A convergent numerical algorithm for the stochastic growth-fragmentation problem

TL;DR

This paper introduces a convergent numerical scheme for simulating the stochastic growth-fragmentation process and accurately approximating its invariant measure, with explicit error bounds and convergence guarantees.

Contribution

It presents a novel numerical algorithm that ensures convergence to the true process and provides quantitative estimates of the invariant measure's approximation error.

Findings

The numerical scheme converges to the continuous process under certain assumptions.

Explicit error bounds are derived for the approximation.

The method accurately estimates the invariant measure of the process.

Abstract

The stochastic growth-fragmentation model describes the temporal evolution of a structured cell population through a discrete-time and continuous-state Markov chain. The simulations of this stochastic process and its invariant measure are of interest. In this paper, we propose a numerical scheme for both the simulation of the process and the computation of the invariant measure, and show that under appropriate assumptions, the numerical chain converges to the continuous growth-fragmentation chain with an explicit error bound. With a triangle inequality argument, we are also able to quantitatively estimate the distance between the invariant measures of these two Markov chains.