Feynman-Kac Formulas for Regime-Switching Jump Diffusions and their Applications

TL;DR

This paper derives Feynman-Kac formulas for complex regime-switching jump diffusions driven by Lévy processes, linking stochastic processes with partial integro-differential equations and demonstrating convergence to the arcsine law.

Contribution

It introduces Feynman-Kac formulas for regime-switching jump diffusions with Lévy jumps and establishes their connection to PDEs under broad conditions.

Findings

Connection between stochastic processes and PDEs established.

Convergence of related random variables to the arcsine law shown.

Framework applicable to initial, terminal, and boundary value problems.

Abstract

This work develops Feynman-Kac formulas for a class of regime-switching jump diffusion processes, in which the jump part is driven by a Poisson random measure associated to a general L\'evy process and the switching part depends on the jump diffusion processes. Under broad conditions, the connections of such stochastic processes and the corresponding partial integro-differential equations are established. Related initial, terminal, and boundary value problems are also treated. Moreover, based on weak convergence of probability measures, it is demonstrated that a sequence of random variables related to the regime-switching jump diffusion process converges in distribution to the arcsine law.

Equations

$$\frac{\partial }{\partial t}u(t,x)=\frac{1}{2}\frac{{\partial }^2}{\partial x^2}u(t,x)-f(x)u(t,x),u(0,x)=g(x),$$

$$u(t,x)=\mathbb{E}\left[g(x+W(t))\exp \left(-{\int }_0^{t}f(x+W(s))ds\right)\right],$$

$$N(t,U):=\sum _{0<s\le t}{I}_U(\Delta {\psi }_s)=\sum _{0<s\le t}{I}_U(\psi (s)-\psi (s-)),$$

$$\tilde{N}(dt,dy):=N(dt,dy)-\nu (dy)dt.$$

$${\int }_{{\mathbb{R}}_0^n}(1\wedge {\left|y\right|}^2)\nu (dy)<\infty ,$$

$$dX(t)=$$

$$+{\int }_{{\mathbb{R}}_0^n}\gamma (X(t-),\alpha (t-),y)\tilde{N}(dt,dy),t\ge 0,$$

$$X(0)=x_0\in {\mathbb{R}}^n,\alpha (0)={\alpha }_0\in \mathcal{M},$$

$$\mathbb{P}\left\{\alpha (t+\delta )=j|X(t)=x,\alpha (t)=i,X(s),\alpha (s),s\le t\right\}=\{\begin{array}{ll}{q}_{ij}(x)\delta +o(\delta ),& ifj\ne i\\ 1+q_{ii}(x)\delta +o(\delta ),& ifj=i,\end{array}$$

$$h(x,i,z)=\sum _{j=1}^m(j-i)I_{\{z\in {\Delta }_{ij}(x)\}}.$$

$$d\alpha (t)={\int }_{\mathbb{R}}h(X(t-),\alpha (t-),z)N_1(dt,dy),$$

$${\left|b(x,i)-b(y,i)\right|}^2+{\left|\sigma (x,i)-\sigma (y,i)\right|}^2$$

$$+{\int }_{{\mathbb{R}}_0^n}{\left|\gamma (x,i,z)-\gamma (y,i,z)\right|}^2\nu (dz)\le \kappa {\left|x-y\right|}^2,$$

$${\int }_{{\mathbb{R}}_0^n}{\left|\gamma (x,i,z)\right|}^2\nu (dz)\le \kappa (1+{\left|x\right|}^2),$$

$$\sup \left\{q_{ij}(x):x\in {\mathbb{R}}^n,i\ne j\in \mathcal{M}\right\}\le \kappa <\infty .$$

$$\displaystyle{\mathcal{D}}_{\mathcal{L}}:=\biggl{\{}g:{\mathbb{R}}^{n}\times{\mathcal{M}}\mapsto{\mathbb{R}},\text{ so that for each }i\in{\mathcal{M}},\text{ we have }g(\cdot,i)\in C^{2}\text{ and }$$

$$\displaystyle\int_{{\mathbb{R}}^{n}_{0}}\left|g(x+\gamma(x,i,z),i)-g(x,i)-D_{x}g(x,i)\cdot\gamma(x,i,z)\right|\nu(dz)<\infty\biggl{\}}.$$

$$\mathcal{L}g(x,i)=$$

$$+{\int }_{{\mathbb{R}}_0^n}[g(x+\gamma (x,i,z),i)-g(x,i)-D_{x}g(x,i)\cdot \gamma (x,i,z)]\nu (dz),(x,i)\in {\mathbb{R}}^d\times \mathcal{M}.$$

$$f(X(t),\alpha (t))-f(X(0),\alpha (0))={\int }_0^t\mathcal{L}f(X(s-),\alpha (s-))ds+M_1^f(t)+M_2^f(t)+M_3^f(t),$$

$$\begin{array}[]{ll}&\!\!\!\displaystyle M_{1}^{f}(t)=\int^{t}_{0}D_{x}f(X(s-),\alpha(s-))\cdot\sigma(X(s-),\alpha(s-))dW(s),\\ &\!\!\!\displaystyle M_{2}^{f}(t)=\int_{0}^{t}\int_{{\mathbb{R}}}\big{[}f(X(s-),\alpha(s-)+h(X(s-),\alpha(s-),z))-f(X(s-),\alpha(s-))\big{]}\widetilde{N}_{1}(ds,dz),\\ &\!\!\!\displaystyle M_{3}^{f}(t)=\int_{0}^{t}\int_{{\mathbb{R}}^{n}_{0}}\!\left[f(X(s-)+\gamma(X(s-),\alpha(s-),z),\alpha(s-))-f(X(s-),\alpha(s-))\right]\widetilde{N}(ds,dz).\end{array}$$

$$\left|\gamma (x,i,z)\right|\le {\rho }_1(x)<\infty \text{ if }\left|z\right|\le 1,$$

$$\left|f(x,i)\right|\le K(1+{\left|x\right|}^2),\text{ for all }(x,i)\in {\mathbb{R}}^n\times \mathcal{M}.$$

$${\int }_{{\mathbb{R}}_0^n}\left|f(x+\gamma (x,i,z),i)-f(x)-D_{x}f(x,i)\cdot \gamma (x,i,z)\right|\nu (dz)<\infty .$$

$$f(x+\gamma (x,i,z),i)-f(x,i)-Df(x,i)\cdot \gamma (x,i,z)=\frac{1}{2}{D}^{2}f(x+\theta \gamma (x,i,z),i)\gamma (x,i,z)\cdot \gamma (x,i,z)$$

$${\int }_{|z|\le 1}$$

$$\le {\int }_{|z|\le 1}{\rho }_2(x){\left|\gamma (x,i,z)\right|}^2\nu (dz)={\rho }_2(x){\int }_{|z|\le 1}{\left|\gamma (x,i,z)\right|}^2\nu (dz)<\infty .$$

$$\begin{array}[]{rl}&\!\!\!\displaystyle\left|f(x+\gamma(x,i,z),i)-f(x,i)-Df(x,i)\cdot\gamma(x,i,z)\right|\\ &\displaystyle\leq\left|f(x+\gamma(x,i,z),i)\right|+\left|f(x,i)\right|+\left|Df(x,i)\right|\left|\gamma(x,i,z)\right|\\ &\displaystyle\leq K(1+\left|x+\gamma(x,i,z)\right|^{2})+\left|f(x,i)\right|+\left|Df(x,i)\right|^{2}+\left|\gamma(x,i,z)\right|^{2}\\ &\displaystyle\leq K_{1}(1+\left|x\right|^{2}+\left|f(x,i)\right|+\left|Df(x,i)\right|^{2}+\left|\gamma(x,i,z)\right|^{2}),\end{array}$$

$$\nu ({\mathbb{R}}_0^n-\left\{z\in {\mathbb{R}}_0^n:|z|\le 1\right\})=\nu (\left\{z:|z|>1\right\})<\infty .$$

$${\int }_{|z|>1}\left|f(x+\gamma (x,i,z),i)-f(x,i)-Df(x,i)\cdot \gamma (x,i,z)\right|\nu (dz)$$

$$\le K_1{\int }_{|z|>1}(1+{\left|x\right|}^2+\left|f(x,i)\right|+{\left|Df(x,i)\right|}^2+{\left|\gamma (x,i,z)\right|}^2)\nu (dz)$$

$$=K_1\left[\left(1+{\left|x\right|}^2+\left|f(x,i)\right|+{\left|Df(x,i)\right|}^2\right)\nu (\left\{z:|z|>1\right\})+{\int }_{|z|>1}|\gamma (x,i,z){|}^2\nu (dz)\right]<\infty .$$

$$\begin{array}[]{ll}\left|f(x,i)\right|\leq K(1+\left|x\right|^{a}),&\text{ for all }(x,i)\in{\mathbb{R}}^{n}\times{\mathcal{M}},\\ \left|\gamma(x,i,z)\right|\leq K\rho_{3}(x)(1+\left|z\right|^{b}),&\text{ for }\left|z\right|\geq 1,\\ \int_{|z|>1}\left|z\right|^{ab}\nu(dz)<\infty.&\end{array}$$

$$\gamma (x,i,z)=g_1(x,i){\left|z\right|}^{b_1}{I}_{\{|z|\le 1\}}+g_2(x,i)|z{|}^{b_2}{I}_{\{|z|>1\}}$$

Full text

Key words. Feynman-Kac formula, partial integro-partial differential equation, arcsine law.

