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Register a new userJoint Hitting-Time Densities for Finite State Markov Processes
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For a finite state Markov process and a finite collection ${ Gamma_k, k in K }$ of subsets of its state space, let $tau_k$ be the first time the process visits the set $Gamma_k$. We derive explicit/recursive formulas for the joint density and tail probabilities of the stopping times ${ tau_k, k in K}$. The formulas are natural generalizations of those associated with the jump times of a simple Poisson process. We give a numerical example and indicate the relevance of our results to credit risk modeling.
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Let 0<alpha<1/2. We show that the mixing time of a continuous-time reversible Markov chain on a finite state space is about as large as the largest expected hitting time of a subset of stationary measure at least alpha of the state space. Suitably modified results hold in discrete time and/or without the reversibility assumption. The key technical tool is a construction of a random set A such that the hitting time of A is both light-tailed and a stationary time for the chain. We note that essentially the same results were obtained independently by Peres and Sousi [arXiv:1108.0133].
In this paper, we consider the optimal stopping problem on semi-Markov processes (SMPs) with finite horizon, and aim to establish the existence and computation of optimal stopping times. To achieve the goal, we first develop the main results of finite horizon semi-Markov decision processes (SMDPs) to the case with additional terminal costs, introduce an explicit construction of SMDPs, and prove the equivalence between the optimal stopping problems on SMPs and SMDPs. Then, using the equivalence and the results on SMDPs developed here, we not only show the existence of optimal stopping time of SMPs, but also provide an algorithm for computing optimal stopping time on SMPs. Moreover, we show that the optimal and -optimal stopping time can be characterized by the hitting time of some special sets, respectively.
This paper describes the structure of solutions to Kolmogorovs equations for nonhomogeneous jump Markov processes and applications of these results to control of jump stochastic systems. These equations were studied by Feller (1940), who clarified in 1945 in the errata to that paper that some of its results covered only nonexplosive Markov processes. We present the results for possibly explosive Markov processes. The paper is based on the invited talk presented by the authors at the International Conference dedicated to the 200th anniversary of the birth of P. L.~Chebyshev.
The approximation of integral functionals with respect to a stationary Markov process by a Riemann-sum estimator is studied. Stationarity and the functional calculus of the infinitesimal generator of the process are used to get a better understanding of the estimation error and to prove a general error bound. The presented approach admits general integrands and gives a unifying explanation for different rates obtained in the literature. Several examples demonstrate how the general bound can be related to well-known function spaces.
We introduce the exit time finite state projection (ETFSP) scheme, a truncation-based method that yields approximations to the exit distribution and occupation measure associated with the time of exit from a domain (i.e., the time of first passage to the complement of the domain) of time-homogeneous continuous-time Markov chains. We prove that: (i) the computed approximations bound the measures from below; (ii) the total variation distances between the approximations and the measures decrease monotonically as states are added to the truncation; and (iii) the scheme converges, in the sense that, as the truncation tends to the entire state space, the total variation distances tend to zero. Furthermore, we give a computable bound on the total variation distance between the exit distribution and its approximation, and we delineate the cases in which the bound is sharp. We also revisit the related finite state projection scheme and give a comprehensive account of its theoretical properties. We demonstrate the use of the ETFSP scheme by applying it to two biological examples: the computation of the first passage time associated with the expression of a gene, and the fixation times of competing species subject to demographic noise.
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