We consider the asymptotic property of the diffusion processes with Markovian switching. For a general case, we prove a large deviation principle for empirical measures of switching diffusion processes with small para...We consider the asymptotic property of the diffusion processes with Markovian switching. For a general case, we prove a large deviation principle for empirical measures of switching diffusion processes with small parameters.展开更多
This work is concerned with the asymptotic behavior of systems of parabolic equations arising from null-recurrent switching diffusions, which are diffusion processes modulated by continuous-time Markov chains. A suffi...This work is concerned with the asymptotic behavior of systems of parabolic equations arising from null-recurrent switching diffusions, which are diffusion processes modulated by continuous-time Markov chains. A sufficient condition for null recurrence is presented. Moreover, convergence rate of the solutions of systems of homogeneous parabolic equations under suitable conditions is established. Then a case study on verifying one of the conditions proposed is provided with the use of a two-state Markov chain. To verify the condition, boundary value problems （BVPs） for parabolic systems are treated, which are not the usual two-point BVP type. An extra condition in the interior is needed resulting in jump discontinuity of the derivative of the corresponding solution.展开更多
This work is concerned with switching diffusion processes, also known as regime-switching diffusions. Our attention focuses on regularity, recurrence, and positive recurrence of the underlying stochastic processes. Th...This work is concerned with switching diffusion processes, also known as regime-switching diffusions. Our attention focuses on regularity, recurrence, and positive recurrence of the underlying stochastic processes. The main effort is devoted to obtaining easily verifiable conditions for the aforementioned properties. Continuous-state-dependent jump processes are considered. First general criteria on regularity and recurrence using Liapunov functions are obtained. Then we focus on a class of problems, in which both the drift and the diffusion coefficients are ＂linearizable＂ with respect to the continuous state, and suppose that the generator of the jump part of the process can be approximated by a generator of an ergodic Markov chain. Sufficient conditions for regularity, recurrence, and positive recurrence are derived, which are linear combination of the averaged coefficients （averaged with respect to the stationary measure of the Markov chain）.展开更多
This work develops asymptotic expansions of systems of partial differential equations associated with multi-scale switching diffusions. The switching process is modeled by using an inhomogeneous continuous- time Marko...This work develops asymptotic expansions of systems of partial differential equations associated with multi-scale switching diffusions. The switching process is modeled by using an inhomogeneous continuous- time Markov chain. In the model, there are two small parameters ε and δ. The first one highlights the fast switching, whereas the other delineates the slow diffusion. Assuming that ε and δ are related in that ε = δγ, our results reveal that different values of γ lead to different behaviors of the underlying systems, resulting in different asymptotic expansions. Although our motivation comes from stochastic problems, the approach is mainly analytic and is constructive. The asymptotic series are rigorously justified with error bounds provided. An example is provided to demonstrate the results.展开更多
This paper develops asymptotic properties of singularly perturbed Markov chains with inclusion of absorbing states.It focuses on both unscaled and scaled occupation measures.Under mild conditions,a mean-square estimat...This paper develops asymptotic properties of singularly perturbed Markov chains with inclusion of absorbing states.It focuses on both unscaled and scaled occupation measures.Under mild conditions,a mean-square estimate is obtained.By averaging the fast components,we obtain an aggregated process.Although the aggregated process itself may be non-Markovian,its weak limit is a Markov chain with much smaller state space.Moreover,a suitably scaled sequence consisting of a component of scaled occupation measures and a component of the aggregated process is shown to converge to a pair of processes with a switching diffusion component.展开更多
This work develops near-optimal controls for systems given by differential equations with wideband noise and random switching.The random switching is modeled by a continuous-time,time-inhomogeneous Markov chain.Under ...This work develops near-optimal controls for systems given by differential equations with wideband noise and random switching.The random switching is modeled by a continuous-time,time-inhomogeneous Markov chain.Under broad conditions,it is shown that there is an associated limit problem,which is a switching jump diffusion.Using near-optimal controls of the limit system,we then build controls for the original systems.It is shown that such constructed controls are nearly optimal.展开更多
基金The authors would like to thank the referees for providing many helpful comments and suggestions. Research of the second author was supported in part by the National Natural Science Foundation of China （Grant No. 11171024）.
