Itô formula and local time for the fractional {B}rownian sheet
The authors consider the two-parameter fractional Brownian motion (fBm) \(W_{s,t}^{\alpha,\beta}\), or fractional Brownian sheet, with Hurst parameters \(\alpha\) and \(\beta\) bigger than \(1/2\); this process can be represented as an integral with respect to a standard Brownian sheet \(W_{s,t}\). They briefly describe the Malliavin calculus for Gaussian processes, and show how this calculus can be used to construct a Skorokhod integral with respect to the two-parameter fBm (this integral can actually be written as a Skorokhod integral with respect to the underlying Brownian sheet). Then they consider the main goal of this article, namely stating and proving an Itô formula for the two-parameter fBm. This is done in several steps, and the formula for \(f(W_{s,t}^{\alpha,\beta})\) involves integrals with respect to \(dW_{u,v}^{\alpha,\beta}\), \(du\,d_vW_{u,v}^{\alpha,\beta}\), \(d_uW_{u,v}^{\alpha,\beta}\,dv\), \(du\,dv\) and a process \(\tilde M_{u,v}\) defined as a double Skorokhod integral. Finally, as an application, they prove a stochastic integral representation formula for the local time of the two-parameter fBm (the local time is defined by means of an occupation density formula).
scientific article; zbMATH DE number 2142120