{d}L\left(s\right)& ={\int }_0^t\beta \left(Z(s-),s\right)\mathrm{d}L\left(s\right)+{\int }_0^{t}C(Z(s-),s)\mathrm{d}t\hfill \\ \hfill & +\sum _{0\le s\le t}\left[H\left(Z\right(s-),\Delta L(s),s)-Z(s-)-\beta \left(Z\right(s-),s)\Delta L\left(s\right)\right],\hfill \end{array}\end{array}$$

(7)

where $C\left(Z\right(s-),s)$ is a vector in ${\mathbb{R}}^d$ with the i-th element ($i=1,2,\cdots ,d$) as

$$\begin{array}{c}\hfill \begin{array}{cc}\hfill C_i(Z(s-),s)=& {\displaystyle \frac{1}{2}}\sum _{m=1}^d\sum _{j=1}^n\sum _{l=1}^n{\beta }_{ml}\left(Z(s-),s\right){\displaystyle \frac{\partial }{\partial x_m}}{\beta }_{ij}\left(Z(s-),s\right)A_{lj}\hfill \\ \hfill =& {\displaystyle \frac{1}{2}}\sum _{j=1}^n\sum _{l=1}^n{\left({\beta }^{\prime }\beta \right)}_{ijl}\left(Z(s-),s\right)A_{lj},\hfill \end{array}\end{array}$$

(8)

where ${\beta }^{\prime }\beta :{\mathbb{R}}^d\times {\mathbb{R}}^{+}\to {\mathbb{R}}^{d\times d\times n}$ is defined as

$$\begin{array}{c}\hfill {\left({\beta }^{\prime }\beta \right)}_{ijl}(x,t)=\sum _{m=1}^d{\beta }_{ml}(x,t){\displaystyle \frac{\partial }{\partial x_m}}{\beta }_{ij}(x,t).\end{array}$$

(9)

Remark 1.

As shown in [5], the Marcus integral operation $“\diamond ”$ satisfies the chain rule of classical calculus, and can be obtained as the limit of Wong–Zakai approximation.

The following Lemma, proof of which is given in Appendix A, will be used in later sections.

Lemma 1.

Suppose that for all $x\in {\mathbb{R}}^d$, the coefficients $\alpha (x,t)$, $\beta (x,t)$ and ${\beta }^{\prime }\beta (x,t)$ satisfy the following two conditions:

(i)

continuous with respect to t;

(ii)

globally Lipschitz continuous with respect to x, i.e., for all $x_1,x_2\in {\mathbb{R}}^d$ and $t\in {\mathbb{R}}^{+}$,

$$\begin{array}{c}\hfill \begin{array}{cc}\hfill & \parallel \alpha (x_1,t)-\alpha (x_2,t)\parallel +\parallel \beta (x_1,t)-\beta (x_2,t)\parallel +\parallel {\beta }^{\prime }\beta (x_1,t)-{\beta }^{\prime }\beta (x_2,t)\parallel \le K\left(t\right)\parallel x_1-x_2\parallel ,\hfill \end{array}\end{array}$$

(10)

where $K:{\mathbb{R}}^{+}\to {\mathbb{R}}^{+},t\to K\left(t\right)$ is locally bounded and measurable.

Then, there exists a unique strong solution to Marcus SDE (1).

If we consider Marcus SDE (1) on the time interval $[0,T]$, the following Lemma naturally follows from Lemma 1.

Lemma 2.

Consider Marcus SDE (1) on the time interval $[0,T]$. Suppose that for all $x\in {\mathbb{R}}^d$, the coefficients $\alpha (x,t)$, $\beta (x,t)$ and ${\beta }^{\prime }\beta (x,t)$ satisfy the following two conditions:

(i)

continuous with respect to t on the time interval $[0,T]$;

(ii)

globally Lipschitz continuous with respect to x, i.e., for all $x_1,x_2\in {\mathbb{R}}^d$ and $t\in [0,T]$,

$$\begin{array}{c}\hfill \begin{array}{cc}\hfill & \parallel \alpha (x_1,t)-\alpha (x_2,t)\parallel +\parallel \beta (x_1,t)-\beta (x_2,t)\parallel +\parallel {\beta }^{\prime }\beta (x_1,t)-{\beta }^{\prime }\beta (x_2,t)\parallel \le K\parallel x_1-x_2\parallel ,\hfill \end{array}\end{array}$$

(11)

where K is a constant.

Then, there exists a unique strong solution to Marcus SDE (1) on the time interval $[0,T]$.

3. Existence and Uniqueness for Marcus SDDEs

Consider the following Marcus SDDE,

$$\begin{array}{c}\hfill \begin{array}{c}\hfill \left\{\begin{array}{c}\mathrm{d}X\left(t\right)=f\left(X\right(t),X(t-\tau \left)\right)\mathrm{d}t+g\left(X\right(t),X(t-\tau \left)\right)\diamond \mathrm{d}L\left(t\right),\mathrm{for}t>0,\hfill \\ X\left(t\right)=\gamma \left(t\right),t\in [-\tau ,0],\hfill \end{array}\right.\end{array}\end{array}$$

(12)

where $X\left(t\right)$ is a ${\mathbb{R}}^d$-valued stochastic process, $L\left(t\right)$ is a ${\mathbb{R}}^n$-valued Lévy process defined on a probability space $(\mathsf{\Omega },\mathcal{F},P)$, $\tau \in {\mathbb{R}}^{+}$ is the time delay, $f:{\mathbb{R}}^d\times {\mathbb{R}}^d\to {\mathbb{R}}^d$, $g={\left(g_{ij}\right)}_{d\times n}:{\mathbb{R}}^d\times {\mathbb{R}}^d\to {\mathcal{M}}^{d\times n}$, and $\gamma :[-\tau ,0]\to {\mathbb{R}}^d$.

