# Thinning and Multilevel Monte Carlo for Piecewise Deterministic (Markov) Processes. Application to a stochastic Morris-Lecar model

## TL;DR

This paper develops thinning-based approximation and multilevel Monte Carlo methods for Piecewise Deterministic Processes, demonstrating improved efficiency in simulating a stochastic Morris-Lecar neuron model.

## Contribution

It introduces a novel MLMC approach using thinning for PDPs and applies it to a complex neuron model, showing enhanced simulation performance.

## Key findings

- MLMC with thinning outperforms classical Monte Carlo in simulations
- Strong and weak error estimates for PDPs and PDMPs are established
- Application to a 2D Morris-Lecar model demonstrates practical benefits

## Abstract

In the first part of this paper we study approximations of trajectories of Piecewise Deter-ministic Processes (PDP) when the flow is not explicit by the thinning method. We also establish a strong error estimate for PDPs as well as a weak error expansion for Piecewise Deterministic Markov Processes (PDMP). These estimates are the building blocks of the Multilevel Monte Carlo (MLMC) method which we study in the second part. The coupling required by MLMC is based on the thinning procedure. In the third part we apply these results to a 2-dimensional Morris-Lecar model with stochastic ion channels. In the range of our simulations the MLMC estimator outperforms the classical Monte Carlo one.