# Feynman-Kac formula for switching diffusions: connections of systems of partial differential equations and stochastic differential equations

- Nicholas A Baran
^{1}, - George Yin
^{1}Email author and - Chao Zhu
^{2}

**2013**:315

https://doi.org/10.1186/1687-1847-2013-315

© Baran et al.; licensee Springer. 2013

**Received: **15 July 2013

**Accepted: **20 September 2013

**Published: **8 November 2013

## Abstract

This work develops Feynman-Kac formulae for switching diffusion processes. It first recalls the basic notion of a switching diffusion. Then the desired stochastic representations are obtained for boundary value problems, initial boundary value problems, and the initial value problems, respectively. Some examples are also provided.

## Keywords

## 1 Introduction

Because of the increasing demands and complexity in modeling, analysis, and computation, significant efforts have been made searching for better mathematical models in recent years. It has been well recognized that many of the systems encountered in the new era cannot be represented by the traditional ordinary differential equation and/or stochastic differential equation models alone. The states of such systems have two components, namely, state = (continuous state, discrete event state). The discrete dynamics may be used to depict a random environment or other stochastic factors that cannot be represented in the traditional differential equation models. Dynamic systems mentioned above are often referred to as hybrid systems. One of the representatives in the class of hybrid system is a switching diffusion process. A switching diffusion process can be thought of as a number of diffusion processes coupled by a random switching process. At a first glance, these processes are seemingly similar to the well-known diffusion processes. A closer scrutiny shows that switching diffusions have very different behavior compared to traditional diffusion processes. Within the class of switching diffusion processes, when the discrete event process or the switching process depends on the continuous state, the problem becomes much more difficult; see [1, 2]. Because of their importance, switching diffusions have drawn much attention in recent years. Many results such as smooth dependence of the initial data, recurrence, positive recurrence, ergodicity, stability, and numerical methods for solution of stochastic differential equations with switching, *etc*., have been obtained. Nevertheless, certain important concepts are yet fully investigated. The Feynman-Kac formula is one of such representatives.

For diffusion processes, the Feynman-Kac formula provides a stochastic representation for solutions to certain second-order partial differential equations (PDEs). These representations are standard in any introductory text to stochastic differential equations (SDEs); see, for example, [3–6], and references therein. The utility of Feynman-Kac formula has enjoyed a wide-range of applications in such areas as stochastic control, mathematical finance, risk analysis, and related fields.

This work aims to derive Feynman-Kac formula for switching diffusions. It provides a probabilistic approach to the study of weakly coupled elliptic systems of partial differential equations (see [7] for weakly coupled systems). Such systems arise in financial mathematics and in the form of the so called diffusion-reaction equations, which describe the concentration of a substance under the influence of diffusion and chemical reactions. The case where the discrete process is a two state process can be found in [[8], Section 5.4]. Our effort is on developing general results, in which the switching process has a finite state space and is continuous-state dependent.

The rest of the paper is organized as follows. We begin by presenting the necessary background materials and problem formulation regarding switching diffusions in Section 2. The setup is in line with that of [1]. Then, using the generalized Itô formula and Dynkin’s formula, we present the Feynman-Kac formula in the context of the Dirichlet problem in Section 4, the initial boundary value problem in Section 5. Finally, we study the Cauchy problem in Section 6.

## 2 Switching diffusions

*i.e.*, ${\mathcal{F}}_{0}$ contains all the null sets and the filtration $\{{\mathcal{F}}_{t}\}$ is right continuous). The probability space $(\mathrm{\Omega},\mathcal{F},P)$ together with the filtration $\{{\mathcal{F}}_{t}\}$ is denoted by $(\mathrm{\Omega},\mathcal{F},\{{\mathcal{F}}_{t}\},P)$. Suppose that $\alpha (\cdot )$ is a stochastic process with right-continuous sample paths (or a pure jump process), finite-state space $\mathcal{M}=\{1,\dots ,{m}_{0}\}$, and

*x*-dependent generator $Q(x)$, so that for a suitable function $f(\cdot ,\cdot )$,

*q*-property [1]. That is, $Q(x)=({q}_{ij}(x))$ satisfies

- (i)
${q}_{ij}(x)$ is Borel measurable and uniformly bounded for all $i,j\in \mathcal{M}$ and $x\in {\mathbb{R}}^{n}$;

- (ii)
${q}_{ij}(x)\ge 0$ for all $x\in {\mathbb{R}}^{n}$ and $j\ne i$; and

- (iii)
${q}_{ii}(x)=-{\sum}_{j\ne i}{q}_{ij}(x)$ for all $x\in {\mathbb{R}}^{n}$ and $i\in \mathcal{M}$.