The “⋄” in (12), as stated before, can be obtained by Wong–Zakai approximation. However, unlike in (1), there seems to be no neat expression for “⋄” in (12) due to the delay $\tau$ appearing in the diffusion coefficient $g\left(X\right(t),X(t-\tau \left)\right)$. We circumvent this problem by associating Equation (12) with the following Marcus SDE without delays,

$$\begin{array}{c}\hfill \left\{\begin{array}{c}\mathrm{d}{X}_1\left(t^{\prime }\right)=f\left(X_1\left(t^{\prime }\right),\gamma (t^{\prime }-\tau )\right)\mathrm{d}{t}^{\prime }+g\left(X_1\left(t^{\prime }\right),\gamma (t^{\prime }-\tau )\right)\diamond \mathrm{d}{L}_1\left(t^{\prime }\right),\hfill \\ \mathrm{d}{X}_2\left(t^{\prime }\right)=f\left(X_2\left(t^{\prime }\right),X_1\left(t^{\prime }\right)\right)\mathrm{d}{t}^{\prime }+g\left(X_2\left(t^{\prime }\right),X_1\left(t^{\prime }\right)\right)\diamond \mathrm{d}{L}_2\left(t^{\prime }\right),\mathrm{for}{t}^{\prime }\in (0,\tau ],\hfill \\ ⋮⋮⋮\hfill \\ \mathrm{d}{X}_k\left(t^{\prime }\right)=f\left(X_k\left(t^{\prime }\right),X_{k-1}\left(t^{\prime }\right)\right)\mathrm{d}{t}^{\prime }+g\left(X_k\left(t^{\prime }\right),X_{k-1}\left(t^{\prime }\right)\right)\diamond \mathrm{d}{L}_k\left(t^{\prime }\right),\hfill \end{array}\right.\end{array}$$

(13)

constrained by the condition that the initial value of $X_1\left(t^{\prime }\right)$ is prescribed, and the final value of $X_i\left(t^{\prime }\right)$ is set to be equal to the initial value of $X_{i+1}\left(t^{\prime }\right)$ for $i=1,2,\cdots ,k-1$, i.e.,

$$X_1\left(0\right)={\gamma }_0\mathrm{and}{X}_i\left(\tau \right)=X_{i+1}\left(0\right).$$

(14)

In (13) and (14), $X_i\left(t^{\prime }\right)\in {\mathbb{R}}^d(i=1,2,\cdots ,k)$ and ${\gamma }_0=\gamma \left(0\right)$ is a constant in ${\mathbb{R}}^d$. Note that condition (14) is different from the conventional initial condition

$$\begin{array}{c}\hfill {\left(X_1^T\left(0\right),X_2^T\left(0\right),\cdots ,X_k^T\left(0\right)\right)}^T={\mathit{x}}_0,\end{array}$$

(15)

where ${\mathit{x}}_0\in {\mathbb{R}}^{kd}$ is a constant. Equation (13) under condition (15), which is component-wise, can also be rewritten in the form of (1), but with higher dimensionality.

Remark 2.

The “⋄” in Marcus SDDE (12) can be equivalently interpreted by the “⋄” in Equation (13), a Marcus SDE without delays. Therefore, the “⋄” for Marcus SDDE is essentially the same as that for Marcus SDE without delays.

Equation (13) under condition (14) can be converted from (12) in the way given below. For each solution to Marcus SDDE (12), we can obtain a solution to (13) and (14) by constructing $X_i\left(t^{\prime }\right)=X(t^{\prime }+(i-1)\tau )$, $L_i\left(t^{\prime }\right)=L(t^{\prime }+(i-1)\tau )-L\left((i-1)\tau \right)$ for $i=1,2,\cdots ,k$. It is straightforward to check that the path of $X_i\left(t^{\prime }\right)$ with $t^{\prime }\in (0,\tau ]$ coincides with the path of $X\left(t\right)$ with $t\in \left(\right(i-1)\tau ,i\tau ]$. Therefore, Marcus SDDE (12) is equivalent to Marcus SDE (13) under condition (14) in the following sense,

$$\begin{array}{c}\hfill X\left(t\right)\stackrel{a.s.}{=}\left\{\begin{array}{cc}{X}_1\left(t\right),\hfill & \mathrm{for}t\in (0,\tau ],\hfill \\ X_2(t-\tau ),\hfill & \mathrm{for}t\in (\tau ,2\tau ],\hfill \\ ⋮\hfill & ⋮\hfill \\ X_k(t-(k-1)\tau ),\hfill & \mathrm{for}t\in \left(\right(k-1)\tau ,k\tau ],\hfill \end{array}\right.\end{array}$$

(16)

or, equivalently,

$$\begin{array}{c}\hfill X_i\left(t^{\prime }\right)\stackrel{\mathrm{a}.\mathrm{s}.}{=}X(t^{\prime }+(i-1)\tau )\mathrm{for}{t}^{\prime }\in (0,\tau ]\mathrm{and}i=1,2,\cdots ,k.\end{array}$$

(17)

The solution to Marcus SDDE (12) can be interpreted by Marcus SDE (13) under condition (14). We are now ready to establish the existence and uniqueness of the solution to (12).

We first introduce some notations that will be used later. Denote ${\mathsf{\Phi }}_k({\gamma }_0,t^{\prime })$, with ${\mathsf{\Phi }}_k:{\mathbb{R}}^d\times [0,\tau ]\to {\mathbb{R}}^{kd},$ $\left({\gamma }_0,t^{\prime }\right)\to {\mathsf{\Phi }}_k({\gamma }_0,t^{\prime })={\left(X_1^T\left(t^{\prime }\right),X_2^T\left(t^{\prime }\right),\cdots ,X_k^T\left(t^{\prime }\right)\right)}^T$, as the solution at time $t^{\prime }$ to (13) under condition (14). Examining the recursive structure of (13) and (14), it is straightforward to check that for $k\ge 2$, ${\mathsf{\Phi }}_{k-1}({\gamma }_0,t^{\prime })$ coincides with the first $k-1$ components of ${\mathsf{\Phi }}_k({\gamma }_0,t^{\prime })$.

Assumption 1.

Suppose that $\gamma \left(t\right)$ is continuous on $[-\tau ,0]$, and the coefficients f, g, and $g^{\prime }g$ are globally Lipschitz, i.e., for all $x_1,x_2,y_1,y_2\in {\mathbb{R}}^d$,

$$\begin{array}{c}\hfill \begin{array}{c}\hfill \parallel f(x_1,y_1)-f(x_2,y_2)\parallel +\parallel g(x_1,y_1)-g(x_2,y_2)\parallel +\parallel g^{\prime }g(x_1,y_1)-g^{\prime }g(x_2,y_2)\parallel \\ \hfill \le C_1\parallel x_1-x_2\parallel +C_2\parallel y_1-y_2\parallel ,\end{array}\end{array}$$

(18)

where $C_1$ and $C_2$ are constants.

