The calculation of expectations for classes of diffusion processes by Lie symmetry methods

Abstract

This paper uses Lie symmetry methods to calculate certain expectations for a large class of Itô diffusions. We show that if the problem has sufficient symmetry, then the problem of computing functionals of the form $E_x(e^{-\hat{I}»X_t-{\int }_0^{t}g(X_s)ds})$ can be reduced to evaluating a single integral of known functions. Given a drift $f$ we determine the functions $g$ for which the corresponding functional can be calculated by symmetry. Conversely, given $g$, we can determine precisely those drifts $f$ for which the transition density and the functional may be computed by symmetry. Many examples are presented to illustrate the method.

Published in at http://dx.doi.org/10.1214/08-AAP534 the Annals of Applied Probability (http://www.imstat.org/aap/) by the Institute of Mathematical Statistics (http://www.imstat.org)