2006/09/25 by Blandine Bérard Bergery, Blandine Berard Bergery, Bergery, Blandine Berard +2
Economics, Econometrics and Finance · Mathematics · #60G44 #60H05 #60H99 #60J55 #60J65 #FOS: Mathematics #Markov Chains and Monte Carlo Methods #Probability (math.PR) #Stochastic processes and financial applications #Stochastic processes and statistical mechanics #math.PR #msc:60G44 #msc:60H05 #msc:60H99 #msc:60J55 #msc:60J65
paper · pdf · doi:10.48550/arxiv.math/0609701
Soumis dans les Comptes rendus - Mathématique
openalex publication_date 2006/09/25 · arxiv created 2007/04/25 · arxiv updated 2009/12/01 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
We give some approximations of the local time process (Ltx)t\geqslant 0 at level x of the real Brownian motion (Xt). We prove that \frac2ε∫0t X(u+ε)\wedge t+ \indi_\Xu \leqslant 0\ du + \frac2ε∫0t X(u+ε) \wedge t- \indi_\Xu>0\ du and \frac4ε∫0t Xu- \indi_\X(u+ε) \wedge t > 0\ du converge in the ucp sense to Lt0, as ε→ 0. We show that \frac1ε∫0t (\indi_\x<Xs+ε\ - \indi_\x<Xs\) (Xs+ε-Xs)ds goes to Ltx in L2(Ω) as ε→ 0, and that the rate of convergence is of order εα, for any α< 1/4.