|
This article is cited in 25 scientific papers (total in 25 papers)
Short Communications
On a property of the moment at which Brownian motion attains its maximum
and some optimal stopping problems
M. A. Urusov M. V. Lomonosov Moscow State University
Abstract:
Let $B=(B_t)_{0\le t\le 1}$ be a standard Brownian motion
and $\theta$ be the moment at which $B$ attains its maximal value,
i.e., $B_\theta=\max_{0\le t\le 1}B_t$.
Denote by $(\mathscr{F}^B_t)_{0\le t\le 1}$ the filtration generated by $B$.
We prove that for any $(\mathscr{F}^B_t)$-stopping time $\tau$ $(0\le\tau\le 1)$,
the following equality holds:
$$
E(B_\theta-B_\tau)^2=E|\theta-\tau|+\frac{1}{2}.
$$
Together with the results
of [S. E. Graversen, G. Peskir, and A. N. Shiryaev,
Theory Probab. Appl., 45 (2000), pp. 41–50] this implies
that the optimal stopping time $\tau_*$ in the problem
$$
\inf_\tauE|\theta-\tau|
$$
has the form
$$
\tau_*=\inf\big\{0\le t\le 1: S_t-B_t\ge z_*\sqrt{1-t}\,\big\},
$$
where $S_t=\max_{0\le s\le t}B_s$, $z_*$ is a unique positive
root of the equation $4\Phi(z)-2z\phi(z)-3=0$, $\phi(z)$
and $\Phi(z)$ are the density and the distribution function
of a standard Gaussian random variable.
Similarly, we solve the optimal stopping problems
$$
\inf_{\tau\in\mathfrak{M}_\alpha}E(\tau-\theta)^+
\quadand\quad
\inf_{\tau\in\mathfrak{N}_\alpha}E(\tau-\theta)^-,
$$
where
$\mathfrak{M}_\alpha=\{\tau\colon\,E(\tau-\theta)^-\le \alpha\}$,
and $\mathfrak{N}_\alpha=\{\tau\colon\,E(\tau-\theta)^+\le\alpha\}$.
The corresponding optimal stopping times are of the
same form as above (with other $z_*$'s).
Keywords:
moment of attaining the maximum, Brownian motion, optimal stopping.
Received: 11.12.2003
Citation:
M. A. Urusov, “On a property of the moment at which Brownian motion attains its maximum
and some optimal stopping problems”, Teor. Veroyatnost. i Primenen., 49:1 (2004), 184–190; Theory Probab. Appl., 49:1 (2005), 169–176
Linking options:
https://www.mathnet.ru/eng/tvp245https://doi.org/10.4213/tvp245 https://www.mathnet.ru/eng/tvp/v49/i1/p184
|
Statistics & downloads: |
Abstract page: | 503 | Full-text PDF : | 163 | References: | 79 |
|