Constructing Strong Markov Processes

2013/03/11 by Robert J. Vanderbei, Vanderbei, Robert J.
Economics, Econometrics and Finance · Mathematics · #47D07 #60J25 #FOS: Mathematics #Markov Chains and Monte Carlo Methods #Probability (math.PR) #Stochastic processes and financial applications #Stochastic processes and statistical mechanics

paper · pdf · doi:10.48550/arxiv.1303.2670

openalex publication_date 2013/03/11 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

The construction presented in this paper can be briefly described as follows: starting from any "finite-dimensional" Markov transition function p_t, on a measurable state space (E,B), we construct a strong Markov process on a certain "intrinsic" state space that is, in fact, a closed subset of a finite dimensional Euclidean space R^d. Of course we must explain the meaning of finite-dimensionality and intrinsity. Starting with p_t, we consider the range of the nonnegative bounded measurable functions under the action of the resolvent. This class of functions induces a uniform structure on E. We say that E is finite-dimensional if this uniformity is finitely generated. In such cases we then map E into R^d. The intrinsic state space is the closure of the range of this mapping. On this enlarged state space we construct a strong Markov process, which corresponds quite naturally to p_t. We give several examples including the usual examples of nonstrong Markov process.

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