Strong convergence of path sensitivities

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.