Almost sure convergence of numerical approximations for Piecewise Deterministic Markov Processes

Abstract

Hybrid systems, and Piecewise Deterministic Markov Processes in particular, are widely used to model and numerically study systems exhibiting multiple time scales in biochemical reaction kinetics and related areas. In this paper an almost sure convergence analysis for numerical simulation algorithms for Piecewise Deterministic Markov Processes is presented. The discussed numerical methods arise through discretisina a constructive method defining these processes. The stochastic problem of simulating the random, path-dependent jump times of such processes is reformulated as a hitting time problem for a system of ordinary differential equations with random threshold. Then deterministic continuous methods (methods with dense output) are serially employed to solve these problems numerically. We show that the almost sure asymptotic convergence rate of the stochastic algorithm is identical to the order of the embedded deterministic method. We illustrate our theoretical findings by numerical examples from mathematical neuroscience were Piecewise Deterministic Markov Processes are used as biophysically accurate stochastic models of neuronal membranes. stochastic simulation; hybrid algorithm; almost sure convergence; Piecewise Deterministic Markov Process. *

Error Bound for Piecewise Deterministic Processes Modeling Stochastic Reaction Systems

“…This explains why such approaches are called piecewise deterministic models in the literature, see, e.g., [10,9,8,37,40] and references therein.…”

“…Such an approach is called a piecewise deterministic model in the literature (cf. [10,9,8,4,37,40]) because between two jumps of the Markov jump process the other part of the systems simply evolves according to an ordinary differential equation. Intuitively it can be expected that in a partial thermodynamic limit (i.e.…”

Numerical Approach to Spatial Deterministic-Stochastic Models Arising in Cell Biology

“…In these algorithms, accurate sampling of stochastic reaction events coupled to continuous variables is achieved by adapting Gillespie’s methods for systems whose transition rates explicitly depend on time. A similar approach was used in a hybrid stochastic algorithm for well-mixed systems with fast and slow components [12]. It should be noted that in adaptive methods, special care is required for ensuring synchronous treatment of the ‘deterministic’ and stochastic subsystems.…”

“…Hybrid algorithms, often proposed heuristically, may appear intuitive, but their rigorous analysis and validation constitute a challenging task [12, 25, 7]. This is particularly true in the context of spatially resolved models.…”

Multiscale Piecewise Deterministic Markov Process in infinite dimension: central limit theorem and Langevin approximation

“…Recently, the asymptotic behavior of PDMPs have been investigated in [4,5,12,37], limit theorems for infinite dimensional PDMPs in [35], control problems in [10,11,22], numerical methods in [33,6], time reversal in [27] and to end up this list with no claim of completeness, estimation of the jump rates for PDMPs in [2]. Hybrid systems are the object of great attention because they offer an accurate description of a large class of phenomena arising in various domains such as physics or biology.…”