Error distributions for random grid approximations of multidimensional stochastic integrals

TL;DR

This paper establishes joint convergence results for approximation errors in multidimensional stochastic integrals with respect to Brownian semimartingales on random, nonuniform grids, providing tools for error analysis and optimization.

Contribution

It introduces new conditions and tools for verifying convergence of errors in stochastic integral approximations on random grids, extending previous results.

Findings

Joint convergence of approximation errors proved

Tools for checking convergence conditions developed

Explicit limit theorem for multidimensional SDE integrals obtained

Abstract

This paper proves joint convergence of the approximation error for several stochastic integrals with respect to local Brownian semimartingales, for nonequidistant and random grids. The conditions needed for convergence are that the Lebesgue integrals of the integrands tend uniformly to zero and that the squared variation and covariation processes converge. The paper also provides tools which simplify checking these conditions and which extend the range for the results. These results are used to prove an explicit limit theorem for random grid approximations of integrals based on solutions of multidimensional SDEs, and to find ways to "design" and optimize the distribution of the approximation error. As examples we briefly discuss strategies for discrete option hedging.