On the problem of stochastic integral representations of functionals of the Brownian motion. I

For functionals $S=S(\omega)$ of the Brownian motion $B$, we propose a method for finding stochastic integral representations based on the Itô formula for the stochastic integral associated with $B$. As an illustration of the method, we consider functionals of the “maximal” type: $S_T$, $S_{T_{-a}}$, $S_{g_T}$, and $S_{\theta_T}$, where $S_T=\max_{t\le T}B_t$ , $S_{T_{-a}}=\max_{t\le T_{-a}}B_t$ with $T_{-a}=\inf\{t>0: B_t=-a\}$, $a>0$, and $S_{g_T}=\max_{t\le g_T} B_t$, $S_{\theta_T}=\max_{t\le \theta_T}B_t$, $g_T$ and $\theta_T$ are \textit{non}-Markov times: $g_T$ is the time of the last zero of Brownian motion on $[0,T]$ and $\theta_T$ is a time when the Brownian motion achieves its maximal value on $[0,T]$.