Two-parameter $p,q$-variation Paths and Integrations of Local Times

In this paper, we prove two main results. The first one is to give a new condition for the existence of two-parameter $p,q$-variation path integrals. Our condition of locally bounded $p,q$-variation is more natural and easy to verify than those of Young. This result can be easily generalized to multi-parameter case. The second result is to define the integral of local time ${\int }_{-\mathrm{\infty }}^{\mathrm{\infty }}{\int }_0^{t}g(s,x)d_{s,x}{L}_s(x)$ pathwise and then give generalized It$\widehat{\mathrm{o}}$'s formula when ${\mathrm{\nabla }}^{-}f(s,x)$ is only of bounded $p,q$-variation in $(s,x)$. In the case that $g(s,x)={\mathrm{\nabla }}^{-}f(s,x)$ is of locally bounded variation in $(s,x)$, the integral ${\int }_{-\mathrm{\infty }}^{\mathrm{\infty }}{\int }_0^t{\mathrm{\nabla }}^{-}f(s,x)d_{s,x}{L}_s(x)$ is the Lebesgue-Stieltjes integral and was used in Elworthy, Truman and Zhao \cite{Zhao}. When $g(s,x)={\mathrm{\nabla }}^{-}f(s,x)$ is of only locally $p,q$-variation, where $p\ge 1$,$q\ge 1$, and $2q+1>2pq$, the integral is a two-parameter Young integral of $p,q$-variation rather than a Lebesgue-Stieltjes integral. In the special case that $f(s,x)=f(x)$ is independent of $s$, we give a new condition for Meyer’s formula and ${\int }_{-\mathrm{\infty }}^{\mathrm{\infty }}{L}_t(x)d_x{\mathrm{\nabla }}^{-}f(x)$ is defined pathwise as a Young integral. For this we prove the local time $L_t(x)$ is of $p$-variation in $x$ for each $t\ge 0$, for each $p>2$ almost surely ($p$-variation in the sense of Lyons and Young, i.e. $\sup\limits_{E:
a finite partition of [-N,N]} \sum\limits_{i=1}^m|L_t(x_i)-L_t(x_{i-1})|^p<\infty$).