  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
$\int_{-\infty}^\infty\int_0^t g(s,x)d_{s,x}L_s(x)$ pathwise and then give
generalized It$\hat {\rm o}$'s formula when $\nabla^-f(s,x)$ is only of bounded
$p,q$-variation in $(s,x)$. In the case that $g(s,x)=\nabla^-f(s,x)$ is of
locally bounded variation in $(s,x)$, the integral
$\int_{-\infty}^\infty\int_0^t \nabla^-f(s,x)d_{s,x}L_s(x)$ is the
Lebesgue-Stieltjes integral and was used in Elworthy, Truman and Zhao
\cite{Zhao}. When $g(s,x)=\nabla^-f(s,x)$ is of only locally $p, q$-variation,
where $p\geq 1$,$q\geq 1$, 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 $\int_{-\infty}^\infty L_t(x)d_x\nabla^-f(x)$
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 $t\geq 0$, for each $p>2$ almost
surely ($p$-variation in the sense of Lyons and Young, i.e. $\sup\limits_{E: \
a finite partition of [-N,N]}
\sum\limits_{i=1}^m|L_t(x_i)-L_t(x_{i-1})|^p<\infty$).
