Markov Stochastic Processes in Biology and Mathematics -- the Same, and yet Different

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Markov Stochastic Processes in Biology and Mathematics -- the Same, and yet Different

Journal Title: JOURNAL OF ADVANCES IN MATHEMATICS - Year 2018, Vol 14, Issue 1

Abstract

Virtually every biological model utilising a random number generator is a Markov stochastic process. Numerical simulations of such processes are performed using stochastic or intensity matrices or kernels. Biologists, however, define stochastic processes in a slightly different way to how mathematicians typically do. A discrete-time discrete-value stochastic process may be defined by a function p : X0 Ã— X â†’ {f : Î¥ â†’ [0, 1]}, where X is a set of states, X0 is a bounded subset of X, Î¥ is a subset of integers (here associated with discrete time), where the function p satisfies 0 < p(x, y)(t) < 1 and  EY p(x, y)(t) = 1. This definition generalizes a stochastic matrix. Although X0 is bounded, X may include every possible state and is often infinite. By interrupting the process whenever the state transitions into the X âˆ’X0 set, Markov stochastic processes defined this way may have non-quadratic stochastic matrices. Similar principle applies to intensity matrices, stochastic and intensity kernels resulting from considering many biological models as Markov stochastic processes. Class of such processes has important properties when considered from a point of view of theoretical mathematics. In particular, every process from this class may be simulated (hence they all exist in a physical sense) and has a well-defined probabilistic space associated with it.

Authors and Affiliations

MI lOS lAWA SOKO

Keywords

Stochastic Matrix Stochastic Kernel Intensity Matrix Intensity Kernel Simulation Of Stochastic Process Probability Space For Markow Stochastic Process

EP ID EP651854
DOI 10.24297/jam.v14i1.7202
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