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

Abstract

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 $p,\break q$-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 by Elworthy, Truman and 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 $\int_{-\infty}^\infty L_{\kern1pt t}(x)d_x\nabla^-f(x)$ is defined pathwise as a Young integral. For this we prove the local time $L_{\kern1pt t}(x)$ 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. ${\mathop {\sup }\limits_{E:\;{\text{a}}\;{\text{finite}}\;{\text{partition}}\;{\text{of}}\;{\left[ { - N,N} \right]}} }{\sum\limits_{i = 1}^m {{\left| {L_{t} {\left( {x_{i} } \right)} - L_{t} {\left( {x_{{i - 1}} } \right)}} \right|}^{p} < \infty } }$).