Mathematics Subject Classification. 60J60, 60J75, 47D08.

1 Introduction

The Feynman-Kac formula establishes natural connections between partial differential equations (PDEs) and stochastic processes. For instance, a simple version of the Feynman-Kac formula [17, Section V.3] indicates that for any bounded functions $f,g:{\mathbb{R}}\mapsto{\mathbb{R}}$ and any bounded solution $u(t,x)$ of the initial value problem

[TABLE]

there is a stochastic representation

[TABLE]

where $W$ is a one-dimensional standard Brownian motion with $W(0)=0$ a.s. Conversely, if we define $u$ to be the right-hand side of (1.2), then under some mild regularity conditions on the functions $f$ and $g$, we can show that $u$ is a classical solution to (1.1). First, the Feynman-Kac formula offers a method of solving certain PDEs by simulating paths of the underlying stochastic processes. In addition, a class of expectations of random processes can be computed by solving the related PDEs. For example, the classical Black-Scholes-Merton PDE can help to determine the arbitrage free price for European call options [21, Section 5.8].

Since the early work of Feynman [14] and Kac [20], the Feynman-Kac formulas have been extended and generalized in different directions. The Feynman-Kac formula for general multi-dimensional diffusion processes can be found in, for instance, [21, Section 5.7]; see also [33] for Feynman-Kac representation formula for variational inequalities, and [7, Section 12.2] and [34, 35] for several versions of Feynman-Kac formulas for jump diffusions. Numerous applications have been found; see, for example, [8, 11, 19] (finance), [3, 29] (DNA breathing dynamics, physics, and computer science), and [10] (statistical physics, biology, and engineering problems). Using switching diffusion models, a recent work [28] incorporated continuous-state dependent switching in optimal stopping with applications to perpetual American put options. This effort may be extended with the use of switching jump diffusion models, which opens up possible considerations of the Feynman-Kac representation for related problems.

Applications demand the treatment of regime-switching diffusions with Poisson type jumps. In many real-world applications, the systems often display discontinuous paths as well as structural changes. Consider, for instance, asset price modeling, in which the commonly used jump diffusion models [7] do not consider the qualitative changes of the volatility, while the regime-switching Black-Scholes models [12, 44] unrealistically assume the continuity of the price evolution. In contrast to the references above, regime-switching jump diffusion processes can naturally capture the features of jump discontinuity as well as random environment changes of the underlying systems. The Poisson jumps and more general Lévy jumps are well-known to incorporate both small and big jumps [2]; while the regime-switching mechanisms provide the structural changes of the systems [30, 42]. Thus, regime-switching diffusion with Lévy jumps provides a uniform and realistic platform for modeling in a wide range of applications. Moreover, as we have seen in [42], adding a switching processes in the modeling is not a simple or trivial extension of the standard models in the literature.

This work aims to develop the Feynman-Kac formulas and to establish connections between a class of coupled systems of partial integro-differential equations and regime-switching jump diffusion processes. We will establish three versions of the Feynman-Kac formula (Theorems 3.1, 3.2, and 3.3), corresponding to initial and terminal value Cauchy problems and boundary value problem, respectively. To the best of our knowledge, such results are not available in the literature. The proofs of these results make essential use of the generalized Itô formula (2.10) and the optional sampling theorem for martingales [21, Theorem 1.3.22], and require careful analysis in handling the (local) martingale terms; see for instance the proof of Theorem 3.1. In particular, in presence of a general Lévy measure $\nu$ and the form of our stochastic differential equation (see (2.1) and (2.2)), the derivation of the Feynman-Kac formulas is not a trivial extension of the counterparts for diffusion or regime-switching diffusion processes; see Remark 3.2. In this work, we provide mild conditions under which the Feynman-Kac formulas for regime-switching jump diffusions are derived rigorously.

Motivated by Kac’s derivation of Lévy’s arcsine law for the occupation time for a one-dimensional Brownian motion using the Feynman-Kac formula as well as a recent result of Khasminskii [22] on arcsine laws for null-recurrent diffusions, we also derive an arcsine law (Proposition 4.1) for a class of one-dimensional regime-switching jump diffusion processes, in which the switching component is singularly perturbed with fast switching. We show that the regime-switching jump diffusion converges weakly to a diffusion process, whose diffusion coefficient is determined by an appropriate average of the diffusion coefficients of the subsystems with respect to the invariant measure of the fast switching component; see Theorem 4.1 for the precise statement. Moreover, we demonstrate by an example that one in general cannot expect $L_{2}$ convergence corresponding to the weak convergence result established in Theorem 4.1. A similar phenomenon was recently observed in [27]. Nevertheless, the weak convergence result, together with [22], will help us to derive the desired arcsine law.

The rest of the paper is arranged as follows. Section 2 presents the formulation of the problem that we wish to study together with some preliminary results. Section 3 concentrates on obtaining the Feynman-Kac formulas. Section 3.1 presents the Feynman-Kac formulas for Cauchy problems while Section 3.2 is devoted to the Feynman-Kac formula for a class of Dirichlet problems. An example on option pricing in incomplete market is provided in Section 3 to demonstrate the utility of our result. Section 4 deals with arcsine laws related to the processes of interests. First, based on two-time-scale formulation, we examine a system in which the switching component is fast varying. This enables us to obtain a limiting diffusion process in the sense of weak convergence, which, in turn, helps us to obtain the desired arcsine law. Finally, we conclude the paper with additional remarks in Section 5.

2 Formulation and Preliminary Results

To facilitate the presentation, we introduce some notation that will be used often in later sections. Throughout the paper, we use $x^{\prime}$ to denote the transpose of $x$, and $x^{\prime}y$ or $x\cdot y$ interchangeably to denote the inner product of the vectors $x$ and $y$. For sufficiently smooth $\phi:{\mathbb{R}}^{n}\to{\mathbb{R}}$, $D_{x_{i}}\phi=\frac{\partial\phi}{\partial x_{i}}$, $D_{x_{i}x_{j}}\phi=\frac{\partial^{2}\phi}{\partial x_{i}\partial x_{j}}$, and we denote by $D_{x}\phi=(D_{x_{1}}\phi,\dots,D_{x_{n}}\phi)^{\prime}\in{\mathbb{R}}^{n}$ and $D^{2}_{xx}\phi=(D_{x_{i}x_{j}}\phi)\in{\mathbb{R}}^{n\times n}$ the gradient and Hessian of $\phi$, respectively. For $k\in\mathbb{N}$, $C^{k}({\mathbb{R}}^{n})$ is the collection of functions $f:{\mathbb{R}}^{n}\mapsto{\mathbb{R}}$ with continuous partial derivatives up to the $k$th order while $C^{k}_{c}({\mathbb{R}}^{n})$ denotes the space of $C^{k}$ functions with compact support. If $B$ is a set, we use $B^{o}$ and $I_{B}$ to denote the interior and indicator function of $B$, respectively. Throughout the paper, we adopt the conventions that $\sup\emptyset=-\infty$ and $\inf\emptyset=+\infty$.

2.1 Formulation

Let $(\Omega,{\mathfrak{F}},\left\{{\mathfrak{F}}_{t}\right\}_{t\geq 0},{\mathbb{P}})$ be a filtered probability space satisfying the usual condition on which is defined an $n$-dimensional standard ${\mathfrak{F}}_{t}$-adapted Brownian motion $W(\cdot)$. Let $\left\{\psi(t)\right\}$ be an ${\mathfrak{F}}_{t}$-adapted Lévy process with Lévy measure $\nu(\cdot)$. Denote by $N(\cdot)$ the corresponding ${\mathfrak{F}}_{t}$-adapted Poisson random measure defined on ${\mathbb{R}}_{+}\times{\mathbb{R}}^{n}_{0}$:

[TABLE]

where $t\geq 0$ and $U$ is a Borel subset of ${\mathbb{R}}^{n}_{0}={\mathbb{R}}^{n}-\left\{0\right\}$. The compensator $\widetilde{N}$ of $N$ is given by

[TABLE]

Assume that $W(\cdot)$ and $N(\cdot)$ are independent and that $\nu(\cdot)$ is a Lévy measure so that

[TABLE]

where $a_{1}\wedge a_{2}=\min\{a_{1},a_{2}\}$ for $a_{1},a_{2}\in{\mathbb{R}}$.

