A pathwise Ito formula for weakly differentiable functions

Abstract

We prove a version of Föllmer's pathwise Itô formula for weakly differentiable functions of continuous paths with finite quadratic variation along a sequence of partitions. For each such path $\omega$, we introduce a path-dependent Sobolev space $W_{\omega ,\pi }^2$ defined through smooth approximation of the weak Hessian in a seminorm generated by discrete weighted occupation measures of the path $\omega$. For $F\in W_{\omega ,\pi }^2$, we construct the pathwise integral $\int \nabla F(\omega )d^{\pi }\omega$ and the covariation $[\nabla F(\omega ),\omega {]}_{\pi }$, and prove the change-of-variable formula $$F(\omega (t))-F(\omega (0))={\int }_0^t\nabla F(\omega (s))d^{\pi }\omega (s)+\frac{1}{2}[\nabla F(\omega ),\omega {]}_{\pi }(t).$$ Our result does not require any assumption on the existence of local time for the path; the Ito term appears as a quadratic covariation. For Brownian motion, we show that functions in $W^{2+,p}({\mathbb{R}}^d)\cap W^{2,1}({\mathbb{R}}^d)$ belong almost surely to the corresponding path-dependent space, outside a polar exceptional set of starting points. If, in addition, $F\in W_{\mathrm{loc}}^{1,2}({\mathbb{R}}^d)$, the pathwise integral agrees with the stochastic Itô integral, yielding a pathwise version of the multidimensional Föllmer--Protter formula.

38 pages