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

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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 $[0,T]$.

Item Type:	Article
Uncontrolled Keywords:	Brownian motion; Markov time; non-Markov time; stochastic integral; stochastic integral representation; Itô formula
Subjects:	MSC 2010, the AMS's Mathematics Subject Classification > 60 Probability theory and stochastic processes
Depositing User:	Ms Lucy van Russelt
Date Deposited:	19 Nov 2007
Last Modified:	20 Oct 2017 14:12
URI:	https://eprints.maths.manchester.ac.uk/id/eprint/910

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