We first derive a stochastic Liouville equation for the moment generator of the stochastic functional, given a particular realization of the underlying discrete Markov process; the latter generates transitions between different dynamical equations for the continuous process. We then analyze the stochastic Liouville equation using methods recently developed for diffusion processes in randomly switching environments. In particular, we obtain dynamical equations for the moment generating function, averaged with respect to realizations of the discrete Markov process. The resulting Feynman-Kac formula takes the form of a differential Chapman-Kolmogorov equation.
We illustrate the theory by calculating the occupation time for a one-dimensional velocity jump process on the infinite or semi-infinite real line.
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Finally, we present an alternative derivation of the Feynman-Kac formula based on a recent path-integral formulation of stochastic hybrid systems. Paul C. Feynman-Kac formula for stochastic hybrid systems Paul C.
Feynman-Kac Formulas, Backward Stochastic Differential Equations and Markov Processes
Bressloff Phys. E 95 , — Published 23 January Abstract We derive a Feynman-Kac formula for functionals of a stochastic hybrid system evolving according to a piecewise deterministic Markov process. 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.
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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.
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. 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.
The setup is in line with that of [ 1 ]. Assume throughout the paper that Q x satisfies the q -property [ 1 ]. 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 ]. 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 have. For details, we refer the reader to [ 9 ] and [ 1 ]. Then it follows that. To proceed, we obtain the following system of Kolmogorov backward equations for switching diffusions; see also [ 2 ].
Remark 1 We illustrate the proof of the theorem using the idea as in [[ 6 ], p. The swings accommodation after Excellent conversation and the able stay that includes through the story from one und to the video, provides from Oct through May.
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Наука и Образование: научно-техническое издание: Feynman formulae for evolution semigroups
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