We consider a stochastic differential equation with regime-switching together with Lévy-type jumps of the form

[TABLE]

with initial conditions

[TABLE]

where $b(\cdot,\cdot):{\mathbb{R}}^{n}\times{\mathcal{M}}\mapsto{\mathbb{R}}^{n}$, $\sigma(\cdot,\cdot):{\mathbb{R}}^{n}\times{\mathcal{M}}\mapsto{\mathbb{R}}^{n\times n}$, and $\gamma(\cdot,\cdot,\cdot):{\mathbb{R}}^{n}\times M\times{\mathbb{R}}^{n}_{0}\mapsto{\mathbb{R}}^{n}$ are measurable functions, and $\alpha(\cdot)$ is a switching process with a finite state space ${\mathcal{M}}:=\left\{1,\dots,m\right\}$ and generator $Q(x)=(q_{ij}(x))\in{\mathbb{R}}^{m\times m}$. That is, $\alpha(\cdot)$ satisfies

[TABLE]

as $\delta\downarrow 0$, where $q_{ij}(x)\geq 0$ for $i,j\in{\mathcal{M}}$ with $j\not=i$ and $q_{ii}(x)=-\sum_{j\not=i}q_{ij}(x)<0$ for each $i\in{\mathcal{M}}$.

The evolution of the discrete component or the switching process $\alpha(\cdot)$ can be represented by a stochastic integral with respect to a Poisson random measure; see, for example, [36]. In fact, for $x\in{\mathbb{R}}^{n}$ and $i,j\in{\mathcal{M}}$ with $j\not=i$, let $\Delta_{ij}(x)$ be the consecutive left-closed, right-open intervals of the real line, each having length $q_{ij}(x)$. Define a function $h:{\mathbb{R}}^{n}\times{\mathcal{M}}\times{\mathbb{R}}\mapsto{\mathbb{R}}$ by

[TABLE]

Then we may write the switching process (2.4) as a stochastic integral

[TABLE]

where ${N_{1}}(dt,dy)$ is a Poisson random measure (corresponding to a random point process $\mathfrak{p}(\cdot)$) with intensity $dt\times\lambda(dy)$, and $\lambda(\cdot)$ is the Lebesgue measure on ${\mathbb{R}}$. Denote the compensated Poisson random measure of $N_{1}(\cdot)$ by $\widetilde{N}_{1}(dt,dy):=N_{1}(dt,dy)-dt\times\lambda(dy)$. Throughout this paper, we assume that the Lévy process $\psi(\cdot)$, the random point process $\mathfrak{p}(\cdot)$, and the Brownian motion $W(\cdot)$ are independent.

The following condition will be used as our standing assumption throughout the paper.

• (A1)

Assume that for some positive constant $\kappa$, we have

[TABLE]

for all $x,y\in{\mathbb{R}}^{n}$ and $i\in{\mathcal{M}}=\left\{1,\dots,m\right\}$, and that

[TABLE]

Under condition (A1), in view of [39, Proposition 2.1], for each initial condition $(x_{0},\alpha_{0})\in{\mathbb{R}}^{n}\times{\mathcal{M}}$, the system represented by (2.2) and (2.4) (or equivalently, (2.2) and (2.6)) has a unique strong solution $(X(\cdot),\alpha(\cdot))=(X^{x_{0},\alpha_{0}}(\cdot),\alpha^{x_{0},\alpha_{0}}(\cdot))$.

Remark 2.1**.**

We note the following facts.

By the Lipschitz continuity (2.7), both $b(\cdot,i)$ and $\sigma(\cdot,i)$ grow at most linearly.

• 2.

Because $Q(x)$ depends on $x$, $(X(\cdot),\alpha(\cdot))$ is a state-dependent regime-switching jump diffusion. In particular, as in (2.4), the evolution of the switching component $\alpha$ depends on the jump diffusion component $X$. Equation (2.2) shows that the coefficients $b,\sigma$ and $\gamma$ depend on $\alpha$. The $X$ component alone is not necessarily Markovian, but the two-component process $(X,\alpha)$ is. Note that the model given in (2.2) and (2.4) is a substantial generalization of the usual Markovian regime-switching jump diffusion. Indeed, if $Q(x)=Q$, a constant matrix, then $\alpha(\cdot)$ is a Markov chain independent of the Brownian motion $W(\cdot)$ and the Poisson random measure $N(\cdot,\cdot)$. The formulation then reduces to the commonly used jump diffusion with Markov switching in the literature. Treating the regime-switching diffusion counterpart, as demonstrated in [42], compared to the usual Markovian regime-switching diffusion considered in [30, 44], the state-dependent regime-switching diffusion provides a more realistic formulation by allowing the the dependence of $\alpha(\cdot)$ on $X(\cdot)$; see, for example, [28, 41, 42] and the references therein for applications of such state-dependent regime-switching diffusion processes in areas such as mathematical finance, risk management, ecosystem modeling, etc. This paper further includes Lévy-type jumps, adding additional versatility to the model and complexity to the problem.

• 3.

In this paper, the jump part or the discontinuity of $X$ is given by the stochastic integral with respect to the compensated Poisson random measure $\widetilde{N}$. As demonstrated in [2, Chapter 6], we can extend our results in relatively straightforward manners to situations where the jump part is given by $\int_{|y|<c}\gamma(X(t-),\alpha(t-),y)\widetilde{N}(dt,dy)+\int_{|y|\geq c}\zeta(X(t-),\alpha(t-),y)N(dt,dy)$ for some $c\in(0,\infty]$ and appropriate functions $\gamma$ and $\zeta$. But for ease of presentation, we choose not to pursue such an extension in this paper.

The generator of $(X(\cdot),\alpha(\cdot))$ is defined as follows. Denote

[TABLE]

For $g\in\mathcal{D}_{\mathcal{L}}$, we define

[TABLE]

Because of the Lévy measure $\nu(\cdot)$, ${\mathcal{L}}g(x,i)$ may not be well-defined if the function $g(\cdot,i)$ is only assumed to be in class $C^{2}$ for each $i\in{\mathcal{M}}$; see Proposition 2.1, Remark 2.2, and Example 2.1 for some sufficient conditions for $f\in\mathcal{D}_{{\mathcal{L}}}$.

2.2 Preliminary Results

This section is devoted to some preliminary results. Similar to diffusions, for every $f\in\mathcal{D_{L}}$, a result known as generalized Itô’s lemma (see [36, 39, 43]) reads

[TABLE]

where $\cal L$ is the operator associated with the process $(X,\alpha)$ defined in (2.9), and

[TABLE]

It is well known that $M_{1}^{f}$ is a local martingale and, using similar arguments as in [43, Lemma 2.4], $M_{2}^{f}$ is a local martingale. Moreover, $M_{3}^{f}$ is a martingale if $f$ is bounded. In addition, we have the following proposition.

Proposition 2.1**.**

Assume that (A1) holds and that the function $\gamma(\cdot)$ satisfies for all $(x,i)\in{\mathbb{R}}^{n}\times{\mathcal{M}}$,

[TABLE]

where $\rho_{1}(x)>0$ depends only on $x$. Let $f:{\mathbb{R}}^{n}\times{\mathcal{M}}\mapsto{\mathbb{R}}$ be such that for each $i\in{\mathcal{M}}$, $f(\cdot,i)\in C^{2}({\mathbb{R}}^{n})$ and that

[TABLE]

Then $f\in\mathcal{D}_{\mathcal{L}}$.

Proof.

We need to verify that for all $(x,i)\in{\mathbb{R}}^{n}\times{\mathcal{M}}$,

[TABLE]

To this end, we will treat the cases $\left|z\right|\leq 1$ and $|z|>1$ separately.

Using a Taylor expansion, for $|z|\leq 1$, we have

[TABLE]

where $\theta\in[0,1]$. Equation (2.12) and the fact that $f(\cdot,i)\in C^{2}$ imply that $\left|D^{2}f(x+\theta\gamma(x,i,z),i)\right|\leq\rho_{2}(x)<\infty$ for some $\rho_{2}(x)>0$. Then it follows from (A1) that

[TABLE]

Next for $|z|>1$, by the quadratic growth condition in (2.13),

[TABLE]

where $K_{1}$ is some positive constant. Observe from (2.1) that

[TABLE]

Then it follows from (2.15) and Assumption (A1) that

[TABLE]

Combining the two cases establishes (2.14). $\Box$

Remark 2.2**.**

Alternatively, one can replace condition (2.13) by the following conditions: There exist positive constants $K$, $a$, $b$, and some $\rho_{3}(x)>0$ such that

[TABLE]

Then under (A1), (2.12), and (2.16), the assertion of Proposition 2.1 still hold. The proof is similar to that of Proposition 2.1 and we shall omit the details here.

