Stochastic Calculus with Respect to Gaussian Processes

Abstract

The aim of this work is to define and perform a study of local times of all Gaussian processes that have an integral representation over a real interval (that maybe infinite). Very rich, this class of Gaussian processes, contains Volterra processes (and thus fractional Brownian motion), multifractional Brownian motions as well as processes, the regularity of which varies along the time. Using the White Noise-based anticipative stochastic calculus with respect to Gaussian processes developed in [Leb17], we first establish a Tanaka formula. This allows us to define both weighted and non-weighted local times and finally to provide occupation time formulas for both these local times. A complete comparison of the Tanaka formula as well as the results on Gaussian local times we present here, is made with the ones proposed in [MV05, LN12, SV14].

Many-server Gaussian limits for overloaded non-Markovian queues with customer abandonment

“…We show that such stochastic integrals can be defined pathwise, and they satisfy an integration-by-parts formula. Integrals with respect to non-Brownian Gaussian processes have been studied by [5] (fractional Brownian motions) and [1,19] (Volterra processes).…”

“…Next, we define continuous-time martingales using the discrete-time martingales in (A. 19). Then we invoke Theorem 7.1.4. on p.339 in [10] to establish convergence and characterize the limit.…”

A New Approach to Stochastic Integration with Respect to Fractional Brownian Motion for No Adapted Processes

“…If Φ is differentiable at every t 0 ∈ I, we said that Φ is differentiable on I. Generally, for every k ∈ N, Φ is C k in (S) ⋆ if the process Φ : 16]). Assume that Φ : I −→ (S) ⋆ is weakly in L 1 (I, m), i.e.…”

“…In section 2, we construct suitable spaces of integrands in order to have a well-defined integral using integral representation. In section 3, We will introduce a new outcome on stochastic integration w.r.t fractional Brownian motion (fbm) for non adapted process by using an idea of Lebovits [16], and we give a new result on stochastic integration w.r.t. fbm for no adapted processes that are 2021] A NEW APPROACH TO STOCHASTIC INTEGRATION W.R.T FBM written as the product of two processes, one is adapted, and the second is instantly independent.…”

Itô’s formula for Gaussian processes with stochastic discontinuities

“…Note that this definition of a stochastic integral with respect to a Gaussian process generalizes the S-transform approach in Bender (2003b) beyond the fractional Brownian motion case. It is in the spirit of the white-noise approach to Hitsuda-Skorokhod integration, see Kuo (1996, Chapter 13) for the Brownian motion case and Lebovits (2017) and the references therein for generalizations to various classes of stochastically continuous Gaussian processes. It also generalizes the notion of an extended Skorokhod integral, which has been studied in the framework of Malliavin calculus e.g.…”