Convergence rate of Markov chains and hybrid numerical schemes to jump-diffusions with application to the Bates model

TL;DR

This paper analyzes the convergence rates of Markov chains approximating jump-diffusion processes, with applications to financial models like the Bates model, and introduces a hybrid numerical scheme combining different techniques for improved accuracy.

Contribution

It provides a general convergence analysis for Markov chain approximations of jump-diffusions and develops a hybrid numerical scheme with proven convergence rates for complex financial models.

Findings

Markov chains can effectively approximate jump-diffusions with quantifiable convergence rates.

Hybrid schemes combining Markov chains with other numerical methods improve approximation accuracy.

Application to the Bates model demonstrates practical effectiveness in financial computations.

Abstract

We study the rate of weak convergence of Markov chains to diffusion processes under suitable but quite general assumptions. We give an example in the financial framework, applying the convergence analysis to a multiple jumps tree approximation of the CIR process. Then, we combine the Markov chain approach with other numerical techniques in order to handle the different components in jump-diffusion coupled models. We study the speed of convergence of this hybrid approach and we provide an example in finance, applying our results to a tree-finite difference approximation in the Heston or Bates model.