Example 2.1**.**

Suppose $n=1$, $\nu(dz)=\left|z\right|^{-1-\beta}dz$ for some $\beta\in(0,2)$ and

[TABLE]

for some $b_{1}>\frac{\beta}{2}$ and $b_{2}<\frac{\beta}{2}$, where $g_{1}$ and $g_{2}$ are continuous functions satisfying

[TABLE]

for some $\kappa>0$ and $(x,i)\in{\mathbb{R}}\times{\mathcal{M}}$. Consider the operator ${\mathcal{L}}$ defined in (2.9), in which for simplicity we assume that $b=\sigma\equiv 0$. We have

[TABLE]

and similarly

[TABLE]

for some $K=K(\kappa,b_{1},b_{2},\beta)>0$. Thus assumption (A1) is satisfied.

Suppose for each $i\in{\mathcal{M}}$, the function $f(\cdot,i)\in C^{2}$ satisfies the first equation of (2.16) for some $a\in(0,\frac{\beta}{b_{2}})$. Clearly both (2.12) and the second equation of (2.16) are satisfied. Moreover, it is easy to show that $\int_{|z|>1}\left|z\right|^{ab_{2}}\nu(dz)=\int_{|z|>1}|z|^{ab_{2}-1-\beta}dz<\infty$, verifying the third equation of (2.16). Thus it follows that $f\in\mathcal{D}_{{\mathcal{L}}}$.

Corollary 2.1**.**

Under Assumption (A1), all functions $f:{\mathbb{R}}^{n}\times{\mathcal{M}}\to{\mathbb{R}}$ such that $f(\cdot,i)\in C_{c}^{2}$ for each $i\in{\mathcal{M}}$ belong to $\mathcal{D}_{{\mathcal{L}}}$ and the Dynkin formula (2.17) holds:

[TABLE]

where $\tau$ is a stopping time with ${\mathbb{P}}_{x,i}\left\{\tau<\infty\right\}=1$.

Proof.

It is easy to see via the Taylor expansion that $f\in\mathcal{D}_{{\mathcal{L}}}$. The Dynkin formula (2.17) then follows from taking expectations on both sides of (2.10) and the optional sampling theorem ([21, Theorem 1.3.22]). $\Box$

In a similar fashion, we can establish the following corollary.

Corollary 2.2**.**

Under Assumption (A1), all functions $f:{\mathbb{R}}^{n}\times{\mathcal{M}}\to{\mathbb{R}}$ such that $f(\cdot,i)\in C^{2}$ with bounded partial derivatives up to the second order for each $i\in{\mathcal{M}}$ belong to $\mathcal{D}_{{\mathcal{L}}}$ and the Dynkin formula (2.17) holds.

We end the section with a brief discussion on the existence and uniqueness for solution to the system represented by (2.2) and (2.4) (or equivalently, (2.2) and (2.6)) when assumption (A1) is only satisfied locally. The global Lipschitz and linear growth conditions in assumption (A1) for the coefficients of (2.2) can be restrictive in many applications. For instance, the mean-reverting model, the logistic growth model, and the Lotka-Volterra model do not satisfy the linear growth condition. Therefore it is vital to relax assumption (A1). The following result gives a set a sufficient conditions under which system represented by (2.2) and (2.4) (or equivalently, (2.2) and (2.6)) still has a unique strong solution even if (A1) is violated.

Proposition 2.2**.**

Suppose that for each bounded open ball $B(\rho)$ centered at [math] with radius $\rho$, Assumption (A1) is satisfied with $\kappa_{\rho}$ replacing the global constant $\kappa$. Assume also that there is a function $V(\cdot,\cdot):{\mathbb{R}}^{n}\times{\mathcal{M}}\mapsto{\mathbb{R}}_{+}$ having continuous partial derivatives with respect to $x$ up to the second order for each $i\in{\mathcal{M}}$ and satisfying for some positive constants $K_{V}$ and $\gamma_{0}$ that

[TABLE]

and

[TABLE]

Then the system represented by (2.2) and (2.4) $($or equivalently, (2.2) and (2.6)$)$ has a unique strong global solution.

Proof.

Note that (2.18) guarantees that $V\in\mathcal{D}_{\mathcal{L}}$. With the given initial condition $(x_{0},\alpha_{0})\in{\mathbb{R}}^{n}\times{\mathcal{M}}$ as in (2.3), since Assumption (A1) is satisfied locally, for any $k\in\mathbb{N}$ with $|x_{0}|<k$, the system given by (2.2) and (2.4) (or equivalently, (2.2) and (2.6)) has a unique strong solution $(X^{(k)},\alpha^{(k)}):=(X^{(k),x_{0},\alpha_{0}},\alpha^{(k),x_{0},\alpha_{0}})$ locally up to the exit time $t\leq\tau_{k}$, where

[TABLE]

Moreover, as in [16, Section 3.4], we can construct a sequence $(X^{(k)}(\cdot),\alpha^{(k)}(\cdot))$ in such a way so that $(X^{(\iota)}(\cdot),\alpha^{(\iota)}(\cdot))$ are identical before exiting the ball $B(k)$ for all $\iota\geq k$. In particular, we have

[TABLE]

where $\tau_{k}$ is defined in (2.22). Therefore we can define a process $(X,\alpha)$ so that $(X(t),\alpha(t)):=(X^{(k)}(t),\alpha^{(k)}(t))$ for $t<\tau_{k}$. Clearly $\tau_{k}$ is an increasing sequence. Denote $\tau_{\infty}:=\lim_{k\to\infty}\tau_{k}$. We need to show that $\tau_{\infty}=\infty$ a.s. Suppose on the contrary that the statement were false. Then there would exist some $T>0$ and $\varepsilon>0$ such that ${\mathbb{P}}_{x,i}\left\{\tau_{\infty}\leq T\right\}>\varepsilon.$ Therefore we could find some $k_{1}\in\mathbb{N}$ such that

[TABLE]

Define

[TABLE]

Then it satisfies $[(\partial/\partial t)+{\mathcal{L}}]U(x,i,t)\leq 0$. Using the generalized Itô formula (2.10), we have

[TABLE]

where

[TABLE]

Clearly ${\mathbb{E}}[M_{1}^{V}(T\wedge\tau_{k})]={\mathbb{E}}[M_{2}^{V}(T\wedge\tau_{k})]=0$. We need to analyze the term $M_{3}^{V}(\tau_{k}\wedge T)$ carefully. Now it follows from (2.19) that

[TABLE]

Thus it follows that ${\mathbb{E}}[M_{3}^{V}(T\wedge\tau_{k})]=0$, as desired. Taking expectations on both sides of (2.24) and using (2.20), we have

[TABLE]

Hence for all $k\geq k_{1}$, by (2.23) and (2.21), we have

[TABLE]

which is a contradiction. Hence we must have $\tau_{\infty}=\infty$ a.s. or the process $(X(\cdot),\alpha(\cdot))$ is a global solution to the system given by (2.2) and (2.4) (or equivalently, (2.2) and (2.6)).

We proceed to establish the uniqueness. Suppose that there is another global solution $(\widetilde{X},\widetilde{\alpha})$ to (2.2) and (2.4) (or equivalently, (2.2) and (2.6)). By the construction of the processes $(X,\alpha)$ and $(\widetilde{X},\widetilde{\alpha})$ and the fact that $\tau_{k}\to\infty$ a.s. as $k\to\infty$, we have We consider

[TABLE]

This shows that the solution to (2.2)–(2.6) is pathwise unique. $\Box$

3 Feynman-Kac Formula

We aim to derive several versions of the Feynman-Kac formulas, corresponding to coupled systems of partial integro-differential equations of Cauchy and Dirichlet types, respectively. Section 3.1 deals with the Cauchy problem and Section 3.2 investigates the Dirichlet problem. To this end, we need the following lemma, which establishes the moment bounds for the regime-switching jump diffusion.

Lemma 3.1**.**

Let $T>0$ be fixed.

• (a)

Then under Assumption (A1), for any positive constant $p\in(0,2]$, we have

[TABLE]

where $K=K(x,T,p)$ is a constant.

• (b)

Suppose Assumption (A1). In addition, if for some $\widetilde{p}>2$ and $\kappa_{2}>0$,

[TABLE]

Then (3.1) is satisfied for all $p\in(0,\widetilde{p}]$.

Proof.

We shall only prove Part (b); Part (a) can be established in a similar manner.

Step 1: Consider first the case $p=\widetilde{p}\geq 2$. Note that

[TABLE]

Using (2.7), taking expectation in (3.3), for the first terms on the right-hand side of (3.3), similar to [42, Proposition 2.3, pp.31-33], we obtain

[TABLE]

where $K_{i}=K_{i}(x,\widetilde{p},T)$ for $i=1,2$.

