In the paper we investigate properties of local time for one class of Gaussian processes. These processes are called by integrators since every function from L 2 ([0; 1]) can be integrated over it. Using the white noise representation one can associate integrators with continuous linear operators in L 2 ([0; 1]). In terms of these operators we discuss existence and properties of local time for integrators. Also, we study the asymptotic behaviour of the Brownian self-intersection local time when its end-point tends to infinity.
Abstract. In present paper we prove an existence and give a moments estimate for the local time of Gaussian integrators. Every Gaussian integrator is associated with a continuous linear operator in the space of square integrable functions via white noise representation. Hence, all properties of such process are completely characterized by properties of the corresponding operator. We describe the sufficient conditions on continuous linear non-invertible operator which allow the local time of the integrator to exists at any real point. Moments estimate for local time is obtained. A continuous dependence of local time of Gaussian integrators on generating them operators is established. The received statement improves our result presented in [10].
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
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