Diffusions, Exit Time Moments and Weierstrass Theorems
Let Xtbe a one-dimensional diffusion with infinitesimal generator given by the operator $L = \frac{1}{2} (a(x)\frac{d}{dx})^{2} + b(x)\frac{d}{dx}$ 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-typ...
| Veröffentlicht in: | Proceedings of the American Mathematical Society. - American Mathematical Society, 1950. - 132(2004), 8, Seite 2465-2474 |
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| Format: | Online-Aufsatz |
| Sprache: | English |
| Veröffentlicht: |
2004
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| Zugriff auf das übergeordnete Werk: | Proceedings of the American Mathematical Society |
| Schlagworte: | Brownian Motion Exit Time Moments Approximation Theory Bessel Functions Physical sciences Mathematics |
| Zusammenfassung: | Let Xtbe a one-dimensional diffusion with infinitesimal generator given by the operator $L = \frac{1}{2} (a(x)\frac{d}{dx})^{2} + b(x)\frac{d}{dx}$ 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 Xtand 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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| ISSN: | 10886826 |