A generalization of the Itô formula
Let \(f\in L^\infty([a,b])\), \(\theta\in C^2({\mathbb R}^2)\) and \(X(t)=\int_a^tf(s)dB(s)\) be the Wiener integral. The white noise calculus is used to derive an (anticipating) version of the Itô\ formula for \(\theta(X(t),F)\), where \(F\in{\mathcal W}^{1/2}\), and \({\mathcal W}^{1/2}\) is a Sobolev space in the Hilbert space \((L^2)=L^2({\mathcal S}'({\mathbb R}),\mu)\), the measure \(\mu\) being the standard Gaussian measure on the space of tempered distributions \({\mathcal S}'({\mathbb R})\). This is a generalization of a well known formula, where \(F=B(c)\) for some \(c\in [a,b]\) [see \textit{H. H. Kuo}, ``White noise distribution theory'' (1996; Zbl 0853.60001)].
scientific article; zbMATH DE number 1814791