Translation invariant diffusions in the space of tempered distributions View Full Text


Ontology type: schema:ScholarlyArticle      Open Access: True


Article Info

DATE

2013-04

AUTHORS

B. Rajeev

ABSTRACT

In this paper we prove existence and pathwise uniqueness for a class of stochastic differential equations (with coefficients σij, bi and initial condition y in the space of tempered distributions) that may be viewed as a generalisation of Ito’s original equations with smooth coefficients. The solutions are characterized as the translates of a finite dimensional diffusion whose coefficients σij ★ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\tilde y$$\end{document}, bi ★ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\tilde y$$\end{document} are assumed to be locally Lipshitz.Here ★ denotes convolution and \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\tilde y$$\end{document} is the distribution which on functions, is realised by the formula \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\tilde y\left( r \right): = y\left( { - r} \right)$$\end{document}. The expected value of the solution satisfies a non linear evolution equation which is related to the forward Kolmogorov equation associated with the above finite dimensional diffusion. More... »

PAGES

231-258

Identifiers

URI

http://scigraph.springernature.com/pub.10.1007/s13226-013-0012-0

DOI

http://dx.doi.org/10.1007/s13226-013-0012-0

DIMENSIONS

https://app.dimensions.ai/details/publication/pub.1039393621


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