Path-dependent Itô formulas under finite $(p,q)$-variation regularity

Abstract

In this work, we establish pathwise functional Itô formulas for non-smooth functionals of real-valued continuous semimartingales. Under finite $(p,q)$-variation regularity assumptions in the sense of two-dimensional Young integration theory, we establish a pathwise local-time decomposition $$F_t(X_t)=F_0(X_0)+{\int }_0^t{\nabla }^h{F}_s(X_s)ds+{\int }_0^t{\nabla }^w{F}_s(X_s)dX(s)-\frac{1}{2}{\int }_{-\infty }^{+\infty }{\int }_0^t({\nabla }_x^w{F}_s){(}^x{X}_s)d_{(s,x)}{\ell }^x(s).$$ Here, $X_t=\{X(s);0\le s\le t\}$ is the continuous semimartingale path up to time $t\in [0,T]$, ${\nabla }^h$ is the horizontal derivative, $({\nabla }_x^w{F}_s){(}^x{X}_s)$ is a weak derivative of $F$ with respect to the terminal value $x$ of the modified path ${}^x{X}_s$ and ${\nabla }^w{F}_s(X_s)=({\nabla }_x^w{F}_s){(}^x{X}_s){|}_{x=X(s)}$. The double integral is interpreted as a space-time 2D-Young integral with differential $d_{(s,x)}{\ell }^x(s)$, where $\ell$ is the local-time of $X$. Under less restrictive joint variation assumptions on $({\nabla }_x^w{F}_t){(}^x{X}_t)$, functional Itô formulas are established when $X$ is a stable symmetric process. Singular cases when $x\mapsto ({\nabla }_x^w{F}_t){(}^x{X}_t)$ is smooth off random bounded variation curves are also discussed. The results of this paper extend previous change of variable formulas in Cont and Fournié and also Peskir, Feng and Zhao and Elworhty, Truman and Zhao in the context of path-dependent functionals. In particular, we provide a pathwise path-dependent version of the classical Föllmer-Protter-Shiryaev formula for continuous semimartingales.

Several typos were corrected

References in corpus (3)

• Two-parameter $p,q$-variation Paths and Integrations of Local Times
• A Maximal Inequality of the 2D Young Integral based on Bivariations
• Pathwise Itô Calculus for Rough Paths and Rough PDEs with Path Dependent Coefficients