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 Teor. Veroyatnost. i Primenen., 2006, Volume 51, Issue 1, Pages 64–77 (Mi tvp146)

On the problem of stochastic integral representations of functionals of the Browning motion. II

S. Graversena, A. N. Shiryaevb, M. Yorc

a University of Aarhus, Department of Mathematical Sciences
b Steklov Mathematical Institute, Russian Academy of Sciences
c Université Pierre & Marie Curie, Paris VI

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 $B=(B_t)_{t\ge 0}$ 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.

Keywords: Brownian motion, Itô integral, max-functionals, stochastic integral representation.

DOI: https://doi.org/10.4213/tvp146

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English version:
Theory of Probability and its Applications, 2007, 51:1, 65–77

Bibliographic databases:

Citation: S. Graversen, A. N. Shiryaev, M. Yor, “On the problem of stochastic integral representations of functionals of the Browning motion. II”, Teor. Veroyatnost. i Primenen., 51:1 (2006), 64–77; Theory Probab. Appl., 51:1 (2007), 65–77

Citation in format AMSBIB
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• http://mi.mathnet.ru/eng/tvp146
• https://doi.org/10.4213/tvp146
• http://mi.mathnet.ru/eng/tvp/v51/i1/p64

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This publication is cited in the following articles:
1. Fotopoulos S.B., Hu X., Munson C.L., “Flexible supply contracts under price uncertainty”, European J. Oper. Res., 191:1 (2008), 253–263
2. Ya. A. Lyulko, “Stochastic representations of max-type functionals of random walk”, Theory Probab. Appl., 54:3 (2010), 516–525
3. O. A. Glonti, O. G. Purtukhiya, “On one integral representation of Brownian functional”, Theory Probab. Appl., 61:1 (2017), 133–139
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