Markov Chain Approximations to Stochastic Differential Equations by Recombination on Lattice Trees

TL;DR

This paper introduces a novel method for approximating stochastic differential equations using Markov chains on lattice trees, enabling efficient, model-agnostic simulations with polynomial growth in states.

Contribution

It presents a universal lattice tree approach that approximates various SDEs with sparse transitions, applicable to both univariate and multivariate cases.

Findings

Sparse transition matrices with polynomial state growth

Universal lattice structure for multiple SDEs

Numerical experiments demonstrating effectiveness

Abstract

We revisit the classical problem of approximating a stochastic differential equation by a discrete-time and discrete-space Markov chain. Our construction iterates Caratheodory's theorem over time to match the moments of the increments locally. This allows to construct a Markov chain with a sparse transition matrix where the number of attainable states grows at most polynomially as time increases. Moreover, the MC evolves on a tree whose nodes lie on a "universal lattice" in the sense that an arbitrary number of different SDEs can be approximated on the same tree. The construction is not tailored to specific models, we discuss both the case of uni-variate and multi-variate case SDEs, and provide an implementation and numerical experiments.