Theorem 1

(Existence and Uniqueness for Marcus SDDE). Under Assumption 1, there exists a unique strong solution to Marcus SDDE (12).

Remark 3.

Theorem 1 here is different from Theorem 6.2 in the literature [17]. While the former is applicable for a general Marcus SDDE (12) which includes delays in both drift and diffusion terms, the latter is applicable for special Marcus SDDEs with delays appearing only in the drift terms.

Proof of Theorem 1.

According to Equations (16) and (17), we only need to show that for any given $k\in \mathbb{N}$, there is a unique strong solution to (13) under condition (14). We shall finish the proof by induction.

First, we argue that the conclusion is true for $k=1$. In fact, for $k=1$, Marcus SDE (13) and condition (14) become

$$\begin{array}{c}\hfill \left\{\begin{array}{c}\mathrm{d}{X}_1\left(t^{\prime }\right)=F_1\left(X_1\left(t^{\prime }\right),t^{\prime }\right)\mathrm{d}{t}^{\prime }+G_1\left(X_1\left(t^{\prime }\right),t^{\prime }\right)\diamond \mathrm{d}{L}_1\left(t^{\prime }\right),t^{\prime }\in (0,\tau ],\hfill \\ X_1\left(0\right)={\gamma }_0,\hfill \end{array}\right.\end{array}$$

(19)

where $F_1\left(X_1\left(t^{\prime }\right),t^{\prime }\right)=f(X_1\left(t^{\prime }\right),\gamma (t^{\prime }-\tau ))$, $G_1\left(X_1\left(t^{\prime }\right),t^{\prime }\right)=g(X_1\left(t^{\prime }\right),\gamma (t^{\prime }-\tau ))$. It is straightforward to check that $F_1$ and $G_1$ satisfy the conditions in Lemma 2, therefore, (19) has a unique strong solution by Lemma 2.

Suppose that the conclusion is true for $k=i$, i.e., Marcus SDE

$$\begin{array}{c}\hfill \left\{\begin{array}{c}\mathrm{d}{X}_1\left(t^{\prime }\right)=f\left(X_1\left(t^{\prime }\right),\gamma (t^{\prime }-\tau )\right)\mathrm{d}{t}^{\prime }+g\left(X_1\left(t^{\prime }\right),\gamma (t^{\prime }-\tau )\right)\diamond \mathrm{d}{L}_1\left(t^{\prime }\right),\hfill \\ \mathrm{d}{X}_2\left(t^{\prime }\right)=f\left(X_2\left(t^{\prime }\right),X_1\left(t^{\prime }\right)\right)\mathrm{d}{t}^{\prime }+g\left(X_2\left(t^{\prime }\right),X_1\left(t^{\prime }\right)\right)\diamond \mathrm{d}{L}_2\left(t^{\prime }\right),\hfill \\ ⋮⋮⋮\hfill \\ \mathrm{d}{X}_i\left(t^{\prime }\right)=f\left(X_i\left(t^{\prime }\right),X_{i-1}\left(t^{\prime }\right)\right)\mathrm{d}{t}^{\prime }+g\left(X_i\left(t^{\prime }\right),X_{i-1}\left(t^{\prime }\right)\right)\diamond \mathrm{d}{L}_i\left(t^{\prime }\right),\hfill \\ X_1\left(0\right)={\gamma }_0,X_2\left(0\right)=X_1\left(\tau \right),\cdots ,X_i\left(0\right)=X_{i-1}\left(\tau \right),\hfill \end{array}\right.\end{array}$$

(20)

has a unique strong solution ${\left(X_1^T\left(t^{\prime }\right),X_2^T\left(t^{\prime }\right),\cdots ,X_i^T\left(t^{\prime }\right)\right)}^T.$

It remains to show that the conclusion is true for $k=i+1$; i.e., there is a unique strong solution to Marcus SDE

$$\begin{array}{c}\hfill \left\{\begin{array}{c}\mathrm{d}{X}_1\left(t^{\prime }\right)=f\left(X_1\left(t^{\prime }\right),\gamma (t^{\prime }-\tau )\right)\mathrm{d}{t}^{\prime }+g\left(X_1\left(t^{\prime }\right),\gamma (t^{\prime }-\tau )\right)\diamond \mathrm{d}{L}_1\left(t^{\prime }\right),\hfill \\ \mathrm{d}{X}_2\left(t^{\prime }\right)=f\left(X_2\left(t^{\prime }\right),X_1\left(t^{\prime }\right)\right)\mathrm{d}{t}^{\prime }+g\left(X_2\left(t^{\prime }\right),X_1\left(t^{\prime }\right)\right)\diamond \mathrm{d}{L}_2\left(t^{\prime }\right),\hfill \\ ⋮⋮⋮\hfill \\ \mathrm{d}{X}_i\left(t^{\prime }\right)=f\left(X_i\left(t^{\prime }\right),X_{i-1}\left(t^{\prime }\right)\right)\mathrm{d}{t}^{\prime }+g\left(X_i\left(t^{\prime }\right),X_{i-1}\left(t^{\prime }\right)\right)\diamond \mathrm{d}{L}_i\left(t^{\prime }\right),\hfill \\ \mathrm{d}{X}_{i+1}\left(t^{\prime }\right)=f\left(X_{i+1}\left(t^{\prime }\right),X_i\left(t^{\prime }\right)\right)\mathrm{d}{t}^{\prime }+g\left(X_{i+1}\left(t^{\prime }\right),X_i\left(t^{\prime }\right)\right)\diamond \mathrm{d}{L}_{i+1}\left(t^{\prime }\right),\hfill \\ X_1\left(0\right)={\gamma }_0,X_2\left(0\right)=X_1\left(\tau \right),\cdots ,X_i\left(0\right)=X_{i-1}\left(\tau \right),X_{i+1}\left(0\right)=X_i\left(\tau \right).\hfill \end{array}\right.\end{array}$$

(21)

