An extension of Itô's formula for anticipating processes (Q1266791)

The authors define a new space of square integrable processes \(\mathbb{L}^F\) which contains both the adapted processes and the processes in the space \(\mathbb{L}^{2,2}\). Let \(\Delta^T_1=\{(s,t)\in [0;T]^2:s\geq t\}\), \(\Delta^T_2 =\{(r,s,t) \in[0;T]^3:r\vee s\geq t\}\), and let \(\mathbb{L}^F\) be the closure of the space of simple processes with respect to the norm \[ \| u\|^2_F=\| u \|^2_2+E\int_{\Delta^T_1}(D_su_t)^2ds dt+ E\int_{\Delta^T_2} (D_rD_su_t)^2 dr ds dt. \] \(\mathbb{L}^F\) is the class of stochastic processes \(u\) such that for each time \(t\), the random variable \(u_t\) is twice weakly differentiable with respect to the Wiener process in the two-dimensional future \(\{(r,s)\in [0;T]^2: r\vee s\geq t\}\). The authors define and generalize the usual properties of the so-called Skorokhod integral \(\delta(u)\) to the space \(\mathbb{L}^F\), and they show an Itô's formula for processes \(X_t=X_0+ \int^t_0 u_sdW_s+ \int^t_0 v_sds\) with \(u\in \mathbb{L}^F\) (Theorem 3).