The process given by (2) and (3) is called a switching diffusion or a regime-switching diffusion. Now, before carrying out our analysis, we state a theorem regarding existence and uniqueness of the solution of the aforementioned stochastic differential equation, which will be important in what follows.

**Theorem 1** (Yin and Zhu [1])

*Let*$x\in {\mathbb{R}}^{n}$, $\mathcal{M}=\{1,\dots ,{m}_{0}\}$,

*and*$Q(x)=({q}_{ij}(x))$

*be an*${m}_{0}\times {m}_{0}$

*matrix satisfying the*

*q*-

*property*.

*Consider the two component process*$Y(t)=(X(t),\alpha (t))$

*given by*(2)

*with initial data*$(x,i)$.

*Suppose that*$Q(\cdot ):{\mathbb{R}}^{n}\to {\mathbb{R}}^{{m}_{0}\times {m}_{0}}$

*is bounded and continuous*,

*and that the functions*$b(\cdot ,\cdot )$

*and*$\sigma (\cdot ,\cdot )$

*satisfy*

*for some constant*$K>0$,

*and for each*$N>1$,

*there exists a positive constant*${M}_{N}$

*such that for all*$i\in \mathcal{M}$

*and all*$x,y\in {\mathbb{R}}^{n}$

*with*$|x|\vee |y|\le {M}_{N}$,

*where* $a\vee b=max(a,b)$ *for* $a,b\in \mathbb{R}$. *Then there exists a unique solution to* (2), *in which the evolution of the discrete component is given by* (3).

Note that (4) and (5) are known as the linear growth and local Lipschitz conditions, respectively. We assume these conditions on $b(\cdot ,\cdot )$ and $\sigma (\cdot ,\cdot )$ for the remainder of the paper.

### 2.1 Itô’s Formula

where $Dg(\cdot ,i)=(\frac{\partial g}{\partial {x}_{1}},\dots ,\frac{\partial g}{\partial {x}_{n}})$, ${D}^{2}g(\cdot ,i)$ denotes the Hessian of $g(\cdot ,i)$, and $Q(x)g(x,\cdot )(i)$ is given by (1). The choice for ℒ will become clear momentarily.

where *h* is an integer-valued function; furthermore, this representation is equivalent to (3). For details, we refer the reader to [9] and [1].

*t*sufficiently small so that $\alpha (t)$ agrees with the initial data. Then it follows that

*t*tend to zero, one gets

*t*is replaced by a stopping time

*τ*satisfying $\tau <\mathrm{\infty}$ w.p.1 (recalling that $g(\cdot ,i)\in {C}_{0}^{2}$), then

Note that if *τ* is the first exit time of the process from a bounded domain satisfying $\tau <\mathrm{\infty}$ w.p.1, then Dynkin’s formula holds for any $g(\cdot ,i)\in {C}^{2}$ and each $i\in \mathcal{M}$ without the compact support assumption. To proceed, we obtain the following system of Kolmogorov backward equations for switching diffusions; see also [2].

**Theorem 2** (Kolmogorov backward equation)

*Suppose that*$g(\cdot ,i)\in {C}_{0}^{2}({\mathbb{R}}^{n})$,

*for*$i\in \mathcal{M}$,

*and define*

*Then*

*u*

*satisfies*

A proof of the theorem can be found in [[2], Theorem 5.2]; see also Theorem 5.1 in the aforementioned reference.

**Remark 1**We illustrate the proof of the theorem using the idea as in [[6], p. 140]. Fix $t>0$. Then using (10) and the Markov property, we have

Thus, by the definition of ℒ, (13) is satisfied.

## 3 The Feynman-Kac formula

We now state the Feynman-Kac formula, which is a generalization of the Kolmogorov backward equation.

**Theorem 3** (The Feynman-Kac formula)

*Suppose that*$g(\cdot ,i)\in {C}_{0}^{2}({\mathbb{R}}^{n})$,

*and let*$c(\cdot ,i)\in C({\mathbb{R}}^{n})$

*be bounded*; $i\in \mathcal{M}$.