By virtue of [31, Lemma 4] (see also [2, Theorem 4.4.23]) together with (2.7) and (3.2),

[TABLE]

Again, $K_{i},i=3,4,5$ depend on $x$, $\widetilde{p}$, and $T$ only. Combining (3.4) and (3.5), we obtain that

[TABLE]

with $K_{4}$ and $K_{5}$ depending on $x$, $\widetilde{p}$, and $T$. The desired result then follows from Gronwall’s inequality.

Step 2: The case when $1\leq p<\widetilde{p}$ follows from Hölder’s inequality.

Step 3: Suppose that $0<p<1$. Since $|x|^{p}=|x|^{p}I_{\{|x|\geq 1\}}+|x|^{p}[1-I_{\{|x|\geq 1\}}]\leq 1+|x|^{1+p}$, using the result in Step 2,

[TABLE]

Combing the above steps gives (3.1). $\Box$

3.1 Cauchy Problems

Theorem 3.1**.**

Assume (A1). Consider the coupled system of partial integro-differential equations of the form

[TABLE]

where ${\mathcal{L}}$ is as in (2.9), $0\leq c(\cdot,i)\in C({\mathbb{R}}^{n})$, and $f(\cdot,i)\in C({\mathbb{R}}^{n})$ for each $i\in{\mathcal{M}}$. If $u$ is a classical solution to (3.7) satisfying

[TABLE]

then it admits a stochastic representation

[TABLE]

Remark 3.1**.**

A smooth function $u:[0,\infty)\times{\mathbb{R}}^{n}\times{\mathcal{M}}\mapsto{\mathbb{R}}$ is said to be a classical solution of (3.7) if (i) $u(\cdot,\cdot,i)\in C^{1,2}$ for each $i\in{\mathcal{M}}$, (ii) $u(t,\cdot,\cdot)$ belongs to the domain $\mathcal{D}_{\mathcal{L}}$ of the generator ${\mathcal{L}}$ for each $t\in[0,\infty)$, and (iii) $u$ satisfies both equations in (3.7) in the classical sense.

Proof.

Suppose $u$ is a classical solution to (3.7). Let $(X(\cdot),\alpha(\cdot))$ be the unique solution to (2.2) and (2.4) (or equivalently, (2.2) and (2.6)) with initial condition $(X(0),\alpha(0))=(x,i)$. For any $t>0$, we define for $0\leq s\leq t$,

[TABLE]

Then it follows from generalized Itô’s formula (2.10) and the first equation of (3.7) that

[TABLE]

where

[TABLE]

Therefore $\left\{M(s),0\leq s\leq t\right\}$ is a local martingale. Put $\tau_{n}:=\inf\left\{t\geq 0:\left|X(t)\right|\geq n\right\}$, $n\in\mathbb{N}$. Then thanks to (2.7), (3.8), and Lemma 3.1, the processes $M_{i}(\cdot\wedge\tau_{n}),i=1,2,3$, are martingales; so is the process $M(\cdot\wedge\tau_{n})$. Therefore, by the definition of $M$ and the optional sampling theorem, we obtain

[TABLE]

Owing to (3.8), the definition of the process $M$, and the assumption that $c\geq 0$, for any $s\in[0,t]$, we have

[TABLE]

Therefore by letting $n\to\infty$, we obtain from the dominated convergence theorem and Lemma 3.1 that ${\mathbb{E}}[M(s)]=u(t,x,i),s\in[0,t]$. In particular, (3.9) follows. $\Box$

Theorem 3.2**.**

Assume (A1) and for some $T>0$. Suppose that $u(\cdot,\cdot,i):[0,T]\times{\mathbb{R}}^{n}\times{\mathcal{M}}\mapsto{\mathbb{R}}$ is of class $C^{1,2}([0,T)\times{\mathbb{R}}^{n})\cap C([0,T]\times{\mathbb{R}}^{n})$ for each $i\in{\mathcal{M}}$ and $u(t,\cdot,\cdot)\in\mathcal{D}_{{\mathcal{L}}}$ for every $t\in[0,T]$ and satisfies the Cauchy problem

[TABLE]

and that satisfies the growth condition

[TABLE]

where for each $i\in{\mathcal{M}}$, the functions $c(\cdot,\cdot,i)\geq 0$, $g(\cdot,\cdot,i)$, and $f(\cdot,i)$ are continuous and satisfy

[TABLE]

and for some $K>0$. Let $(X,\alpha)=(X^{t,x,i},\alpha^{t,x,i})$ be the solution to (2.2) and (2.4) $($or equivalently, (2.2) and (2.6)$)$ with $(X(t),\alpha(t))=(x,i)$. Then we have

[TABLE]

Proof.

Define $\tau_{n}$ as in the proof of Theorem 3.1. Then as before, we apply Itô’s formula to the process $u(s,X(s),\alpha(s))\exp\left\{-\int_{t}^{s}c(r,X(r),\alpha(r))dr\right\},s\in[t,T]$ and then take expectations to obtain

[TABLE]

Since $c(\cdot)\geq 0$ and $\tau_{n}\to\infty$ a.s. as $n\to\infty$, the first term of (3.14) converges to

[TABLE]

by (3.12), Lemma 3.1, and the dominated convergence theorem. Similarly, we can show that as $n\to\infty$, the second term of (3.14) converges to

[TABLE]

Let us analyze the third term of (3.14). Owing to (3.11), it is bounded in absolute value by

[TABLE]

Certainly we have ${\mathbb{P}}_{t,x,i}\left\{\tau_{n}\leq T\right\}\to 0$ as $n\to\infty$. Furthermore, since $p<2$ in (3.11), by the Hölder inequality, Lemma 3.1, and the Chebyshev inequality, we have

[TABLE]

as $n\to\infty$, where $K_{1}$ and $K_{2}$ are positive constants independent of $n$. Thus the third term of (3.14) goes to [math] as $n\to\infty$. Finally we obtain (3.13) by combining the above estimates into (3.14). $\Box$

Remark 3.2**.**

If the Lévy measure $\nu(\cdot)\equiv 0$, then the process $X(\cdot)$ is in fact a regime-switching diffusion and has continuous sample paths [42]. In such a case, similar to [21, Theorem 5.7.6], we can relax assumption (3.11) to

[TABLE]

where $\mu\geq 2$. Indeed, since $|X(\tau_{n})|=n$, one can bound the third term of (3.14) by $K(1+n^{\mu}){\mathbb{P}}_{t,x,i}\{\tau_{n}\leq T\}$. Furthermore, in view of [42, Proposition 2.2.3], for any $\iota>\mu$, we have

[TABLE]

where $K>0$ is independent of $n$. This shows that the third term of (3.14) converges to [math] as $n\to\infty$. However, in the presence of a Lévy measure $\nu(\cdot)$, the inequality $|X(\tau_{n})|\leq n$ is not necessarily true. Thus in general we cannot relax (3.11) and apply the arguments of [21] directly.

If we relax (3.11) to (3.15) for some $\mu\geq 2$ and suppose also (3.12) is replaced by

[TABLE]

where $K$ is a positive constant and $\widetilde{p}>\mu$. Then the stochastic representation (3.13) is still valid if we assume in addition that (3.2) holds for $\widetilde{p}>\mu$.

In fact, if (3.2) holds for $\widetilde{p}>\mu$, then Lemma 3.1, part (b) reveals that we still have the moment bound ${\mathbb{E}}[\sup_{0\leq s\leq T}|X(s)|^{\widetilde{p}}]\leq K<\infty$, where $K=K(T,\widetilde{p})$. Thus similar to the proof of Theorem 3.2, we compute

[TABLE]

as $n\to\infty$. This, together with almost the same argument as that in the proof of Theorem 3.2, helps us to establish the stochastic representation (3.13). We summarize the above discussion into the following corollary.

Corollary 3.1**.**

Assume (A1). Suppose that (3.15), (3.16), and (3.2) hold for some positive constants $\widetilde{p}>\mu\geq 2$. Then the conclusion of Theorem 3.2 continues to hold.

