An integral representation of the local time of the Brownian motion via the Clark–Ocone formula

Let $\left(L^B(t,x),t\ge 0,x\in R\right)$ be the local time of $\left(B_t,t\ge 0\right),$ the real-valued one-dimensional Brownian motion. In this paper, in case of $g,$ a strictly increasing and bijective function, we propose some integral representations of $L^{g\left(B\right)}(t,x),$ of the form: $R(t,x)+{\int }_0^{t}K(t,x,B_s)d{B}_s,$ where $R(t,x)$ is a deterministic function and $K(t,x,B_s)$ is a random function depending on $t$ and $F,$ the cumulative distribution function of the standard normal distribution $N(0,1)$ and some Brownian functionals with no Malliavin derivative. Our study is based on the case $L^B(t,x).$ An exact formula of the expectation $E\left[L^B(t,x)\right]$ is given in this paper.