A pathwise Ito formula for weakly differentiable functions

Abstract

We prove a version of F\"ollmer's pathwise It\^o 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 $$ F(\omega(t))-F(\omega(0)) = \int_{0}^{t}\nabla F(\omega(s))\,d^{\pi}\omega(s) + \frac12[\nabla F(\omega),\omega]_{\pi}(t). $$ 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 It\^o integral, yielding a pathwise version of the multidimensional F\"ollmer--Protter formula.