Example 3.1**.**

In this example, we demonstrate that Theorem 3.2 can be applied in mathematical finance. We consider a generalized Black-Scholes market that consists of two assets: a bond and a stock. The price of the bond evolves according to the equation

[TABLE]

where $\alpha(\cdot)$ is a continuous-time Markov chain with generator $Q=(q_{ij})$ and a finite state space ${\mathcal{M}}=\left\{1,\dots,m\right\}$, and $r:{\mathcal{M}}\mapsto{\mathbb{R}}_{+}$. Hence the discounted process is given by $D(t)=\exp\{-\int_{0}^{t}r(\alpha(r))dr\}.$ Suppose that under some risk neutral measure $\mathbb{Q}$, the price of the stock is modeled by the stochastic differential equation

[TABLE]

where $W^{\mathbb{Q}}$ is a one-dimensional Brownian motion and $N^{\mathbb{Q}}$ is a Poisson random measure with compensator $\widetilde{N}^{\mathbb{Q}}(t,E):=N^{\mathbb{Q}}(t,E)-t\nu^{\mathbb{Q}}(E)$ under the risk neutral measure $\mathbb{Q}$, in which $\nu^{\mathbb{Q}}$ is a Lévy measure satisfying $\int_{{\mathbb{R}}_{0}}(1\wedge|z|^{2})\nu^{\mathbb{Q}}(dz)<\infty$. Such a risk neutral measure $\mathbb{Q}$ can be found using the Esscher transform ([12, 15]). For simplicity, we assume that $\alpha(\cdot)$, $W^{\mathbb{Q}}(\cdot)$, and $N^{\mathbb{Q}}(\cdot)$ are independent. For $i\in{\mathcal{M}}$, $r_{i}=r(i),\sigma_{i}=\sigma(i)$ are positive constants and $\gamma(i,\cdot)$ is a real-valued function satisfying $\int_{{\mathbb{R}}_{0}}\gamma(i,z)^{2}\nu^{\mathbb{Q}}(dz)\leq K<\infty$. Furthermore, we assume that $\gamma(i,z)>-1$ for all $i\in{\mathcal{M}}$ and $z\in{\mathbb{R}}_{0}$.

Note that for this example, assumption (A1) is satisfied. Therefore (3.18) has a unique strong solution and the moment bound (3.1) holds. Using the generalized Itô formula (2.10), we obtain $S(t)=S(0)\exp(X(t)),$ where

[TABLE]

For a European type contingent claim with payoff $h(S(T),\alpha(T))$ at the time of expiration $T>0$, according to the fundamental theorem of asset pricing [9],

[TABLE]

gives an arbitrage free price of the derivative at time $t\in[0,T]$, where $\mathbb{F}=\left\{{\mathfrak{F}}_{t},0\leq t\leq T\right\}$ is the natural filtration of $S$. Suppose that for each $i\in{\mathcal{M}}$, the function $h(\cdot,i)$ is continuous and satisfies the growth condition $0\leq h(s,i)\leq K(1+s^{2})$ for some $K>0$ and all $(s,i)\in(0,\infty)\times{\mathcal{M}}$. By the Markov property, we can write $V_{t}=V(t,S(t),\alpha(t))$, where

[TABLE]

By virtue of Theorem 3.2, any solution satisfying the growth condition (3.11) of the terminal value problem

[TABLE]

can be represented by the right hand side of (3.19), where ${\mathcal{L}}^{\mathbb{Q}}$ is defined as follows:

[TABLE]

in which $D_{s}f(s,i)$ and $D_{ss}f(s,i)$ denote the first and second order partial derivatives of $f$ with respect to the variable $s$, respectively. On the other hand, if one can show that the function $V$ defined in (3.19) is a classical (or even a viscosity) solution to (3.20), then we can find the arbitrage free price $V(t,S(t),\alpha(t))$ by solving (3.20).

3.2 A Dirichlet Problem

In this subsection, we work with a bounded and open domain $D\subset{\mathbb{R}}^{n}$ with boundary $\partial D$ and closure $\overline{D}=D\cup\partial D$. Similar to Section 3.1, our goal is to obtain a stochastic representation for the classical solution $u$ (in the sense of Remark 3.1) to the Dirichlet problem

[TABLE]

where for each $i\in{\mathcal{M}}$, the functions $c(\cdot,i)\geq 0$, $\xi(\cdot,i)$, and $\eta(\cdot,i)$ are continuous on their domains. To proceed, we first present the following lemma, which asserts that starting from a point $(x,i)\in D\times{\mathcal{M}}$, the regime-switching jump diffusion given by (2.2) and (2.4) (or equivalently, (2.2) and (2.6)) will exit from $D$ in finite time a.s.

Lemma 3.2**.**

Assume (A1). In addition, suppose that for some $\ell\in\left\{1,2,\dots,n\right\}$,

[TABLE]

Let $\tau$ be the first exit time of the regime-switching jump diffusion of (2.2) and (2.4) $($or equivalently, (2.2) and (2.6)$)$ from $D$:

[TABLE]

Then $\tau<\infty$ a.s. and ${\mathbb{E}}[\tau]<\infty$.

Proof.

We proceed to use the idea in [38]. Since $D$ is bounded, there exists an $R>0$ such that $|x|\leq R$ for all $x\in D$. Furthermore, let us choose positive constants $\beta>3R$ and $K>\beta^{2p}$, where $p>1$ is to be specified later. Consider a function ${\varphi}\in C_{c}^{2}({\mathbb{R}})$ such that

[TABLE]

Moreover, we can pick ${\varphi}$ so that it is nonnegative, symmetric about [math], and nonincreasing on $[0,\infty)$. For any $(t,x,i)\in[0,\infty)\times{\mathbb{R}}^{n}\times{\mathcal{M}},$ set $\Phi(t,x,i):=e^{\lambda t}\phi(x,i)$ with $\phi(x,i):={\varphi}(x_{\ell}-2R)$, where $x_{\ell}$ is the $\ell$th component of $x$ and $\lambda>0$ is a constant to be determined. Note that $\phi\in\mathcal{D}_{{\mathcal{L}}}$ and that for all $x\in\overline{D}$, we have

[TABLE]

Hence $\left|x_{\ell}-2R\right|\leq 3R<\beta$. Thus for all $(x,i)\in D\times{\mathcal{M}},$ $\phi(x,i)=K-(x_{\ell}-2R)^{2p}$, which, in turn, implies that

[TABLE]

We claim that for all $(x,i)\in D\times{\mathcal{M}}$,

[TABLE]

To see this, we consider two cases. If $|x_{\ell}+\gamma_{\ell}(x,i,z)-2R|\leq\beta$, then $\phi(x+\gamma(x,i,z),i)={\varphi}(x_{\ell}+\gamma_{\ell}(x,i,z)-2R)=K-(x_{\ell}+\gamma_{\ell}(x,i,z)-2R)^{2p}$. Using a Taylor expansion, we have

[TABLE]

for some $\theta\in[0,1]$, where ${\varphi}_{x_{\ell}}=(d/dx_{\ell}){\varphi}$ and ${\varphi}_{x_{\ell}x_{\ell}}=(d^{2}/dx_{\ell}^{2}){\varphi}$ are the first-order and second-order derivatives of ${\varphi}$, respectively. Since $|x_{\ell}-2R|\leq\beta$ and $|x_{\ell}+\gamma_{\ell}(x,i,z)-2R|\leq\beta$, we must have $|x_{\ell}-2R+\theta\gamma_{\ell}(x,i,z)|\leq\beta$. Thus

[TABLE]

Therefore we arrive at

[TABLE]

On the other hand, if $|x_{\ell}+\gamma_{\ell}(x,i,z)-2R|>\beta$, by the monotonicity of ${\varphi}$ on $[0,\infty)$, we have

[TABLE]

Note that for all $(x,i)\in D\times{\mathcal{M}}$, since $|x_{\ell}-2R|\leq 3R<\beta$,

[TABLE]

Then it follows that

[TABLE]

Furthermore, a simple contradiction argument reveals that for all $(x,i)\in D\times{\mathcal{M}}$,

[TABLE]

Then it follows from (2.7) that

[TABLE]

This, together with (3.26) and (3.27), implies (3.25). Putting (3.25) into (3.24), and using (3.22) and (3.23), we deduce

[TABLE]

by selecting $p>1$ sufficiently large, where $\rho=\rho(p,R,\beta,\overline{D})>0$ is a constant.

Now we apply the generalized Itô formula to the process $\Phi(X(t),\alpha(t))=e^{\lambda t}\phi(X(t),\alpha(t))$ to get

[TABLE]

where $M_{k}^{\Phi}(t),k=1,2,3$ are defined as

[TABLE]

Using the definition of $\phi(\cdot)$, we have ${\mathbb{E}}[M_{k}^{\Phi}(t\wedge\tau)]=0$ for $k=1,2,3$. Moreover, for all $s\geq 0$, $\phi(X(s-),\alpha(s-))\leq K<\infty$, with $K$ being the constant in the definition of the function ${\varphi}(\cdot)$. Note that for $\lambda>0$ sufficiently small, we have $\lambda K-\rho<0$. Thus it follows from (3.28) that

[TABLE]

Now since the function $\phi(\cdot)$ is nonnegative and bounded above by $K$, we have

[TABLE]

Passing to the limit as $t\to\infty$ yields that ${\mathbb{E}}[e^{\lambda\tau}]<\infty$. Thus, we conclude that $\tau<\infty$ a.s. and ${\mathbb{E}}[\tau]<\infty$. $\Box$

To proceed, we further impose the following condition.

