A limit theorem for boundary local time of a symmetric reflected diffusion

Abdelatif Benchérif Madani
Theory of Stochastic Processes
Vol.22 (38), no.1, 2017, pp.41-61
Let X be a symmetric diffusion reflecting in a C³-bounded domain D ⊂ R^d, d≥1, with a C²-bounded and non-degenerate matrix a. For t>0 and n,k ∈ N let N(n,t) be the number of dyadic intervals I_n,k of length 2^-n, k≥0, that contain a time s ≤ t s.t. X(s) ∈ ∂D. For a suitable normalizing factor H(t) we prove, extending the one dimensional case, that a.s. for all t>0 the entropy functional N(n,t)/H(2^-n) converges to the boundary local time L(t) as n→∞. Applications include boundary value problems in PDE theory, efficient Monte Carlo simulations and Finance.