Path-dependent It\^o formulas under finite $(p,q)$-variation regularity

TL;DR

This paper develops pathwise Itô formulas for non-smooth functionals of continuous semimartingales under finite variation regularity, extending classical change of variable formulas to path-dependent cases with local-time and Young integral components.

Contribution

It introduces new pathwise Itô formulas for non-smooth, path-dependent functionals under finite $(p,q)$-variation assumptions, extending previous results to more general settings.

Findings

Established pathwise local-time decomposition for non-smooth functionals.

Extended Itô formulas to stable symmetric processes with less restrictive assumptions.

Discussed singular cases with smoothness off bounded variation curves.

Abstract

In this work, we establish pathwise functional It\^o 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 $$F_t(X_t) = F_0(X_0)+ \int_0^t\nabla^hF_s(X_s)ds + \int_0^t\nabla^wF_s(X_s)dX(s) - \frac{1}{2}\int_{-\infty}^{+\infty}\int_0^t(\nabla^w_xF_s)(^{x}X_s)d_{(s,x)}\ell^x(s).$$ 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, $(\nabla^w_x F_s)(^{x}X_s)$ is a weak derivative of $F$ with respect to the terminal value $x$ of the modified path $^{x}X_s$ and $\nabla^w F_s(X_s) = (\nabla^w_x F_s)(^{x} X_s)|_{x=X(s)}$. 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 $(\nabla^w_x F_t)(^{x} X_t)$, functional It\^o formulas are established when $X$ is a stable symmetric process. Singular cases when $x\mapsto (\nabla^w_x F_t)(^{x}X_t)$ is smooth off random bounded variation curves are also discussed. The results of this paper extend previous change of variable formulas in Cont and Fourni\'e 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 F\"{o}llmer-Protter-Shiryaev formula for continuous semimartingales.