Note that the existence and uniqueness for (20) imply that the solution

$${\left(X_1^T\left(t^{\prime }\right),X_2^T\left(t^{\prime }\right),\cdots ,X_i^T\left(t^{\prime }\right)\right)}^T$$

at $t^{\prime }=\tau$ is a fixed value equal to ${\mathsf{\Phi }}_i({\gamma }_0,\tau )$, i.e.,

$$\begin{array}{c}\hfill \begin{array}{c}\hfill {\left(X_1^T\left(\tau \right),X_2^T\left(\tau \right),\cdots ,X_i^T\left(\tau \right)\right)}^T={\mathsf{\Phi }}_i({\gamma }_0,\tau ).\end{array}\end{array}$$

(22)

With Equation (22), Equation (21) can be written as

$$\begin{array}{c}\hfill \left\{\begin{array}{c}\mathrm{d}{X}_1\left(t^{\prime }\right)=f\left(X_1\left(t^{\prime }\right),\gamma (t^{\prime }-\tau )\right)\mathrm{d}{t}^{\prime }+g\left(X_1\left(t^{\prime }\right),\gamma (t^{\prime }-\tau )\right)\diamond \mathrm{d}{L}_1\left(t^{\prime }\right),\hfill \\ \mathrm{d}{X}_2\left(t^{\prime }\right)=f\left(X_2\left(t^{\prime }\right),X_1\left(t^{\prime }\right)\right)\mathrm{d}{t}^{\prime }+g\left(X_2\left(t^{\prime }\right),X_1\left(t^{\prime }\right)\right)\diamond \mathrm{d}{L}_2\left(t^{\prime }\right),\hfill \\ ⋮⋮⋮\hfill \\ \mathrm{d}{X}_i\left(t^{\prime }\right)=f\left(X_i\left(t^{\prime }\right),X_{i-1}\left(t^{\prime }\right)\right)\mathrm{d}{t}^{\prime }+g\left(X_i\left(t^{\prime }\right),X_{i-1}\left(t^{\prime }\right)\right)\diamond \mathrm{d}{L}_i\left(t^{\prime }\right),\hfill \\ \mathrm{d}{X}_{i+1}\left(t^{\prime }\right)=f\left(X_{i+1}\left(t^{\prime }\right),X_i\left(t^{\prime }\right)\right)\mathrm{d}{t}^{\prime }+g\left(X_{i+1}\left(t^{\prime }\right),X_i\left(t^{\prime }\right)\right)\diamond \mathrm{d}{L}_{i+1}\left(t^{\prime }\right),\hfill \end{array}\right.\end{array}$$

(23)

with initial condition

$$\begin{array}{c}\hfill \begin{array}{c}\hfill {\left(X_1^T\left(0\right),X_2^T\left(0\right),\cdots ,X_i^T\left(0\right),X_{i+1}^T\left(0\right)\right)}^T={\left({\gamma }_0^T,{\mathsf{\Phi }}_i^T({\gamma }_0,\tau )\right)}^T.\end{array}\end{array}$$

(24)

Equations (23) and (24) can be rewritten in form of Equation (1) as

$$\begin{array}{c}\hfill \begin{array}{c}\hfill \left\{\begin{array}{c}\mathit{X}\left(t^{\prime }\right)=F_{i+1}\left(\mathit{X}\left(t^{\prime }\right),t^{\prime }\right)\mathrm{d}{t}^{\prime }+G_{i+1}\left(\mathit{X}\left(t^{\prime }\right),t^{\prime }\right)\diamond \mathrm{d}\mathit{L}\left(t^{\prime }\right),\hfill \\ \mathit{X}\left(0\right)={\mathit{x}}_0,\hfill \end{array}\right.\end{array}\end{array}$$

(25)

where

$$\begin{array}{c}\hfill \mathit{X}\left(t^{\prime }\right)=\left(\begin{array}{c}{X}_1\left(t^{\prime }\right)\\ X_2\left(t^{\prime }\right)\\ ⋮\\ X_i\left(t^{\prime }\right)\\ X_{i+1}\left(t^{\prime }\right)\end{array}\right),F_{i+1}(\mathit{X}\left(t^{\prime }\right),t^{\prime })=\left(\begin{array}{c}f(X_1\left(t^{\prime }\right),\gamma (t^{\prime }-\tau ))\\ f(X_2\left(t^{\prime }\right),X_1\left(t^{\prime }\right))\\ ⋮\\ f(X_i\left(t^{\prime }\right),X_{i-1}\left(t^{\prime }\right))\\ f(X_{i+1}\left(t^{\prime }\right),X_i\left(t^{\prime }\right))\end{array}\right),\end{array}$$

$$\begin{array}{c}\hfill \mathit{L}\left(t^{\prime }\right)=\left(\begin{array}{c}{L}_1\left(t^{\prime }\right)\\ L_2\left(t^{\prime }\right)\\ ⋮\\ L_i\left(t^{\prime }\right)\\ L_{i+1}\left(t^{\prime }\right)\end{array}\right),{\mathit{x}}_0=\left(\begin{array}{c}{\gamma }_0\\ {\mathsf{\Phi }}_i({\gamma }_0,\tau )\end{array}\right),\end{array}$$

and

$$\begin{array}{c}\hfill G_{i+1}(\mathit{X}\left(t^{\prime }\right),t^{\prime })=\left(\begin{array}{ccccc}g(X_1\left(t^{\prime }\right),\gamma (t^{\prime }-\tau ))& & & & \\ & g(X_2\left(t^{\prime }\right),X_1\left(t^{\prime }\right))& & & \\ & & \ddots & & \\ & & & & g(X_{i+1}\left(t^{\prime }\right),X_i\left(t^{\prime }\right))\end{array}\right).\end{array}$$

It is straightforward to show that $F_{i+1}$ and $G_{i+1}$ satisfy the conditions in Lemma 2; therefore, Equation (25) has a unique strong solution by Lemma 2. □

4. Representation Formula for the Density of Marcus SDDE

Definition 2.

For $k\in \mathbb{N}$, define ${\mathcal{Q}}_k$: ${\mathbb{R}}^{kd}\times [0,\tau ]\times {\mathbb{R}}^{kd}\times [0,\tau ]\to [0,\infty )$, $(\mathit{u},t^{\prime },\mathit{v},s)\to {\mathcal{Q}}_k(\mathit{u};t^{\prime }|\mathit{v};s)$ such that $\forall \mathit{u},\mathit{v}\in {\mathbb{R}}^{kd}$ and $0\le s<t^{\prime }\le \tau$, ${\mathcal{Q}}_k(\mathit{u};t^{\prime }|\mathit{v};s)$ represents the probability density for the solution ${\left(X_1^T\left(t^{\prime }\right),X_2^T\left(t^{\prime }\right),\cdots ,X_k^T\left(t^{\prime }\right)\right)}^T$ to Marcus SDE (13) at $\mathit{u}$ under the condition ${\left(X_1^T\left(s\right),X_2^T\left(s\right),\cdots ,X_k^T\left(s\right)\right)}^T=\mathit{v}$.