文摘We consider the asymptotic property of the diffusion processes with Markovian switching. For a general case, we prove a large deviation principle for empirical measures of switching diffusion processes with small parameters.
基金the National Science Foundation under DMS-0603287the National Security Agency,MSPF-068-029 the National Natural Science Foundation of China under No.60574069
文摘This work is concerned with the asymptotic behavior of systems of parabolic equations arising from null-recurrent switching diffusions, which are diffusion processes modulated by continuous-time Markov chains. A sufficient condition for null recurrence is presented. Moreover, convergence rate of the solutions of systems of homogeneous parabolic equations under suitable conditions is established. Then a case study on verifying one of the conditions proposed is provided with the use of a two-state Markov chain. To verify the condition, boundary value problems （BVPs） for parabolic systems are treated, which are not the usual two-point BVP type. An extra condition in the interior is needed resulting in jump discontinuity of the derivative of the corresponding solution.
基金This research is supported in part by the National Science Foundation under DMS-0624849, in part by the National Security Agency under MSPF-068-029, and in part by the National Natural Science Foundation of China under Grant No. 60574069.
文摘This work is concerned with switching diffusion processes, also known as regime-switching diffusions. Our attention focuses on regularity, recurrence, and positive recurrence of the underlying stochastic processes. The main effort is devoted to obtaining easily verifiable conditions for the aforementioned properties. Continuous-state-dependent jump processes are considered. First general criteria on regularity and recurrence using Liapunov functions are obtained. Then we focus on a class of problems, in which both the drift and the diffusion coefficients are ＂linearizable＂ with respect to the continuous state, and suppose that the generator of the jump part of the process can be approximated by a generator of an ergodic Markov chain. Sufficient conditions for regularity, recurrence, and positive recurrence are derived, which are linear combination of the averaged coefficients （averaged with respect to the stationary measure of the Markov chain）.
基金supported in part by the Air Force Office of Scientific Research under FA9550-15-1-0131
文摘This work develops asymptotic expansions of systems of partial differential equations associated with multi-scale switching diffusions. The switching process is modeled by using an inhomogeneous continuous- time Markov chain. In the model, there are two small parameters ε and δ. The first one highlights the fast switching, whereas the other delineates the slow diffusion. Assuming that ε and δ are related in that ε = δγ, our results reveal that different values of γ lead to different behaviors of the underlying systems, resulting in different asymptotic expansions. Although our motivation comes from stochastic problems, the approach is mainly analytic and is constructive. The asymptotic series are rigorously justified with error bounds provided. An example is provided to demonstrate the results.
文摘This paper develops asymptotic properties of singularly perturbed Markov chains with inclusion of absorbing states.It focuses on both unscaled and scaled occupation measures.Under mild conditions,a mean-square estimate is obtained.By averaging the fast components,we obtain an aggregated process.Although the aggregated process itself may be non-Markovian,its weak limit is a Markov chain with much smaller state space.Moreover,a suitably scaled sequence consisting of a component of scaled occupation measures and a component of the aggregated process is shown to converge to a pair of processes with a switching diffusion component.
基金supported in part by the National Science Foundation under DMS-1207667supported in part by NSFC and RFDP
文摘This work develops near-optimal controls for systems given by differential equations with wideband noise and random switching.The random switching is modeled by a continuous-time,time-inhomogeneous Markov chain.Under broad conditions,it is shown that there is an associated limit problem,which is a switching jump diffusion.Using near-optimal controls of the limit system,we then build controls for the original systems.It is shown that such constructed controls are nearly optimal.
