2007.173: On the problem of stochastic integral representations of functions of the Brownian motion II

2007.173: S. Graversen, A. N. Shiryaev and M. Yor (2007) On the problem of stochastic integral representations of functions of the Brownian motion II. Theory of Probability and its Applications, 51 (1). pp. 65-77. ISSN 1095-7219

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DOI: 10.1137/S0040585X97982190

Abstract

In the first part of this paper [A. N. Shiryaev and M. Yor, Theory Probab. Appl., 48 (2004), pp. 304–313], a method of obtaining stochastic integral representations of functionals $S(\omega)$ of Brownian motion $B=(B_t)_{t\ge0}$ was stated. Functionals $\max_{t\le T}B_t$ and $\max_{t\le T_{-a}}B_t$, where $T_{-a}=\inf\{t: B_t=-a\}$, $a>0$, were considered as an illustration. In the present paper we state another derivation of representations for these functionals and two proofs of representation for functional $\max_{t\le g_T}B_t$, where (non-Markov time) $g_T=\sup\{0\le t\le T: B_t=0\}$ are given.

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