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FeynmanKac formula for switching diffusions: connections of systems of partial differential equations and stochastic differential equations
Advances in Difference Equations volume 2013, Article number: 315 (2013)
Abstract
This work develops FeynmanKac 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.
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 wellknown 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 FeynmanKac formula is one of such representatives.
For diffusion processes, the FeynmanKac formula provides a stochastic representation for solutions to certain secondorder 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 FeynmanKac formula has enjoyed a widerange of applications in such areas as stochastic control, mathematical finance, risk analysis, and related fields.
This work aims to derive FeynmanKac 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 diffusionreaction 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 continuousstate 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 FeynmanKac 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
Let (\mathrm{\Omega},\mathcal{F},P) be a probability space, and let \{{\mathcal{F}}_{t}\} be a filtration on this space satisfying the usual condition (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 rightcontinuous sample paths (or a pure jump process), finitestate space \mathcal{M}=\{1,\dots ,{m}_{0}\}, and xdependent generator Q(x), so that for a suitable function f(\cdot ,\cdot ),
Assume throughout the paper that Q(x) satisfies the qproperty [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}.
Let w(\cdot ) be an {\mathbb{R}}^{n}valued standard Brownian motion defined on (\mathrm{\Omega},\mathcal{F},\{{\mathcal{F}}_{t}\},P), b(\cdot ,\cdot ):{\mathbb{R}}^{n}\times \mathcal{M}\to {\mathbb{R}}^{n}, and \sigma (\cdot ,\cdot ):{\mathbb{R}}^{n}\times \mathcal{M}\to {\mathbb{R}}^{n}\times {\mathbb{R}}^{n} such that the twocomponent process (X(\cdot ),\alpha (\cdot )) satisfies
and
The process given by (2) and (3) is called a switching diffusion or a regimeswitching 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 qproperty. 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
Consider (X(t),\alpha (t)) given in (2), and let a(x,i)=\sigma (x,i){\sigma}^{\prime}(x,i), where {\sigma}^{\prime}(x,i) denotes the transpose of \sigma (x,i). Given any function g(\cdot ,i)\in {C}^{2}({\mathbb{R}}^{n}) with i\in \mathcal{M}, define ℒ by
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.
It turns out that the evolution of the discrete component can be represented as a stochastic integral with respect to a Poisson random measure \mathfrak{p}(dt,dz), whose intensity is dt\times m(dz), where m(\cdot ) is the Lebesgue measure on ℝ. We have
where h is an integervalued function; furthermore, this representation is equivalent to (3). For details, we refer the reader to [9] and [1].
We now state (generalized) Itô’s formula. For each i\in \mathcal{M} and g(\cdot ,i)\in {C}^{2}({\mathbb{R}}^{n}), we have
where
The compensated or centered Poisson measure \mu (ds,dz)=\mathfrak{p}(ds,dz)ds\times m(dz) is a martingale measure. For t\ge 0, and g(\cdot ,i)\in {C}_{0}^{2} (the collection of {C}^{2} functions with compact support) for each i\in \mathcal{M},
where {E}^{x,i} denotes the expectation with initial data (X(0),\alpha (0))=(x,i). The above equation is known as Dynkin’s formula. The condition g\in {C}_{0}^{2} ensures that
Furthermore, one can show that ℒ agrees with its classical interpretation, as the (infinitesimal) generator of the process (X(t),\alpha (t)) given by
To see this, pick t sufficiently small so that \alpha (t) agrees with the initial data. Then it follows that
by continuity. Hence by multiplying by {t}^{1}, then letting t tend to zero, one gets
and, consequently, (10). Noting (9), when the deterministic time 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 FeynmanKac formula
We now state the FeynmanKac formula, which is a generalization of the Kolmogorov backward equation.
Theorem 3 (The FeynmanKac 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
Now, following the argument in Remark 1, we fix t>0. We have
First, clearly,
Furthermore, we claim that
To verify this claim, first, note that
by continuity. Now, if we let
for 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
Hence f is differentiable at the origin and
This in turn yields that
Furthermore, the assumptions on the functions c(\cdot ,i) and g(\cdot ,i) ensure that this forms a bounded sequence, so we may apply the bounded convergence theorem to yield
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
Let O\subset {\mathbb{R}}^{n}, be a bounded open set, and consider the following Dirichlet problem:
where ∂O denotes the boundary of O. To proceed, we impose assumption (A1).
(A1) The following conditions hold:

