Reverting Processes

TL;DR

This paper studies history-dependent stochastic processes that are not Markovian but can be represented as directed Markov processes subordinate to a random process, with applications to reverting random walks and branching processes.

Contribution

It introduces a subclass of non-Markovian processes that can be analyzed via subordinate directed Markov processes, expanding understanding of history-dependent stochastic models.

Findings

Representation of certain non-Markovian processes as subordinate Markov processes

Analysis of reverting random walks and branching processes

Detailed exploration of properties of the directing process

Abstract

We consider random processes that are history-dependent, in the sense that the distribution of the next step of the process at any time depends upon the entire past history of the process. In general, therefore, the Markov property cannot hold, but it is shown that a suitable sub-class of such processes can be seen as directed Markov processes, subordinate to a random non-Markov directing process whose properties we explore in detail. This enables us to describe the behaviour of the subordinated process of interest. Some examples, including reverting random walks and a reverting branching process, are given.