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

TL;DR

This paper introduces a Lie symmetry approach to efficiently compute expectations of certain functionals for a broad class of Itô diffusion processes, simplifying complex calculations when symmetry conditions are met.

Contribution

It develops a method to identify symmetries in diffusion processes that allow explicit calculation of expectations, linking drifts and functions for which the method applies.

Findings

Symmetry-based reduction simplifies expectation calculations.

Explicit conditions for drifts and functions g are derived.

Numerous examples demonstrate the method's applicability.

Abstract

This paper uses Lie symmetry methods to calculate certain expectations for a large class of It\^{o} diffusions. We show that if the problem has sufficient symmetry, then the problem of computing functionals of the form $E_x(e^{-\lambda X_t-\int_0^tg(X_s) ds})$ 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.