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 |
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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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