Abstract

To draw approximate samples from a distribution p on a continuous state-space, a time-honoured approach is to simulate a Markov process X_t with p as its invariant measure. Historically, this has been achieved by use of stochastic differential equations (SDEs), but in recent years, there has been an increasing amount of attention paid to Piecewise-Deterministic Markov Processes (PDMPs). These are an alternative class of Markov processes which use a combination of deterministic dynamics and discrete jumps to suppress random-walk behaviour and reach equilibrium rapidly.

Although the PDMP framework accommodates a wide range of underlying dynamics in principle, existing approaches have tended to use quite simple dynamics, such as straight lines and elliptical orbits. In this work, I present a procedure which elucidates how one can use a general dynamical system in the PDMP framework to sample from a given measure. The procedure makes use of `trajectorial reversibility’, a generalisation of `detailed balance’ which allows for tractable computation with otherwise non-reversible processes. Correctness of the procedure is established in a general setting, and specific, constructive recommendations are made for how to implement the resulting algorithms in practice.

No background in stochastic simulation will be assumed, and emphasis will be placed on outlining and understanding the key mechanisms which dictate the behaviour of PDM Ps.