Strong convergence of path sensitivities

TL;DR

This paper proves that the Euler-Maruyama method's strong convergence rate of O(h^{1/2}) extends to the approximation of path sensitivities in stochastic differential equations, filling a gap in stochastic numerical analysis.

Contribution

It establishes the strong convergence rate for path sensitivities in SDE discretisation, assuming sufficient smoothness, which was previously unproven.

Findings

Euler-Maruyama approximates path sensitivities with O(h^{1/2}) error

The result applies under bounded derivatives of the coefficients

Fills a gap in stochastic numerical analysis literature

Abstract

It is well known that the Euler-Maruyama discretisation of an autonomous SDE using a uniform timestep $h$ has a strong convergence error which is $O(h^{1/2})$ when the drift and diffusion are both globally Lipschitz. This note proves that the same is true for the approximation of the path sensitivity to changes in a parameter affecting the drift and diffusion, assuming the appropriate number of derivatives exist and are bounded. This seems to fill a gap in the existing stochastic numerical analysis literature.