The following three notations, ${\mathcal{P}}_{\mathcal{A}}$, ${\mathcal{Q}}_k$, and p, are used in this paper to represent probability densities.

(i)

${\mathcal{P}}_{\mathcal{A}}$: the density for the solution $X\left(t\right)$ defined in Marcus SDDE (12). For example, ${\mathcal{P}}_{\mathcal{A}}(x,t)$ represents the density for $X\left(t\right)$ at $X\left(t\right)=x$. ${\mathcal{P}}_{\mathcal{A}}(x,3\tau |y,\tau ;z,2\tau )$ represents the conditional density for $X\left(3\tau \right)$ at $X\left(3\tau \right)=x$ given $X\left(\tau \right)=y$ and $X\left(2\tau \right)=z$.

(ii)

${\mathcal{Q}}_k$: as given in Definition 2, ${\mathcal{Q}}_k$ is the transitional density of the ${\mathbb{R}}^{kd}$-valued solution $\left(X_1^T\left(t^{\prime }\right),X_2^T\left(t^{\prime }\right),\cdots ,\right.$ ${\left.X_k^T\left(t^{\prime }\right)\right)}^T$ to Marcus SDE (13). For example, ${\mathcal{Q}}_2(x,y;t^{\prime }|w,z;s)$ with $0\le s<t^{\prime }\le \tau$ represents the density of ${\left(X_1^T\left(t^{\prime }\right),X_2^T\left(t^{\prime }\right)\right)}^T$ at $X_1\left(t^{\prime }\right)=x$ and $X_2\left(t^{\prime }\right)=y$ given $X_1\left(s\right)=w$ and $X_2\left(s\right)=z$.

(iii)

p: the density in general cases. For example, $p(X=x;Y=y)$ represents the density of $(X,Y)$ at $X=x$ and $Y=y$; $p(X=x;Y=y|Z=z)$ represents the density of $(X,Y)$ at $X=x$ and $Y=y$ given $Z=z$. Note that ${\mathcal{P}}_{\mathcal{A}}$ and ${\mathcal{Q}}_k$ can be expressed in terms of p.

Both ${\mathcal{P}}_{\mathcal{A}}$ and ${\mathcal{Q}}_k$ can be expressed by p. For instance,

$$\begin{array}{cc}\hfill {\mathcal{P}}_{\mathcal{A}}(x,t)& =p\left(X\left(t\right)=x\left|X\right(0\right)={\gamma }_0),\hfill \end{array}$$

(26)

$$\begin{array}{cc}\hfill {\mathcal{P}}_{\mathcal{A}}(x,3\tau |y,\tau ;z,2\tau )& =p(X\left(3\tau \right)=x|X\left(0\right)={\gamma }_0;X\left(\tau \right)=y;X\left(2\tau \right)=z),\hfill \end{array}$$

(27)

$$\begin{array}{cc}\hfill {\mathcal{Q}}_2(x,y;t^{\prime }|u,v;s)& =p(X_1\left(t^{\prime }\right)=x;X_2\left(t^{\prime }\right)=y|X_1\left(0\right)={\gamma }_0;X_1\left(s\right)=u;X_2\left(s\right)=v).\hfill \end{array}$$

(28)

To proceed, we need the following assumptions.

Assumption 2.

Suppose $\forall {\mathit{x}}_0\in {\mathbb{R}}^{kd}$ with $k\in \mathbb{N}$; the Marcus SDE defined by (13) under initial condition (15) has a unique strong solution.

Assumption 3.

Suppose $\forall \mathit{u},\mathit{v}\in {\mathbb{R}}^{kd}$ with $k\in \mathbb{N}$ and $0\le s<t^{\prime }\le \tau$; the probability density ${\mathcal{Q}}_k(\mathit{u},t^{\prime }|\mathit{v},s)$, which represents the density of $\mathit{X}\left(t^{\prime }\right)$ at $\mathit{u}$ given $\mathit{X}\left(s\right)=\mathit{v}$, exists and is strictly positive.

Lemma 2 gives a sufficient condition for Assumption 2 to be true. As for Assumption 3, there exist sufficient conditions for the existence and regularity of the probability density for some SDEs driven by Lévy processes with certain restrictions imposed on the jumping measures; see [1,21] and the references therein, among others. However, sufficient conditions for SDEs driven by general Lévy processes are still not available. There are also some sufficient conditions available for the density to be strictly positive in cases with Gaussian white noise. The strictly positive property of the density for a general class of SDEs can be concluded from the heat kernel estimations [22,23]. A more general sufficient condition for strictive positiveness of density is presented in [24].

Now, we use the transition density ${\mathcal{Q}}_k$ of Marcus SDE (13) to represent the density ${\mathcal{P}}_{\mathcal{A}}$ of Marcus SDDE (12).

Theorem 2

(Representation formula for the density of Marcus SDDEs). Suppose that Assumptions 2 and 3 hold. Then $\forall t>0$, the probability density function ${\mathcal{P}}_{\mathcal{A}}(x,t)$ for the solution $X\left(t\right)$ defined by Marcus SDDE (12) exists. Moreover, $\forall x\in {\mathbb{R}}^d$, and the following statements are true.