• (A2)

Assume $\partial D\in C^{2}$ and that for each $i\in{\mathcal{M}}$,

[TABLE]

Theorem 3.3**.**

Assume (A1), (A2), (3.22), and

[TABLE]

Then the solution of the system of boundary value problem (3.21) is given by

[TABLE]

Proof.

We consider the process $M_{t}:=\exp\{-\int_{0}^{t}c(X(s),\alpha(s))ds\}u(X(t),\alpha(t))$. Since $u\in{\mathcal{D}}_{{\mathcal{L}}}$ for any $t\geq 0$, we can apply the generalized Itô formula (2.10) to obtain

[TABLE]

where

[TABLE]

Since $D$ is bounded, we have ${\mathbb{E}}[M_{1}(t\wedge\tau)]={\mathbb{E}}[M_{2}(t\wedge\tau)]=0$ and, thanks to (3.29), we can show ${\mathbb{E}}[M_{3}(t\wedge\tau)^{2}]<\infty$ and hence ${\mathbb{E}}[M_{3}(t\wedge\tau)]=0$.

Now it follows from (3.21) that

[TABLE]

Since $\xi(\cdot)$ is bounded and $c(\cdot)\geq 0$, Lemma 3.2 and the bounded convergence theorem lead to

[TABLE]

as $t\to\infty$. On the other hand, using the continuity of the function $u$ and boundedness of the function $\eta(\cdot)$, we have

[TABLE]

where $K_{1}:=\max\{|u(x,i)|:x\in\overline{D},i\in{\mathcal{M}}\}$ and $K_{2}:=\sup\{|\eta(x,i)|:x\in D^{c},i\in{\mathcal{M}}\}$. Thus we can again apply the bounded convergence theorem and Lemma 3.2 to derive

[TABLE]

Putting the above equations together leads to (3.30). $\Box$

4 Arcsine Laws

Paul Lévy’s celebrated arcsine law [26] states that for all $a\in(0,1)$,

[TABLE]

The arcsine law plays a crucial role in the theory of fluctuations in random walks. In fact, it was shown in [13] that for any random walk $\{S_{n}\}_{n=1}^{\infty}$ with zero mean and unit variance,

[TABLE]

Later, the arcsine laws were generalized in [22, 25, 37], to name just a few.

We aim to derive an arcsine law for regime-switching jump diffusion processes considered in this paper. To be precise, we consider a one-dimensional regime-switching jump diffusion process, in which for simplicity, we assume that the drift term $b$ is identically zero. Moreover, motivated by applications in mathematical finance [4], risk modeling [40], queueing networks [41], etc., we suppose that the switching process is singularly perturbed with fast switching (see Section 4.1 for the precise formulation). We show that as $\varepsilon\to 0$, the regime-switching jump diffusion process converges weakly and that the limiting process has the arcsine law. To this purpose, we first show in Section 4.1 that under certain assumptions, the regime-switching jump diffusion process with fast switching converges weakly to a diffusion process. This result, together with the arcsine law for null recurrent diffusion established in [22], helps us to derive the desired arcsine law for regime-switching jump diffusion processes in Proposition 4.1, which is in Section 4.2. Finally we remark in Section 4.3 that in general there is no $L_{2}$ convergence associated with the weak convergence result established in Section 4.1.

4.1 Weak Limit of Switching Jump Diffusions with Fast Switching

For one-dimensional diffusions, it is well-known (see, e.g., [21, Section 5.5]) that through a proper transformation, one can remove the drift term. For simplicity, we consider a one-dimensional regime-switching jump diffusion without drift of the following form

[TABLE]

where ${\mathbb{R}}_{0}:={\mathbb{R}}-\left\{0\right\}$ (consistent with the definition ${\mathbb{R}}^{n}_{0}$ with $n=1$) and $\varepsilon>0$ is a small parameter. We assume $x_{0}$ to be independent of $\varepsilon$ and non-random for simplicity. Throughout this section, we assume that the switching process is a continuous-time Markov chain $\alpha^{\varepsilon}(t)$ taking values in ${\mathcal{M}}=\left\{1,\dots,m\right\}$. Moreover, the Markov chain is time-homogeneous with a generator $Q^{\varepsilon}=Q/\varepsilon$ such that $Q$ is weakly irreducible [41, p. 23]. Denote the associated quasi-stationary distribution by $\nu=(\nu_{1},\dots,\nu_{m})\in{\mathbb{R}}^{1\times m}$. As in Section 2, we assume that the Markov chain $\alpha^{\varepsilon}(\cdot)$, the Brownian motion $W(\cdot)$, and the Poisson random measure $N(\cdot)$ are independent.

Because of the assumption that $Q^{\varepsilon}=Q/\varepsilon$, $\alpha^{e}(\cdot)$ is the so-called singularly perturbed process with fast switching. The motivation for considering such systems stems from a wide range of applications in flexible manufacturing systems, production planning, queueing networks, mathematical finance, risk modeling, etc. The main goal is to obtain limiting systems with reduced complexity that still have good approximation properties to the original systems. For example, in [41], starting with the motivation of a production planning problem in Section 1.1, Sections 3.3-3.5 present a number of examples of queues with finite capacity, system reliability, competing risk models, stochastic optimization problems, and linear quadratic control problems among others. It was demonstrated that a two-time-scale approach using $Q^{\varepsilon}=Q/\varepsilon$ leads to simpler limiting systems with good approximating properties. Additionally, using such a two-time-scale approach, as in [32], one can consider time-inhomogeneous Markovian models with slowly varying rates as well. The reader is referred to the aforementioned references for further reading.

We proceed to obtain the weak convergence theorem of this section. Let $X^{\varepsilon}(\cdot)$ be the solution to (4.1). Then Theorem 4.1 indicates that as $\varepsilon\to 0$, $X^{\varepsilon}(\cdot)$ converges weakly to $X(\cdot)$, where $X(\cdot)$ is a solution to the following stochastic differential equation

[TABLE]

with

[TABLE]

Remark 4.1**.**

Define

[TABLE]

By virtue of [21, Theorems 5.5.4 and 5.5.7],

• •

the SDE (4.2) has a non-exploding weak solution for every initial distribution $\mu$ if and only if $I(\overline{\sigma})\subset Z(\overline{\sigma})$,

• •

the SDE (4.2) has a solution that is unique in the sense of probability law if and only if $I(\overline{\sigma})=Z(\overline{\sigma})$.

Suppose that $\sigma(\cdot,i)$ is continuous and $\nu_{i}>0$ for each $i\in{\mathcal{M}}$. Then it is straightforward to show that $I(\overline{\sigma})\subset Z(\overline{\sigma})$ and hence (4.2) has a non-exploding weak solution. Suppose also that $Z(\overline{\sigma})=\emptyset$ (i.e., the diffusion is non-degenerate). Then the inclusion $Z(\overline{\sigma})\subset I(\overline{\sigma})$ is trivially satisfied and thus (4.2) has a unique weak solution.

Theorem 4.1**.**

Assume condition (A1) and suppose that $Q$ is weakly irreducible and equation (4.2) has a unique $($in the sense of probability law$)$ solution for each initial condition. Then for $0<T<\infty$, $\{X^{\varepsilon}(\cdot)\}$ is tight in $D([0,T]:{\mathbb{R}})$ and any weakly convergent subsequence has limit $X(\cdot)$, which is the solution of (4.2).

Proof.

The proof is divided into several steps.

Step 1: We first prove the tightness of $\{X^{\varepsilon}(\cdot)\}$ in $D([0,T]:{\mathbb{R}})$. Note that by virtue of Lemma 3.1, ${\mathbb{E}}\sup_{t\in[0,T]}|X^{\varepsilon}(t)|^{2}<\infty$. For any $\delta>0$ and any $0<t,s<T$ with $s\leq\delta$, we have

[TABLE]

Use ${\mathbb{E}}^{\varepsilon}_{t}$ to denote the conditioning on the $\sigma$-algebra generated by ${\cal F}^{\varepsilon}_{t}=\{X^{\varepsilon}(u):u\leq t\}$. Then by the Hölder inequality and the Lipschitz continuity of $\sigma(\cdot,i)$ for each $i$, we have

[TABLE]

where

[TABLE]

Using Lemma 3.1, (4.6) implies that

[TABLE]

It then follows that there is a $\widetilde{\mu}^{\varepsilon}(\delta)$ such that

[TABLE]

and that

[TABLE]

Therefore, by the Kurtz tightness criterion [24, Theorem 3, p.47], $\{X^{\varepsilon}(\cdot)\}$ is tight in $D([0,T],{\mathbb{R}})$.

