Existence and asymptotic behaviour of some time-inhomogeneous diffusions

2009/11/18 by Mihai Gradinaru, Gradinaru, Mihai, Yoann Offret +1
Business, Management and Accounting · Economics, Econometrics and Finance · Mathematics · #Advanced Queuing Theory Analysis #FOS: Mathematics #Probability (math.PR) #Stochastic processes and financial applications #Stochastic processes and statistical mechanics #math.PR

paper · pdf · doi:10.48550/arxiv.0911.3534

31 pages

openalex publication_date 2009/11/18 · arxiv created 2012/04/23 · arxiv updated 2012/04/24 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

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

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