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Diffusions, exit time moments and Weierstrass theorems

Authors: Victor H. de la Peña and Patrick McDonald
Journal: Proc. Amer. Math. Soc. 132 (2004), 2465-2474
MSC (2000): Primary 60J65, 40A30
Published electronically: March 24, 2004
MathSciNet review: 2052427
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Abstract: Let $X_t$ be a one-dimensional diffusion with infinitesimal generator given by the operator $L = \frac12 (a(x) \frac{d}{dx})^2 + b(x) \frac{d}{dx}$ where $a(x)$ is a smooth, positive real-valued function and the ratio of $a(x)$ and $b(x)$ is a constant. Given a compact interval, we prove a Weierstrass-type theorem for the exit time moments of $X_t$ and their corresponding (naturally weighted) first derivatives, and we provide an algorithm that produces uniform approximations of arbitrary continuous functions by exit time moments. We investigate analogues of these results in higher-dimensional Euclidean spaces. We give expansions for several families of special functions in terms of exit time moments.

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Additional Information

Victor H. de la Peña
Affiliation: Department of Statistics, Columbia University, New York, New York 10027

Patrick McDonald
Affiliation: Department of Mathematics, New College of Florida, Sarasota, Florida 34243

Keywords: Brownian motion, exit time moments, approximation theory, Bessel functions
Received by editor(s): August 13, 2002
Published electronically: March 24, 2004
Communicated by: Richard C. Bradley
Article copyright: © Copyright 2004 American Mathematical Society

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