Diffusions, exit time moments and Weierstrass theorems
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- by Victor H. de la Peña and Patrick McDonald PDF
- Proc. Amer. Math. Soc. 132 (2004), 2465-2474 Request permission
Abstract:
Let $X_t$ be a one-dimensional diffusion with infinitesimal generator given by the operator $L = \frac 12 (a(x) \frac {d}{dx})^2 + b(x) \frac {d}{dx}$ 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.References
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Additional Information
- Victor H. de la Peña
- Affiliation: Department of Statistics, Columbia University, New York, New York 10027
- MR Author ID: 268889
- Email: vp@stat.columbia.edu
- Patrick McDonald
- Affiliation: Department of Mathematics, New College of Florida, Sarasota, Florida 34243
- Email: ptm@virtu.sar.usf.edu
- Received by editor(s): August 13, 2002
- Published electronically: March 24, 2004
- Communicated by: Richard C. Bradley
- © Copyright 2004 American Mathematical Society
- Journal: Proc. Amer. Math. Soc. 132 (2004), 2465-2474
- MSC (2000): Primary 60J65, 40A30
- DOI: https://doi.org/10.1090/S0002-9939-04-07196-5
- MathSciNet review: 2052427