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 FeynmanKac 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
Clearly V(\cdot ,i)\in {C}^{\mathrm{\infty}}(O) and since V is independent of i\in \mathcal{M},
and, thus,
Note that as long as \lambda >\frac{{b}_{1}}{2{a}_{11}}, it follows that \mathcal{L}V(x,i)<0. Hence, by choosing 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
where {E}^{x,i} denotes the expectation taken with (X(0),\alpha (0))=(x,i). This yields that
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
To simplify notation, we let
We have
Taking the limit as t\to \mathrm{\infty} and noting the boundary conditions, (17) follows. □
5 The initial boundary value problem
Consider next the initial boundary value problem given by
where O is the same as before and
We will use assumption (A2).
(A2) The following conditions hold:

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.
Under (A2), it follows that the system of initial boundary value problems has a unique solution; see [3] or [5]. In order to get a stochastic representation for the solution, we also require the drift and diffusion coefficients of u to be Lipschitz continuous in the time variable; namely we require
in addition to (4) and (5).
Now, for (x,t,i)\in O\times [0,T)\times \mathcal{M}, consider the switching SDE given by
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
To simplify notation, we let
We have
If we note that
then by replacing the correct value for
in the above derivation, one gets (21). □
6 The Cauchy problem
If we let O={\mathbb{R}}^{n} in the initial value problem (17) of the previous section, we get the Cauchy problem
To proceed, we impose assumption (A3).
(A3) The following conditions hold:

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:
Where Q(x)=\left(\begin{array}{cc}{q}_{11}(x)& {q}_{12}(x)\\ {q}_{21}(x)& {q}_{22}(x)\end{array}\right) satisfies the qproperty. Such systems are studied in [10]. It follows that this Dirichlet problem has the unique solution
where B(t) is a standard, ndimensional Browning motion, and \alpha (t) is a twostate, discrete process with generator Q(x).
Example 2 Let
and consider the following stationary system; found in [8].
It follows that the solution of the above problem has the form:
where \tilde{q}(x,i)={q}_{ii}(x)+{q}_{ij}(x) and \alpha (t) is a twostate process satisfying:
Hence if the generator Q(x)=\left(\begin{array}{cc}{q}_{11}(x)& {q}_{12}(x)\\ {q}_{21}(x)& {q}_{22}(x)\end{array}\right) satisfies the qproperty, then it follows that \tilde{q}(x,i)=0 for all x, so the solution reduces to the form:
which agrees with the solution to the Dirichlet problem given by:
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 multiindex. Put \gamma ={\gamma}_{1}+\cdots +{\gamma}_{n}, and define {D}_{x}^{\gamma} as
Let us state another condition.
(A0) For each i\in \mathcal{M}, b(\cdot ,i) and \sigma (\cdot ,i) have continuous partial derivatives with respect to the variable x up to the second order and that
where {K}_{0} and β are positive constants and γ is a multiindex 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.
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Acknowledgements
The research of N. Baran and G. Yin was supported in part by the Army Research Office under grant W911NF1210223. The research of C. Zhu was supported in part by the National Science Foundation under DMS1108782, and a grant from the UWM Research Growth Initiative.
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Baran, N.A., Yin, G. & Zhu, C. FeynmanKac formula for switching diffusions: connections of systems of partial differential equations and stochastic differential equations. Adv Differ Equ 2013, 315 (2013). https://doi.org/10.1186/168718472013315
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DOI: https://doi.org/10.1186/168718472013315
Keywords
 switching diffusion
 FeynmanKac formula
 Dirichlet problem
 Cauchy problem