2007.172: On the problem of stochastic integral representations of functionals of the Browian motion II

2007.172: A. N. Shiryaev and M. Yor (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

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DOI: 10.1137/S0040585X97980427

Abstract

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 $[0,T]$.

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