(i)

$For$ $t\in (0,\tau ]$,

$${\mathcal{P}}_{\mathcal{A}}(x,t)={\mathcal{Q}}_1(x;t|{\gamma }_0;0).$$

(29)

(ii)

$For$ $t\in \left(\right(k-1)\tau ,k\tau )$ $with$ $k\ge 2$ $and$ $k\in \mathbb{N}$,

$$\begin{array}{cc}\hfill {\mathcal{P}}_{\mathcal{A}}(x,t)& ={\int }_{{\mathbb{R}}^{2(k-1)d}}{\mathcal{Q}}_{k-1}\left(x_1,\cdots ,x_{k-1};\tau |y_1,\cdots ,y_{k-1};t-(k-1)\tau \right)\hfill \\ \hfill & \times {\mathcal{Q}}_k\left(y_1,\cdots ,y_{k-1},x;t-(k-1)\tau |{\gamma }_0,x_1,\cdots ,x_{k-1};0\right)\prod _{i=1}^{k-1}\mathrm{d}{x}_i\prod _{i=1}^{k-1}\mathrm{d}{y}_i.\hfill \end{array}$$

(30)

(iii)

$For$ $t=k\tau$ with $k\ge 2$ and $k\in \mathbb{N}$,

$${\mathcal{P}}_{\mathcal{A}}(x,t)={\int }_{{\mathbb{R}}^{(k-1)d}}{\mathcal{Q}}_k(x_1,\cdots ,x_{k-1},x;\tau |{\gamma }_0,x_1,\cdots ,x_{k-1};0)\prod _{i=1}^{k-1}\mathrm{d}{x}_i.$$

(31)

Remark 4.

Theorem 2 shows that the density for Marcus SDDEs can be expressed in terms of that for Marcus SDEs without delays. Governing equations for the density of Marcus SDEs without delays have been established; see [7].

Remark 5.

Note that (31) can be absorbed into (30). In fact, $\forall \mathit{u},\mathit{v}\in {\mathbb{R}}^{kd}$, ${\mathcal{Q}}_k(\mathit{u},s|\mathit{v},s)=\underset{t^{\prime }\to s}{\lim }{\mathcal{Q}}_k(\mathit{u},t^{\prime }|\mathit{v},s)=\delta (\mathit{u}-\mathit{v})$ and $f\left(\mathit{u}\right)={\int }_{{\mathbb{R}}^{kd}}\delta (\mathit{u}-\mathit{v})f\left(\mathit{v}\right)\mathrm{d}\mathit{v}.$ Therefore, for $t=k\tau$, Equation (30) becomes Equation (31).

Proof of Theorem 2.

First, we prove statement (i) of Theorem 2. In fact, by Equations (16) or (17), the density of $X\left(t\right)$ for $0<t\le \tau$ defined by Marcus SDDE (12) is the same as the density of $X_1\left(t\right)$ defined by Marcus SDE (13) with initial value ${\gamma }_0\in {\mathbb{R}}^d$ and $k=1$. Therefore, (i) is true.

Next, we prove statement (ii) of Theorem 2. For $t\in \left(\right(k-1)\tau ,k\tau )$ with $k\ge 2$ and $k\in \mathbb{N}$, as shown in Appendix B, if Assumption 3 holds (i.e., ${\mathcal{Q}}_k$ exists), then ${\mathcal{P}}_{\mathcal{A}}(x,t|$$x_1,\tau ;\cdots ;x_{k-1},(k-1)\tau ;x_k,k\tau )$, ${\mathcal{P}}_{\mathcal{A}}(x_k,k\tau |x_1,\tau ;\cdots ;x_{k-1},(k-1)\tau )$, and ${\mathcal{P}}_{\mathcal{A}}\left(x_1,\tau ;\cdots ;x_{k-1},\right.\left.\left(\right(k-1\left)\tau \right)\right)$ exist, and can be respectively expressed as

$$\begin{array}{c}\hfill \begin{array}{cc}\hfill & {\mathcal{P}}_{\mathcal{A}}(x,t|x_1,\tau ;\cdots ;x_{k-1},(k-1)\tau ;x_k,k\tau )\hfill \\ \hfill =& {\int }_{{\mathbb{R}}^{(k-1)d}}{\mathcal{Q}}_k(x_1,\cdots ,x_{k-1},x_k;\tau |y_1,\cdots ,y_{k-1},x;t-(k-1)\tau )\hfill \\ \hfill & \times {\displaystyle \frac{{\mathcal{Q}}_k(y_1,\cdots ,y_{k-1},x;t-(k-1)\tau |{\gamma }_0,x_1,\cdots ,x_{k-1};0)}{{\mathcal{Q}}_k(x_1,\cdots ,x_k;\tau |{\gamma }_0,x_1,\cdots ,x_{k-1};0)}}\prod _{i=1}^{k-1}\mathrm{d}{y}_i,\hfill \end{array}\end{array}$$

(32)

$$\begin{array}{c}\hfill {\mathcal{P}}_{\mathcal{A}}(x_k,k\tau |x_1,\tau ;\cdots ;x_{k-1},(k-1)\tau )={\displaystyle \frac{{\mathcal{Q}}_k(x_1,\cdots ,x_k;\tau |{\gamma }_0,x_1,\cdots ,x_{k-1};0)}{{\mathcal{Q}}_{k-1}(x_1,\cdots ,x_{k-1};\tau |{\gamma }_0,x_1,\cdots ,x_{k-2};0)}},\end{array}$$

(33)

and

$$\begin{array}{c}\hfill {\mathcal{P}}_{\mathcal{A}}(x_1,\tau ;\cdots ;x_{k-1},(k-1)\tau )={\mathcal{Q}}_{k-1}\left(x_1,\cdots ,x_{k-1};\tau |{\gamma }_0,x_1,\cdots ,x_{k-2};0\right).\end{array}$$

(34)

Furthermore, the condition density ${\mathcal{P}}_{\mathcal{A}}(x,t|x_1,\tau ;\cdots ;x_{k-1},(k-1)\tau )$ exists due to the identity

$$\begin{array}{c}\hfill \begin{array}{cc}\hfill & {\mathcal{P}}_{\mathcal{A}}(x,t|x_1,\tau ;\cdots ;x_{k-1},(k-1)\tau )\hfill \\ \hfill =& {\int }_{{\mathbb{R}}^d}{\mathcal{P}}_{\mathcal{A}}(x,t|x_1,\tau ;\cdots ;x_{k-1},(k-1)\tau ;x_k,k\tau )\times {\mathcal{P}}_{\mathcal{A}}(x_k,k\tau |x_1,\tau ;\cdots ;x_{k-1},(k-1)\tau )\mathrm{d}{x}_k.\hfill \end{array}\end{array}$$

(35)

