The isometry of symmetric-Stratonovich integrals w.r.t. Fractional Brownian motion $H< \frac{1}{2}$

Abstract: In this work, we present a detailed analysis on the exact expression of the $L^2$-norm of the symmetric-Stratonovich stochastic integral driven by a multi-dimensional fractional Brownian motion $B$ with parameter $\frac{1}{4} < H < \frac{1}{2}$. Our main result is a complete description of a Hilbert space of integrand processes which realizes the $L^2$-isometry where none regularity condition in the sense of Malliavin calculus is imposed. The main idea is to exploit the regularity of the conditional expectation of the tensor product of the increments $B_{t-\delta,t+\delta}\otimes B_{s-\epsilon,s+\epsilon}$ onto the Gaussian space generated by $(B_s,B_t)$ as $(\delta,\epsilon)\downarrow 0$. The Hilbert space is characterized in terms of a random Radon $\sigma$-finite measure on $[0,T]^2$ off diagonal which can be characterized as a product of a non-Markovian version of the stochastic Nelson derivatives. As a by-product, we present the exact explicit expression of the $L^2$-norm of the pathwise rough integral in the sense of Gubinelli.