The Annals of Applied Probability

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

Mark Craddock and Kelly A. Lennox

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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 Ex(eλXt0tg(Xs) 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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Ann. Appl. Probab., Volume 19, Number 1 (2009), 127-157.

First available in Project Euclid: 20 February 2009

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Primary: 35C05: Solutions in closed form 35K15: Initial value problems for second-order parabolic equations 60H99: None of the above, but in this section 60G99: None of the above, but in this section

Lie symmetry groups fundamental solutions diffusion processes transition densities expectations and functionals


Craddock, Mark; Lennox, Kelly A. The calculation of expectations for classes of diffusion processes by Lie symmetry methods. Ann. Appl. Probab. 19 (2009), no. 1, 127--157. doi:10.1214/08-AAP534.

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