A pathwise Ito formula for weakly differentiable functions

Abstract

We extend Föllmer's pathwise Itô formula to weakly differentiable functions of continuous paths with finite quadratic variation along a sequence of partitions. For each such path $ω$, we introduce a path-dependent Sobolev space $W_{ω,π}^{2}$ defined through smooth approximation of the weak Hessian in a seminorm generated by discrete weighted occupation measures of the path $ω$. For $F\in W_{ω,π}^{2}$, we construct the pathwise integral $\int \nabla F(ω)\,d^πω$ and the covariation $[\nabla F(ω),ω]_π$, and prove the change-of-variable formula $$ F(ω(t))-F(ω(0)) = \int_{0}^{t}\nabla F(ω(s))\,d^πω(s) + \frac12[\nabla F(ω),ω]_π(t). $$ 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.