Stochastic integration with respect to Volterra processes
For \(f\in L^1([0,1])\) and \(\gamma >0\), define the fractional integral \[ (I^\gamma_{0^+}f)(x) = {1 \over \Gamma (\gamma )} \int_0^x f(t) (x-t)^{\gamma -1} dt, \qquad x\geq 0 . \] Consider a Hilbert-Schmidt operator \(K\) on \(L^2([0,1])\) which is continuous and one-to-one from \(L^2([0,1])\) into the Besov-Liouville space \({\mathcal I}_{\alpha+1/2,2}\) for some \(\alpha >0\), and triangular in the sense that \(\{ f \;: \;f(t)=0, \;t\leq \lambda \}\) is invariant by \(K\) for all \(\lambda \in [0,1]\). Assume moreover that for any \(T\in [0,1]\) the map \(I_{0^+}^1 \circ K\) is densely defined and closable on \(L^2([0,T])\) and that its domain contains a dense subset \({\mathcal D}\) stable by the maps \(f \mapsto p_Tf\) for all \(T\in [0,1]\), where \(p_Tf = f1_{[0,T]}\). Assume furthermore that the adjoint \({\mathcal K}_1^*\) of \({\mathcal K}\) is continuous from \({\mathcal I}_{1/2-\alpha,p}\) into \(L^p([0,T])\) for any \(p\geq 2\), and that there exists processes \(u\) such that \[ R_h(u) : = h^{-1} \int_0^1 ( {\mathcal K}_1^* p_{t+h}u(s)^2 - {\mathcal K}_1^* p_t u(s)^2 ) ds, \] has a finite limit in \(L^1(\Omega)\) as \(h\) goes to zero. A stochastic calculus is constructed for Volterra processes of the form \(X_t = \int_0^t K(t,s) dB_s ,\) where \((B_s)_{s\in {\mathbb R}_+}\) is a standard Brownian motion. In particular, a Stratonovich type integral is introduced, its pathwise regularity and relation to the Skorokhod integral are studied, and an Itô formula is obtained under the above hypotheses. ~ This class of Volterra processes includes fractional Brownian motions.
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