Feynman-Kac formula for switching diffusions: connections of systems of partial differential equations and stochastic differential equations
© Baran et al.; licensee Springer. 2013
Received: 15 July 2013
Accepted: 20 September 2013
Published: 8 November 2013
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.
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  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 [, 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 . 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
is Borel measurable and uniformly bounded for all and ;
for all and ; and
for all and .
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 )
where for . 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 and for the remainder of the paper.
2.1 Itô’s Formula
where , denotes the Hessian of , and is given by (1). The choice for ℒ will become clear momentarily.
Note that if τ is the first exit time of the process from a bounded domain satisfying w.p.1, then Dynkin’s formula holds for any and each without the compact support assumption. To proceed, we obtain the following system of Kolmogorov backward equations for switching diffusions; see also .
Theorem 2 (Kolmogorov backward equation)
A proof of the theorem can be found in [, Theorem 5.2]; see also Theorem 5.1 in the aforementioned reference.
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)
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).
for some , and all , ,
and are uniformly Lipschitz continuous in for each ,
and is uniformly Hölder continuous in for each ,
is uniformly continuous in , and is continuous on ∂O, both for each .
It follows that under (A1), the system of boundary value problems has a unique solution; see  or . 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 . That is, τ is the first exit time from the open set O of the switching diffusion given in (2) and (3). Then w.p.1.
Taking the limit as , and using the monotone convergence theorem yields , which in turn leads to w.p.1. □
Taking the limit as and noting the boundary conditions, (17) follows. □
5 The initial boundary value problem
We will use assumption (A2).
, for each and for (),
, are uniformly Lipschitz continuous in , for each ,
and are uniformly Hölder continuous in , for each ,
is continuous on , is continuous on , for each , where ∂O denotes the boundary of O,
, for .
in addition to (4) and (5).
with initial data . If we let be the square root of , then the following is true.
in the above derivation, one gets (21). □
6 The Cauchy problem
To proceed, we impose assumption (A3).
The functions , are bounded in and uniformly Lipschitz continuous in in compact subsets of , for each .
The functions are Hölder continuous in x, uniformly with respect to in , for each .
The function is bounded in and uniformly Hölder continuous in in compact subsets of , for each .
- 4.The function is continuous in , for each , Hölder continuous in x with respect to , and
The function is continuous in , for each , and , where K and p are positive constants.
Now, proceeding as in the proof of the initial boundary value problem, we get (23). □
This section presents a couple of examples.
where is a standard, n-dimensional Browning motion, and is a two-state, discrete process with generator .
Let us state another condition.
where and β are positive constants and γ is a multi-index with .
In Theorems 2 and 3, we used the approach in  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  for further details.
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.
- Yin G, Zhu C: Hybrid Switching Diffusions: Properties and Applications. Springer, New York; 2010.View ArticleMATHGoogle Scholar
- Yin G, Zhu C: Properties of solutions of stochastic differential equations with continuous-state-dependent switching. J. Differ. Equ. 2010, 249: 2409-2439. 10.1016/j.jde.2010.08.008MATHMathSciNetView ArticleGoogle Scholar
- Friedman A: Stochastic Differential Equations and Applications. Dover, New York; 2006.MATHGoogle Scholar
- Khasminskii RZ: Stochastic Stability of Differential Equations. 2nd edition. Springer, Berlin; 2012.MATHView ArticleGoogle Scholar
- Mao X: Stochastic Differential Equations and Their Applications. Horwood, Chichester; 1997.MATHGoogle Scholar
- Øksendal B: Stochastic Differential Equations: An Introduction with Applications. Springer, Berlin; 2010.MATHGoogle Scholar
- Mitidieri E, Sweers G: Weakly coupled elliptic systems and positivity. Math. Nachr. 1995, 173(1):259-286. 10.1002/mana.19951730115MATHMathSciNetView ArticleGoogle Scholar
- Freidlin M Annals of Mathematics Studies 109. In Functional Integration and Partial Differential Equations. Princeton University Press, Princeton; 1985.Google Scholar
- Skorokhod A Amer Mathematical Society 78. Asymptotic Methods in the Theory of Stochastic Differential Equations 2008.Google Scholar
- Weinberger HF: Some remarks on invariant sets for systems. Pitman Research Notes in Math 173. Maximum Principles and Eigenvalue Problems in Partial Differential Equations 1988, 189-207.Google Scholar
This article is published under license to BioMed Central Ltd. This is an open access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/2.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.