# On the problem of stochastic integral representations of functionals of the Browian motion II

Shiryaev, A. N. and Yor, M. (2004) On the problem of stochastic integral representations of functionals of the Browian motion II. Theory of Probability and its Applications, 48 (2). pp. 304-313. ISSN 1095-7219

For functionals $S=S(\omega)$ of the Brownian motion~$B$, we propose a method for finding stochastic integral representations based on the It\^o 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 $T_{-a}=\inf\{{t>0:}\allowbreak B_t=-a\}$, $a>0$, 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 {\em 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 $[0,T]$.