Step 2: Characterize the limiting process. Since $\{X^{\varepsilon}(\cdot)\}$ is tight, by Prohorov’s theorem, we can extract a weakly convergent subsequence. With a slight abuse of notation, let’s still index such a subsequence by $\{X^{\varepsilon}(\cdot)\}$ with limit denoted by $X(\cdot)$. In view of the Skorohod representation theorem, we may assume without loss of generality that $X^{\varepsilon}(\cdot)$ converges to $X(\cdot)$ a.s., and the convergence is uniform on any bounded interval. We proceed to characterize the limiting process using a martingale problem formulation. We pick out any $\rho(\cdot)\in C^{3}_{0}$ (class of $C^{3}$ functions with compact support), which implies that $D^{2}_{x}\rho(\cdot)$ is Lipschitz. For any $t,s>0$,

[TABLE]

Moreover,

[TABLE]

Choose an arbitrary real-valued, bounded and continuous function $\varphi(\cdot)$, positive integer $\kappa$, $t>0$, $s>0$, and $t_{l}\leq t$ with $l\leq\kappa$, we have

[TABLE]

by the weak convergence and the Skorohod representation, together with the continuity of $\varphi(\cdot)$ and $\rho(\cdot)$. Next, in view of the moment bounds of the jump term, the weak convergence, and the Skorohod representation, we also have

[TABLE]

Dividing $[t,t+s]$ into subintervals of length $\Delta_{\varepsilon}=\varepsilon^{(1-\Delta)/2}$ for some $0<\Delta<1$. We have

[TABLE]

where

[TABLE]

Using the boundedness and smoothness of $\rho(\cdot)$, the Lipschitz continuity of $\sigma(\cdot,\alpha)$, and (4.12), it is easily seen that

[TABLE]

Concentrating on the term in the second line of (4.11), we have

[TABLE]

Using similar estimates as in [41, Lemma 5.35 (a)],

[TABLE]

By the boundedness of $D^{2}_{x}\rho(\cdot)$, the linear growth condition of $\sigma(\cdot,i)$, and the estimate of the second moment ${\mathbb{E}}\sup_{0\leq t\leq T}|X^{\varepsilon}(t)|^{2}<\infty,$ we have

[TABLE]

Next, we obtain

[TABLE]

Combining (4.9)–(4.15), we obtain that

[TABLE]

That is,

[TABLE]

Equivalently, $X(\cdot)$ is a solution of the martingale problem with operator

[TABLE]

The desired result thus follows. $\Box$

4.2 Arcsine Laws

The arcsine law due to [22] is concerned with null recurrent diffusions. Such null recurrent Markov processes were studied in details in [16] and refined and more verifiable conditions were given in [23]. Note that a necessary and sufficient condition for the one-dimensional diffusion process $X(t)$ given in (4.2) to be null recurrent is

[TABLE]

• (A3)

The following conditions hold:

• $\bullet$

$\overline{\sigma}^{2}(x)\geq\sigma_{0}>0$ for all $x\in{\mathbb{R}}$, and for some $p_{\pm}\in{\mathbb{R}}$,

[TABLE]

• $\bullet$

for a bounded and piecewise continuous function $f:{\mathbb{R}}\mapsto{\mathbb{R}}$, there exist $f_{+}\not=f_{-}$ such that

[TABLE]

Now let us state the arcsine law given in [22].

Lemma 4.1**.**

Assume (A3). Consider (4.2) with $\overline{\sigma}(x)$ given in (4.3) and define

[TABLE]

Then the following results hold.

• (i)

When $p_{+}=p_{-}$, the limiting distribution is the arcsine law

[TABLE]

• (ii)

When $p_{+}\not=p_{-}$, the limiting distribution coincides with the distribution of a random variable $\Xi$ such that for all $z>0$ and $A=\sqrt{p_{+}/p_{-}}$,

[TABLE]

The distribution of $\Xi$ is uniquely determined by (4.19).

Lemma 4.1 presents an arcsine law for the limiting diffusion process. We further obtain a limiting distributional result. By virtue of Theorem 4.1, $X^{\varepsilon}(\cdot)$ converges weakly to $X(\cdot)$ in $D[0,\infty)$ (that is, for any $T<\infty$, $X^{\varepsilon}(\cdot)$ converges weakly to $X(\cdot)$ in $D[0,T]$). Furthermore, using perturbed Lyapunov function argument (see [24, Chapter 4]), it is not difficult to demonstrate that $\{X^{\varepsilon}(t):t>0,\ \varepsilon>0\}$ is tight. Define

[TABLE]

and

[TABLE]

Since $f(\cdot)$ is a bounded and piecewise continuous function, Theorem 4.1 and [6, Corollary 5.2, p.31] yield that $f(X^{\varepsilon}(\cdot))$ converges weakly to $f(X(\cdot))$ in $D[0,T]$ for any $T<\infty$. Thus, $\eta^{\varepsilon}(\cdot)$ converges weakly to $\eta(\cdot)$. By Lemma 4.1 together with the tightness of $\{X^{\varepsilon}(t):$ $t>0$, $\varepsilon>0\}$, $\eta(T)$ as random variables converge in distribution to a random variable $\widehat{\eta}$ such that either (i) or (ii) in Lemma 4.1 holds. We summarize this into the following result.

Proposition 4.1**.**

Under the conditions of Theorem 4.1 and Lemma 4.1,

• (i)

when $p_{+}=p_{-}$,

[TABLE]

• (ii)

*when $p_{+}\not=p_{-}$, the limiting distribution of $\eta^{\varepsilon}(T)$ *$($as $\varepsilon\to 0$ and then ${T\to\infty})$ is the same as the distribution of a random variable $\Xi$ given in (4.19) for all $z>0$ with $A=\sqrt{p_{+}/p_{-}}$. The distribution of $\Xi$ is uniquely determined by (4.19).

4.3 There Is No $L_{2}$ Convergence

We have established weak convergence in Theorem 4.1. One natural question is: Can we get a stronger convergence in the sense of $L_{2}$? The following example gives a negative answer to the question.

Example 4.1**.**

Consider a regime-switching diffusion

[TABLE]

where $\sigma(1)=\sigma_{1}\not=\sigma(2)=\sigma_{2}$, and $\alpha^{\varepsilon}\in\left\{1,2\right\}$ is a continuous-time Markov chain generated by $Q/\varepsilon$ with $Q=\begin{pmatrix}-q_{1}&q_{1}\\ q_{2}&-q_{2}\end{pmatrix}$ and $q_{1},q_{2}>0$. By virtue of Theorem 4.1, $X^{\varepsilon}$ converges weakly to $X$, the solution to

[TABLE]

where $\overline{\sigma}=\sqrt{\sigma_{1}^{2}\nu_{1}+\sigma^{2}_{2}\nu_{2}}$, and $\nu_{1}=\frac{q_{2}}{q_{1}+q_{2}}$, $\nu_{2}=\frac{q_{1}}{q_{1}+q_{2}}$. Note that

[TABLE]

Using the Kolmogorov forward equation, one can find that

[TABLE]

Then it follows that

[TABLE]

Consequently we have

[TABLE]

Note that under the condition $\sigma_{1}\not=\sigma_{2}$, one can immediately verify that $\overline{\sigma}-(\sigma_{1}\nu_{1}+\sigma_{2}\nu_{2})>0$ and hence ${\mathbb{E}}_{x,1}\left|X^{\varepsilon}(t)-X(t)\right|^{2}\not\to 0$ as $\varepsilon\to 0$.

5 Further Remarks

This paper has been devoted to revealing the connections of regime-switching jump diffusions with a class of coupled systems of partial integro-differential equations. Under broad conditions, we have obtained serval versions of the Feynman-Kac formulas together with the associated initial, terminal, and boundary value problems. Moreover, certain limiting results have been obtained for processes with fast switching. In addition, arcsine laws for a limiting process enables us to draw conclusion for certain systems with two-time-scale formulation.

In this work, the jump part is driven by a Poisson random measure associated with a Lévy process. A worthwhile future effort is to treat systems in which the random driving force is an alpha-stable process that has finite $p$th moment with $p<2$. This requires more work and careful consideration. In this paper, when the switching component $\alpha$ switches at time $\tau$ or $\alpha(\tau)\not=\alpha(\tau-)$, it is assumed the $X$ component is fixed or $X(\tau)=X(\tau-)$. A relevant question is: Can we allow the $X$ component to take place from $x$ in one plane to $y$ in another plane at the instant of a switching? Mathematically, the switching part will also be represented by an integral operator as in the formulation of [18]. This adds another fold of difficulty. With the aid of the Feynman-Kac formulas, we may proceed to treat many stochastic control problems. For example, combining real options, game theory, and a regime-switching formulation with jumps, we may consider an irreversible investment problem with Stackelberg leader-follower competition and market regime changes (see the related work using switching diffusion formulation [5]). The treatment of the real options and the related problems with competition have received resurgent attention lately. Furthermore, effort may also be directed to applications and extensions to ratchet theory for molecular motor, and stochastic dynamics of electrical membrane with voltage-dependent ion channel fluctuations. All of these will be worthwhile future efforts.