On the local time of Gaussian and Lévy processes

Abstract

Abstract The local time (LT) of a given stochastic process { X t : t ≥ 0 } \{X_{t}:t\geq 0\} is defined informally as L X ⁢ ( t , x ) = ∫ 0 t δ x ⁢ ( X s ) ⁢ d s , L_{X}(t,x)=\int_{0}^{t}\delta_{x}(X_{s})\,ds, where δ x \delta_{x} denotes the Dirac function; actually, it counts the duration of the process’s stay at 𝑥 up to time 𝑡. Using an approximation approach, we study the existence and the regularity of the LT process for two kinds of stochastic processes. The first type is the stochastic process defined by the indefinite Wiener integral X t := ∫ 0 t f ⁢ ( u ) ⁢ d B u X_{t}:=\int_{0}^{t}f(u)\,dB_{u} for a given deterministic function f ∈ L 2 ( [ 0 , + ∞ [ ) f\in L^{2}([0,+\infty[) , and secondly, for Lévy type processes, i.e. ones that are stationary and with independent increments.