*Define*

*Then*

*v*

*satisfies*

*Proof*To simplify the notation, let

*r*sufficiently small. Denote the first jump time of $\alpha (\cdot )$ by ${\tau}_{1}$. With $\alpha (0)=i$, for any $t\in [0,{\tau}_{1})$, $\alpha (t)=i$. It follows that

*f*is differentiable at the origin and

as claimed. This completes the proof. □

So we have seen that the functions given by (12) and (14) necessarily satisfy certain initial value problems. The remainder of the paper will be dedicated to giving stochastic representations for solutions to certain partial differential equations (PDEs) related to the operator ℒ.

## 4 The Dirichlet problem

where *∂O* denotes the boundary of *O*. To proceed, we impose assumption (A1).

- 1.
$\partial O\in {C}^{2}$,

- 2.
for some $1\le j\le r$, and all $i\in \mathcal{M}$, ${min}_{x\in \overline{O}}{a}_{jj}(x,i)>0$,

- 3.
$a(\cdot ,i)$ and $b(\cdot ,i)$ are uniformly Lipschitz continuous in $\overline{O}$ for each $i\in \mathcal{M}$,

- 4.
$c(x,i)\le 0$ and $c(\cdot ,i)$ is uniformly Hölder continuous in $\overline{O}$ for each $i\in \mathcal{M}$,

- 5.
$\psi (\cdot ,i)$ is uniformly continuous in $\overline{O}$, and $\phi (\cdot ,i)$ is continuous on

*∂O*, both for each $i\in \mathcal{M}$.

It follows that under (A1), the system of boundary value problems has a unique solution; see [3] or [5]. Our goal is to derive a stochastic representation for this problem, similar to the Feynman-Kac formula. In order to achieve this, we need the following lemma.

**Lemma 1** *Suppose that* $\tau =inf\{t\ge 0:{X}^{x}(t)\notin O\}$. *That is*, *τ* *is the first exit time from the open set* *O* *of the switching diffusion given in* (2) *and* (3). *Then* $\tau <\mathrm{\infty}$ *w*.*p*.1.

*Proof*We use the idea as in [3]. Consider a function $V:{\mathbb{R}}^{n}\times \mathcal{M}\to \mathbb{R}$ defined by

*V*is independent of $i\in \mathcal{M}$,

*λ*and $A=A(\lambda )$ sufficiently large, we can make $\mathcal{L}V(x,i)\le -1$ for each $i\in \mathcal{M}$. As the function $V(\cdot ,i)$ and its derivatives w.r.t.

*x*are bounded on $\overline{O}$, we may apply Dynkin’s formula to yield

Taking the limit as $t\to \mathrm{\infty}$, and using the monotone convergence theorem yields ${E}^{x,i}\tau <\mathrm{\infty}$, which in turn leads to $\tau <\mathrm{\infty}$ w.p.1. □

**Theorem 4**

*Suppose that*(A1)

*holds*.

*Then with*

*τ*

*as in the previous lemma*,

*the solution of the system of boundary value problems*(16)

*is given by*

*Proof*We apply Itô’s formula to the switching process

Taking the limit as $t\to \mathrm{\infty}$ and noting the boundary conditions, (17) follows. □

## 5 The initial boundary value problem

*O*is the same as before and

We will use assumption (A2).

- 1.
$\u3008a(x,t,i)y,y\u3009\ge \kappa {|y|}^{2}$, for each $i\in \mathcal{M}$ and for $y\in {\mathbb{R}}^{n}$ ($\kappa >0$),

- 2.
${a}_{lk}(\cdot ,\cdot ,i)$, ${b}_{l}(\cdot ,\cdot ,i)$ are uniformly Lipschitz continuous in $\overline{O}\times [0,T]$, for each $i\in \mathcal{M}$,

- 3.
$c(\cdot ,\cdot ,i)$ and $\psi (\cdot ,\cdot ,i)$ are uniformly Hölder continuous in $\overline{O}\times [0,T]$, for each $i\in \mathcal{M}$,

- 4.
$\phi (\cdot ,i)$ is continuous on $\overline{O}$, $\varphi (\cdot ,\cdot ,i)$ is continuous on $\partial O\times [0,T]$, for each $i\in \mathcal{M}$, where

*∂O*denotes the boundary of*O*, - 5.
$\phi (x,i)=\varphi (x,T,i)$, for $x\in \partial O$.

*u*to be Lipschitz continuous in the time variable; namely we require

in addition to (4) and (5).

with initial data $(X(t),\alpha (t))=(x,i)$. If we let $\sigma (x,t,i)$ be the square root of $a(x,t,i)$, then the following is true.

