Stochastic Stratonovich calculus fBm for fractional Brownian motion with Hurst parameter less than \(1/2\) (Q5950098)

Let \(B^H\) be fractional Brownian motion with Hurst parameter \(H\in(0,1)\). The authors study the existence of the `Stratonovich' integral NEWLINE\[NEWLINE\delta^B_S(u)\doteq P-\lim_{\varepsilon\to 0} (2\varepsilon)^{-1} \int^T_0 u_s(B^H_{(s+ \varepsilon)\wedge T}- B^H_{(s- \varepsilon)\wedge 0}) dsNEWLINE\]NEWLINE defined for a certain class of processes \(u\). In particular, they show that NEWLINE\[NEWLINE\delta^B_S(u)= \delta^B(u)+ \text{Tr }Du,NEWLINE\]NEWLINE where \(\delta^B(u)\) is so-called Skorokhod integral of the process \(u\) with respect to the fractional Brownian motion, \(D\) is the Malliavin derivative of the process \(u\) and Tr is the trace. Let \(F\) be a function with growth condition \(\max\{|F(x)|,|F'(x)|\}\leq ce^{\lambda x^2}\), where \(c\) and \(\lambda\) are positive constants and \(\lambda< T^{-2H}/4\). Moreover, if in addition to the assumptions mentioned above \(H> 1/4\), then the following makes sense: NEWLINE\[NEWLINE\delta^B_S(F(B))= \delta^B(F(B))+ H \int^T_0 F'(B_t) t^{2H- 1} dt.NEWLINE\]NEWLINE After discussing with the help of some examples, when the integrals exist, the authors give an Itô formula in this context. The paper ends with an application to stochastic differential equations driven by fractional Brownian motion with \(H\in (1/4,1/2)\).