Substituting Equations (32) and (33) into (35), we obtain

$$\begin{array}{c}\hfill \begin{array}{cc}\hfill & {\mathcal{P}}_{\mathcal{A}}(x,t|x_1,\tau ;\cdots ;x_{k-1},(k-1)\tau )\hfill \\ \hfill =& {\int }_{{\mathbb{R}}^{(k-1)d}}{\int }_{{\mathbb{R}}^d}{\mathcal{Q}}_k(x_1,\cdots ,x_{k-1},x_k;\tau |y_1,\cdots ,y_{k-1},x;t-(k-1)\tau )\mathrm{d}{x}_k\hfill \\ \hfill & \times {\displaystyle \frac{{\mathcal{Q}}_k(y_1,\cdots ,y_{k-1},x;t-(k-1)\tau |{\gamma }_0,x_1,\cdots ,x_{k-1};0)}{{\mathcal{Q}}_{k-1}(x_1,\cdots ,x_{k-1};\tau |{\gamma }_0,x_1,\cdots ,x_{k-2};0)}}\prod _{i=1}^{k-1}\mathrm{d}{y}_i\hfill \\ \hfill =& {\int }_{{\mathbb{R}}^{(k-1)d}}{\mathcal{Q}}_{k-1}(x_1,\cdots ,x_{k-1};\tau |y_1,\cdots ,y_{k-1},x;t-(k-1)\tau )\hfill \\ \hfill & \times {\displaystyle \frac{{\mathcal{Q}}_k(y_1,\cdots ,y_{k-1},x;t-(k-1)\tau |{\gamma }_0,x_1,\cdots ,x_{k-1};0)}{{\mathcal{Q}}_{k-1}(x_1,\cdots ,x_{k-1};\tau |{\gamma }_0,x_1,\cdots ,x_{k-2};0)}}\prod _{i=1}^{k-1}\mathrm{d}{y}_i\hfill \\ \hfill =& {\int }_{{\mathbb{R}}^{(k-1)d}}{\mathcal{Q}}_{k-1}(x_1,\cdots ,x_{k-1};\tau |y_1,\cdots ,y_{k-1};t-(k-1)\tau )\hfill \\ \hfill & \times {\displaystyle \frac{{\mathcal{Q}}_k(y_1,\cdots ,y_{k-1},x;t-(k-1)\tau |{\gamma }_0,x_1,\cdots ,x_{k-1};0)}{{\mathcal{Q}}_{k-1}(x_1,\cdots ,x_{k-1};\tau |{\gamma }_0,x_1,\cdots ,x_{k-2};0)}}\prod _{i=1}^{k-1}\mathrm{d}{y}_i.\hfill \end{array}\end{array}$$

(36)

To derive the last identity in (36), we use

$$\begin{array}{c}\hfill {\mathcal{Q}}_{k-1}(x_1,\cdots ,x_{k-1};\tau |y_1,\cdots ,y_{k-1},x;t-(k-1)\tau )\\ \hfill ={\mathcal{Q}}_{k-1}(x_1,\cdots ,x_{k-1};\tau |y_1,\cdots ,y_{k-1};t-(k-1)\tau ),\end{array}$$

which follows from the fact that ${\left(X_1^T\left(\tau \right),X_2^T\left(\tau \right),\cdots ,X_{k-1}^T\left(\tau \right)\right)}^T$ in (13) only depends on

$$\begin{array}{c}\hfill {\left(X_1^T(t-(k-1)\tau ),X_2^T(t-(k-1)\tau ),\cdots ,X_{k-1}^T(t-(k-1)\tau )\right)}^T\end{array}$$

and is independent of $X_k(t-(k-1)\tau )$.

By using Equations (34) and (36), the density ${\mathcal{P}}_{\mathcal{A}}(x,t)$ of Marcus SDDE (12) exists by the identity

$$\begin{array}{cc}\hfill {\mathcal{P}}_{\mathcal{A}}\left(x,t\right)=& {\int }_{{\mathbb{R}}^{(k-1)d}}{\mathcal{P}}_{\mathcal{A}}\left(x,t|x_1,\tau ;\cdots ;x_{k-1},(k-1)\tau \right)\times {\mathcal{P}}_{\mathcal{A}}\left(x_1,\tau ;\cdots ;x_{k-1},(k-1)\tau \right)\prod _{i=1}^{k-1}\mathrm{d}{x}_i\hfill \\ \hfill =& {\int }_{{\mathbb{R}}^{2(k-1)d}}{\mathcal{Q}}_{k-1}\left(x_1,\cdots ,x_{k-1};\tau |y_1,\cdots ,y_{k-1};t-(k-1)\tau \right)\hfill \\ \hfill & \times {\mathcal{Q}}_k\left(y_1,\cdots ,y_{k-1},x;t-(k-1)\tau |{\gamma }_0,x_1,\cdots ,x_{k-1};0\right)\prod _{i=1}^{k-1}\mathrm{d}{x}_i\prod _{i=1}^{k-1}\mathrm{d}{y}_i.\hfill \end{array}$$

(37)

Therefore, (ii) is true.

Finally, we show that statement (iii) of Theorem 2 is true. In fact, for $t=k\tau$ with $k\ge 2$ and $k\in \mathbb{N}$,

$$\begin{array}{cc}\hfill & {\mathcal{P}}_{\mathcal{A}}(x,k\tau )={\int }_{{\mathbb{R}}^d}{\mathcal{P}}_{\mathcal{A}}(x,k\tau |x_1,\tau ;\cdots ;x_{k-1},(k-1)\tau )\times {\mathcal{P}}_{\mathcal{A}}(x_1,\tau ;\cdots ;x_{k-1},(k-1)\tau )\prod _{i=1}^{k-1}\mathrm{d}{x}_i.\hfill \end{array}$$

(38)

Substituting Equations (33) and (34) into (38), we obtain Equation (31). Therefore, (iii) is true. □

To verify Theorem 2, a toy example which has an analytical solution is presented below. Consider a special case of Marcus SDDE (12) with $f(x,y)=y$, $g(x,y)=1$ and $L\left(t\right)=B\left(t\right)$ being a Brownian motion:

$$\begin{array}{c}\hfill \begin{array}{c}\hfill \left\{\begin{array}{c}\mathrm{d}X\left(t\right)=X(t-\tau )\mathrm{d}t+\mathrm{d}B\left(t\right),\mathrm{for}t>0,\hfill \\ X\left(t\right)=0,t\in [-\tau ,0],\hfill \end{array}\right.\end{array}\end{array}$$

