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arXiv: 0911.3534

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Asymptotic behaviour of a family of time-inhomogeneous diffusions

Mihai Gradinaru (  Corresponding author ) 1, Yoann Offret 1

(2009-11-18)

Let $X$ a solution of the time-inhomogeneous stochastic differential equation driven by a Brownian motion with drift coefficient $b(t,x)=\rho\,{\rm sgn}(x)\frac{|x|^\alpha}{t^\beta}$. This process can be viewed as a distorted Brownian motion in a potential possibly singular, depending on time. After obtaining results on the existence and the uniqueness of solution, we study its asymptotic behaviour and made a precise description, in terms of parameters $\rho,\alpha$ and $\beta$, of the recurrence, transience and convergence. More precisely, asymptotic distributions, iterated logarithm type laws and rates of transience are proved for such processes.

1:	Institut de Recherche Mathématique de Rennes (IRMAR)
	CNRS : UMR6625 – Université de Rennes 1 – École normale supérieure de Cachan - ENS Cachan – Institut National des Sciences Appliquées (INSA) : - RENNES – Université de Rennes II - Haute Bretagne

Subject

:

Mathematics/Probability

Keyword(s): time-inhomogeneous diffusions – singular stochastic differential equations – explosion times – scaling transformations and changes of time – recurrence and transience – iterated logarithm type laws – asymptotic distributions

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From: Mihai Gradinaru <>

Submitted on: Wednesday, 18 November 2009 16:12:36

Updated on: Wednesday, 18 November 2009 17:12:57