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Stochastic Stratonovich calculus fBm for fractional Brownian motion with Hurst parameter less than \(1/2\). (English) Zbl 0989.60054
Let \(B^H\) be fractional Brownian motion with Hurst parameter \(H\in(0,1)\). The authors study the existence of the ‘Stratonovich’ integral \[ \delta^B_S(u)\doteq P-\lim_{\varepsilon\to 0} (2\varepsilon)^{-1} \int^T_0 u_s(B^H_{(s+ \varepsilon)\wedge T}- B^H_{(s- \varepsilon)\wedge 0}) ds \] defined for a certain class of processes \(u\). In particular, they show that \[ \delta^B_S(u)= \delta^B(u)+ \text{Tr }Du, \] 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: \[ \delta^B_S(F(B))= \delta^B(F(B))+ H \int^T_0 F'(B_t) t^{2H- 1} dt. \] 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 \(H\in (1/4,1/2)\).

60H05 Stochastic integrals
60H07 Stochastic calculus of variations and the Malliavin calculus
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