(39)

It is straightforward to check that (39) has an analytical solution given by

$$\begin{array}{c}\hfill X\left(t\right)=\left\{\begin{array}{cc}B\left(t\right),\hfill & \mathrm{for}t\in (0,\tau ],\hfill \\ (t-\tau )B(t-\tau )+B\left(t\right)-{\int }_0^{t-\tau }s\mathrm{d}B\left(t\right),\hfill & \mathrm{for}t\in (\tau ,2\tau ],\hfill \end{array}\right.\end{array}$$

(40)

and the probability density for the solution $X\left(t\right)$ is

$$\begin{array}{c}\hfill {\mathcal{P}}_{\mathcal{A}}(x,t)=\left\{\begin{array}{cc}{\displaystyle \frac{1}{\sqrt{2\pi t}}}\exp \left(-{\displaystyle \frac{x^2}{2t}}\right).\hfill & \mathrm{for}t\in (0,\tau ],\hfill \\ {\displaystyle \frac{\sqrt{3}}{\sqrt{2\pi \left({(t-\tau )}^3+3\tau -1\right)}}}\exp \left(-{\displaystyle \frac{3x^2}{2({(t-\tau )}^3+3\tau -1)}}\right),\hfill & \mathrm{for}t\in (\tau ,2\tau ].\hfill \end{array}\right.\end{array}$$

(41)

To verify Theorem 2, we show that the same result as in (41) can be obtained by using Theorem 2. Considering the special case given by (39), it follows from Theorem 2 that for $t\in (0,\tau ]$,

$${\mathcal{P}}_{\mathcal{A}}(x,t)={\mathcal{Q}}_1(x;t|0;0),$$

(42)

and for $t\in (\tau ,2\tau )$,

$$\begin{array}{c}\hfill {\mathcal{P}}_{\mathcal{A}}(x,t)={\int }_{\mathbb{R}}{\int }_{\mathbb{R}}{\mathcal{Q}}_1\left(x_1;\tau |y_1;t-\tau \right){\mathcal{Q}}_2\left(y_1,x;t-\tau |0,x_1;0\right)\mathrm{d}{x}_1\mathrm{d}{y}_1,\end{array}$$

(43)

where ${\mathcal{Q}}_1(x;t|y;s)$ satisfies the Fokker–Planck equation

$$\begin{array}{c}\hfill {\displaystyle \frac{\partial }{\partial t}}{\mathcal{Q}}_1(x;t|y;s)={\displaystyle \frac{1}{2}}{\displaystyle \frac{{\partial }^2}{\partial x^2}}{\mathcal{Q}}_1(x;t|y;s),\mathrm{for}t>s,\end{array}$$

(44)

with initial condition

$$\begin{array}{c}\hfill \underset{t\to s}{lim}{\mathcal{Q}}_1(x;t|y;s)=\delta (x-y),\end{array}$$

(45)

and ${\mathcal{Q}}_2(x_1,x_2;t|y_1,y_2;s)$ with $0\le s<t\le \tau$ satisfies the Fokker–Planck equation

$$\begin{array}{c}{\displaystyle \frac{\partial }{\partial t}}{\mathcal{Q}}_2(x_1,x_2;t|y_1,y_2;s)=-x_1{\displaystyle \frac{\partial }{\partial x_2}}{\mathcal{Q}}_2(x_1,x_2;t|y_1,y_2;s)\hfill \\ +{\displaystyle \frac{1}{2}}{\displaystyle \frac{{\partial }^2}{\partial x_1^2}}{\mathcal{Q}}_2(x_1,x_2;t|y_1,y_2;s)+{\displaystyle \frac{1}{2}}{\displaystyle \frac{{\partial }^2}{\partial x_2^2}}{\mathcal{Q}}_2(x_1,x_2;t|y_1,y_2;s)\mathrm{for}t>s\hfill \end{array}$$

(46)

with initial condition

$$\begin{array}{c}\hfill \underset{t\to s}{lim}{\mathcal{Q}}_2(x_1,x_2;t|y_1,y_2;s)=\delta (x_1-y_1)\delta (x_2-y_2).\end{array}$$

(47)

It is straightforward to check that the solutions to the above two Fokker–Planck equations can be expressed respectively as

$$\begin{array}{c}\hfill {\mathcal{Q}}_1(x;t|y;s)={\displaystyle \frac{1}{\sqrt{2\pi (t-s)}}}exp\left\{-{\displaystyle \frac{{(x-y)}^2}{2(t-s)}}\right\},\end{array}$$

(48)

and

$$\begin{array}{cc}\hfill & {\mathcal{Q}}_2(x_1,x_2;t|y_1,y_2;s)={\displaystyle \frac{1}{2\pi (t-s)\sqrt{\frac{{(t-s)}^2+12}{12}}}}\hfill \\ \hfill & \times exp\left\{-{\displaystyle \frac{2{(t-s)}^2+6}{{(t-s)}^2+12}}\left[{\displaystyle \frac{{(y_1-x_1)}^2}{t-s}}-{\displaystyle \frac{3(x_1-y_1)(x_2-y_2-y_1(t-s))}{{(t-s)}^2+3}}\right.\right.\hfill \\ \hfill & +\left.\left.{\displaystyle \frac{3{(x_2-y_2-y_1(t-s))}^2}{{(t-s)}^3+3(t-s)}}\right]\right\}.\hfill \end{array}$$

(49)

Substituting Equations (48) and (49) into (42) and (43), respectively, we can obtain Equation (41) by some direct computation.

5. Conclusions

Marcus SDEs have been widely used in physics and engineering fields. It is challenging to study the existence, uniqueness, and probability density of Marcus SDEs with memory, due to the fact that the delays cause very complicated correction terms. We establish the existence and uniqueness of the solution of Marcus SDDEs in the form of (12) and present a representation formula for the probability density associated with (12). The formula, which is given in Equations (29)–(31), expresses the density for Marcus SDDEs in terms of the known density for Marcus SDEs without delays. The result in this paper makes it possible to compute the density of Marcus SDDEs by numerically solving the density of Marcus SDEs without delay. This has many important applications, especially in some typical climate models such as EI-Nino Southern Oscillation (ENSO) [25,26], which will be further explored in our future work.