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Teor. Veroyatnost. i Primenen., 2003, Volume 48, Issue 2, Pages 375–385 (Mi tvp290)  

This article is cited in 8 scientific papers (total in 8 papers)

On the problem of stochastic integral representations of functionals of the Brownian motion. I

A. N. Shiryaeva, M. Yorb

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

Abstract: For functionals $S=S(\omega)$ of the Brownian motion $B$, we propose a method for finding stochastic integral representations based on the Itô 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 $T_{-a}=\inf\{t>0: B_t=-a\}$, $a>0$, 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 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 $[0,T]$.

Keywords: Brownian motion, Markov time, non-Markov time, stochastic integral, stochastic integral representation, Itô formula.

DOI: https://doi.org/10.4213/tvp290

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English version:
Theory of Probability and its Applications, 2004, 48:2, 304–313

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Received: 01.12.2002

Citation: A. N. Shiryaev, M. Yor, “On the problem of stochastic integral representations of functionals of the Brownian motion. I”, Teor. Veroyatnost. i Primenen., 48:2 (2003), 375–385; Theory Probab. Appl., 48:2 (2004), 304–313

Citation in format AMSBIB
\by A.~N.~Shiryaev, M.~Yor
\paper On the problem of stochastic integral representations of functionals of the Brownian motion.~I
\jour Teor. Veroyatnost. i Primenen.
\yr 2003
\vol 48
\issue 2
\pages 375--385
\jour Theory Probab. Appl.
\yr 2004
\vol 48
\issue 2
\pages 304--313

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    This publication is cited in the following articles:
    1. S. Graversen, A. N. Shiryaev, M. Yor, “On the problem of stochastic integral representations of functionals of the Browning motion. II”, Theory Probab. Appl., 51:1 (2007), 65–77  mathnet  crossref  crossref  mathscinet  zmath  isi  elib
    2. Renaud J.-F., Remillard B., “Explicit martingale representations for Brownian functionals and applications to option hedging”, Stochastic Analysis and Applications, 25:4 (2007), 801–820  crossref  mathscinet  zmath  isi  scopus
    3. V. Jaoshvili, O. G. Purtukhiya, “An Extension of the Ocone–Haussmann–Clark Formula for the Compensated Poisson Processes”, Theory Probab. Appl., 53:2 (2009), 316–321  mathnet  crossref  crossref  zmath  isi
    4. Fotopoulos S.B., Hu X., Munson C.L., “Flexible supply contracts under price uncertainty”, European Journal of Operational Research, 191:1 (2008), 253–263  crossref  mathscinet  zmath  isi  scopus
    5. Ya. A. Lyulko, “Stochastic representations of max-type functionals of random walk”, Theory Probab. Appl., 54:3 (2010), 516–525  mathnet  crossref  crossref  mathscinet  isi
    6. O. A. Glonti, O. G. Purtukhiya, “On one integral representation of Brownian functional”, Theory Probab. Appl., 61:1 (2017), 133–139  mathnet  crossref  crossref  mathscinet  zmath  isi  elib
    7. Feng R., “Stochastic Integral Representations of the Extrema of Time-homogeneous Diffusion Processes”, Methodol. Comput. Appl. Probab., 18:3 (2016), 691–715  crossref  mathscinet  zmath  isi  elib  scopus
    8. Aurzada F., Buck M., Kilian M., “Penalizing Fractional Brownian Motion For Being Negative”, Stoch. Process. Their Appl., 130:11 (2020), 6625–6637  crossref  mathscinet  isi
  • Теория вероятностей и ее применения Theory of Probability and its Applications
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