Quasi-invariant theorem on the Gaussian path space
Qinpin Chen, Jian Sun, Bo Wu
TL;DR
The paper extends quasi-invariant analysis from the standard Wiener space to a broad class of Gaussian path spaces via $F$-Gaussian processes $B^F_t=\int_0^t F(t,s)\,dB_s$, unifying Brownian and fractional Brownian settings. It develops the Cameron–Martin framework with operators $K_F$, $K_F^*$ and $\mathscr K_F^*$, constructs stochastic integration against $B^F$, and proves a quasi-invariant theorem under shifts by $h^F\in\mathbb H^F$, with a density involving $(R_{\hat F})^{-1}$ and $B^{\hat F}$. Core contributions include invertibility results for $K_F$ and $(K_F^*)^{-1}$ under (A1)–(A4), an integration theory and Girsanov-type result for $B^F$, a Clark–Ocone–Haussmann formula, damped logarithmic Sobolev inequalities, and the Bismut–Elworthy–Li derivative formula, plus martingale representations and financial applications modeling volatility via Gaussian kernels. Collectively, these results provide a unified framework for stochastic calculus on generalized Gaussian path spaces and enable new analytic tools for finance models driven by non-Brownian noise.
Abstract
In this article, we will first introduce a class of Gaussian processes, and prove the quasi-invariant theorem with respect to the Gaussian Wiener measure, which is the law of the associated Gaussian process. In particular, it includes the case of the fractional Brownian motion. As applications, we will establish the integration by parts formula and Bismut-Elworthy-Li formula on the Gaussian path space, and by which some logarithmic Sobolev inequalities will be presented. Moreover, we will also provides some applications in the field of financial mathematics.