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

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

2009-02-27

arxiv.org/abs/0902.4806v1

Annals of Applied Probability 2009, Vol. 19, No. 1, 127-157

Mathematics

Probability

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

Scientific paper

10.1214/08-AAP534

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.

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