2017/01/12 by Arash Jamshidpey, Jamshidpey, Arash
Economics, Econometrics and Finance · Mathematics · #Mathematical Dynamics and Fractals #Stochastic processes and financial applications #Stochastic processes and statistical mechanics #math.PR
paper · pdf · doi:10.48550/arxiv.1701.03224
arxiv created 2017/01/12 · arxiv updated 2017/01/13
The Fleming-Viot (FV) process is a measure-valued diffusion that models the evolution of type frequencies in a countable population which evolves under resampling (genetic drift), mutation, and selection. In the classic FV model the fitness (strength) of types is given by a measurable function. In this paper, we introduce and study the Fleming-Viot process in random environment (FVRE), when by random environment we mean the fitness of types is a stochastic process with càdlàg paths. We identify FVRE as the unique solution to a so called quenched martingale problem and derive some of its properties via martingale and duality methods. We develop the duality methods for general time-inhomogeneous and quenched martingale problems. In fact, some important aspects of the duality relations only appears for time-inhomogeneous (and quenched) martingale problems. For example, we see that duals evolve backward in time with respect to the main Markov process whose evolution is forward in time. Using a family of function-valued dual processes for FVRE, we prove that, as the number of individuals N tends to ∞, the measure-valued Moran process μNeN (with fitness process eN) converges weakly in Skorokhod topology of càdlàg functions to the FVRE process μe (with fitness process e), if eN → e a.s. in Skorokhod topology of càdlàg functions. We also study the long-time behaviour of FVRE process (μte)t≥ 0 joint with its fitness process e=(et)t≥ 0 and prove that the joint FV-environment process (μte,et)t≥ 0 is ergodic under the assumption of weak ergodicity of e.