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In this work, we establish pathwise functional It\^o formulas for non-smoothfunctionals of real-valued continuous semimartingales. Under finite$(p,q)$-variation regularity assumptions in the sense of two-dimensional Youngintegration 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 totime $t\in [0,T]$, $\nabla^h$ is the horizontal derivative, $(\nabla^w_xF_s)(^{x}X_s)$ is a weak derivative of $F$ with respect to the terminal value$x$ of the modified path $^{x}X_s$ and $\nabla^w F_s(X_s) = (\nabla^w_xF_s)(^{x} X_s)|_{x=X(s)}$. The double integral is interpreted as a space-time2D-Young integral with differential $d_{(s,x)}\ell^x(s)$, where $\ell$ is thelocal-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 $x\mapsto (\nabla^w_xF_t)(^{x}X_t)$ is smooth off random bounded variation curves are alsodiscussed. The results of this paper extend previous change of variableformulas 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, weprovide a pathwise path-dependent version of the classicalF\"{o}llmer-Protter-Shiryaev formula for continuous semimartingales.

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