A General Fractional White Noise Theory And Applications To Finance

Abstract

We present a new framework for fractional Brownian motion in which processes with all indices can be considered under the same probability measure. Our results extend recent contributions by Hu, Øksendal, Duncan, Pasik-Duncan, and others. As an application we develop option pricing in a fractional Black-Scholes market with a noise process driven by a sum of fractional Brownian motions with various Hurst indices.

A generalized white noise space approach to stochastic integration for a class of Gaussian stationary increment processes

“…By continuity, we obtain that . The resulting T m was defined in [9] and denoted M H there. In the white noise space, the fractional Brownian motion can be defined by the continuous version of the process { ω, M H 1 t } t≥0 .…”

“…In the present work, this inner product is determined by the spectrum of the process through the operator T m . We note that when m(ξ) = |ξ| 1−2H , and H ∈ ( 1 2 , 1), the operator T m reduces, up to a multiplicative constant, to the operator M H defined in [9] and in [5]. The set L 2 φ presented in [8, equation (3)] is the domain of T m and the functional C m was used with the Bochner-Minlos theorem in [6, (3.5), p. 49].…”

No-arbitrage, leverage and completeness in a fractional volatility model

“…These results might be avoided either by restricting the class of trading strategies [16], introducing transaction costs [17] or replacing pathwise integration by a different type of integration [18] [19]. However this is not free of problems because the Skorohod integral approach requires the use of a Wick product either on the portfolio or on the self-financing condition, leading to unreasonable situations from the economic point of view (for example positive portfolio with negative Wick value, etc.)…”

Optimal Long-Term Investment Model with Memory

“…To name some related contributions, let us mention here Comte and Renault [7,8], Rogers [30], Heyde [16], Willinger et al [32], Barndorff-Nielsen and Shephard [5], BarndorffNielsen et al [4], Hu and Øksendal [18], Hu et al [19], Elliott and van der Hoek [9], and Heyde and Leonenko [17]. In most of these references, driving noise processes are assumed to have stationary increments since this is a natural requirement of simplicity.…”