**Theorem 5**

*Suppose that*(A2)

*holds*.

*Then the solution of the system of initial value problems in*(18)

*is given by*

*Proof*Proceeding similarly to the previous theorem, we apply Itô’s formula to the process

in the above derivation, one gets (21). □

## 6 The Cauchy problem

To proceed, we impose assumption (A3).

- 1.
The functions ${a}_{lk}(\cdot ,\cdot ,i)$, ${b}_{l}(\cdot ,\cdot ,i)$ are bounded in ${\mathbb{R}}^{n}\times [0,T]$ and uniformly Lipschitz continuous in $(x,t,i)$ in compact subsets of ${\mathbb{R}}^{n}\times [0,T]\times \mathcal{M}$, for each $i\in \mathcal{M}$.

- 2.
The functions ${a}_{lk}(\cdot ,\cdot ,i)$ are Hölder continuous in

*x*, uniformly with respect to $(x,t,i)$ in ${\mathbb{R}}^{n}\times [0,T]\times \mathcal{M}$, for each $i\in \mathcal{M}$. - 3.
The function $c(\cdot ,\cdot ,i)$ is bounded in ${\mathbb{R}}^{n}\times [0,T]$ and uniformly Hölder continuous in $(x,t,i)$ in compact subsets of ${\mathbb{R}}^{n}\times [0,T]\times \mathcal{M}$, for each $i\in \mathcal{M}$.

- 4.The function $\psi (\cdot ,\cdot ,i)$ is continuous in ${\mathbb{R}}^{n}\times [0,T]$, for each $i\in \mathcal{M}$, Hölder continuous in
*x*with respect to $(x,t,i)\in {\mathbb{R}}^{n}\times [0,T]\times \mathcal{M}$, and$|\psi (x,t,i)|\le K(1+{|x|}^{p}),\phantom{\rule{1em}{0ex}}\text{in}{\mathbb{R}}^{n}\times [0,T]\times \mathcal{M}.$ - 5.
The function $\phi (\cdot ,i)$ is continuous in ${\mathbb{R}}^{n}$, for each $i\in \mathcal{M}$, and $|\phi (x,i)|\le K(1+{|x|}^{p})$, where

*K*and*p*are positive constants.

Under (A3), it follows that the Cauchy problem has a unique solution; see [3] or [5]. Moreover, the following is true.

**Theorem 6**

*Suppose that*(A3)

*holds*.

*Then the solution of the Cauchy problem in*(22)

*is given by*

*Proof*As before, by Itô’s formula, one has

Now, proceeding as in the proof of the initial boundary value problem, we get (23). □

**Remark 2**Note by taking $c=\psi =0$, we see that the Kolmogorov backward equation is a special case of the Cauchy problem by replacing

*u*by

### 6.1 Examples

This section presents a couple of examples.

**Example 1**Let $O\subset {\mathbb{R}}^{n}$ be an open set, and consider the following weakly coupled system:

*q*-property. Such systems are studied in [10]. It follows that this Dirichlet problem has the unique solution

where $B(t)$ is a standard, *n*-dimensional Browning motion, and $\alpha (t)$ is a two-state, discrete process with generator $Q(x)$.

**Example 2**Let

*q*-property, then it follows that $\tilde{q}(x,i)=0$ for all

*x*, so the solution reduces to the form:

**Remark 3**In closing, we make the following remark. Recall that a vector $\gamma =({\gamma}_{1},\dots ,{\gamma}_{n})$ with nonnegative integer components is referred to as a multi-index. Put $|\gamma |={\gamma}_{1}+\cdots +{\gamma}_{n}$, and define ${D}_{x}^{\gamma}$ as

Let us state another condition.

*x*up to the second order and that

where ${K}_{0}$ and *β* are positive constants and *γ* is a multi-index with $|\gamma |\le 2$.

In Theorems 2 and 3, we used the approach in [6] to derive the desired equations. If we assume that (A0) holds, then the functions defined by the stochastic representations (12) and (14) are smooth and classical solutions to the systems of parabolic equations (13) and (15), respectively; see [2] for further details.

## Declarations

### Acknowledgements

The research of N. Baran and G. Yin was supported in part by the Army Research Office under grant W911NF-12-1-0223. The research of C. Zhu was supported in part by the National Science Foundation under DMS-1108782, and a grant from the UWM Research Growth Initiative.

## Authors’ Affiliations

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