|
This article is cited in 10 scientific papers (total in 10 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.
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
Linking options:
https://www.mathnet.ru/eng/tvp290https://doi.org/10.4213/tvp290 https://www.mathnet.ru/eng/tvp/v48/i2/p375
|
Statistics & downloads: |
Abstract page: | 531 | Full-text PDF : | 192 | References: | 104 |
|