Clark formula for local time for one class of Gaussian processes

Abstract

In the article we present chaotic decomposition and analog of the Clark formula for the local time of Gaussian integrators. Since the integral with respect to Gaussian integrator is understood in Skorokhod sense, then there exist more than one Clark representation for the local time. We present different representations and discuss the representation with the minimal L 2 -norm.2010 Mathematics Subject Classification 60G15, 60H05, 60H40, 60J55

A representation for the Kantorovich--Rubinstein distance on the abstract Wiener space

“…where • v denotes the total variation norm. Among integral representations (1.3) of a random variable α there is a unique representation with a minimal L 2 (X, µ; H)-norm [2]. In the next lemma the needed properties of this representation are gathered.…”

“…This work was partially motivated by results of [2] where several integral representations for functionals from the Gaussian white noise were derived. Namely, for every random variable α ∈ L 2 (W, µ) the equation…”

Clark Representation for Local Times of Self-Intersection of Gaussian Integrators

“…So one can ask about Clark representation with Skorokhod integral. In the paper [3] the version of Clark representation with Skorokhod integral was obtained for the local time of integrators. In the present article we consider Clark representation for the self-intersection local time of one dimensional Gaussian integrators.…”

“…It was mentioned in [3] that Clark representation for this local time has a form Here p t is a density of normal distribution with mean zero and variance t. If α has a stochastic derivative [4], then the process η can be expressed through Clark-Ocone formula [5]. But there are interesting cases when α is not stochastically differentiable.…”