Malliavin calculus for difference approximations of multidimensional diffusions: Truncated local limit theorem
For difference approximations of multidimensional diffusions, the truncated local limit theorem is proved. Under very mild conditions on the distributions of difference terms, this theorem states that the transition probabilities of these approximations, after truncation of some asymptotically neg...
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| Datum: | 2008 |
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| Hauptverfasser: | , |
| Format: | Artikel |
| Sprache: | Englisch |
| Veröffentlicht: |
Institute of Mathematics, NAS of Ukraine
2008
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| Online Zugang: | https://umj.imath.kiev.ua/index.php/umj/article/view/3160 |
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| Назва журналу: | Ukrains’kyi Matematychnyi Zhurnal |
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Ukrains’kyi Matematychnyi Zhurnal| Zusammenfassung: | For difference approximations of multidimensional diffusions, the truncated local limit theorem is proved.
Under very mild conditions on the distributions of difference terms, this theorem states that the transition
probabilities of these approximations, after truncation of some asymptotically negligible terms, possess
densities that converge uniformly to the transition probability density for the limiting diffusion and
satisfy certain uniform diffusion-type estimates. The proof is based on a new version of the
Malliavin calculus for the product of a finite family of measures that may contain non-trivial singular components.
Applications to the uniform estimation of mixing and convergence rates for
difference approximations of stochastic differential equations and to the convergence of difference approximations of local times of multidimensional diffusions are given. |
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