Taiwanese Journal of Mathematics

A SHARP MAXIMAL INEQUALITY FOR A GEOMETRIC BROWNIAN MOTION

Cloud Makasu

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Abstract

Let $X=(X_t)_{t\geq0}$ be a geometric Brownian motion with drift $\mu$ and volatility $\sigma\gt 0$, and let $Y=(Y_t)_{t\geq0}$ be the associated maximum process of $X$. Under certain conditions, we prove a sharp maximal inequality for the geometric Brownian motion. The method of proof is essentially based on explicit forms of the following optimal stopping problem: Find a stopping time $\tau^*$, if it exists, such that \begin{eqnarray*} \Phi(x,y):=\sup_{\tau}\mathbf{E}^{x,y}\left[Y_\tau-c\int_0^\tau X^\theta_sds\right],\quad c,\theta\gt 0 \end{eqnarray*} where the supremum is taken over all stopping times $\tau$ for the process $X$. The present result complements and extends a similar result proved by Graversen and Peskir [1].

Article information

Source
Taiwanese J. Math., Volume 19, Number 2 (2015), 585-591.

Dates
First available in Project Euclid: 4 July 2017

Permanent link to this document
https://projecteuclid.org/euclid.twjm/1499133648

Digital Object Identifier
doi:10.11650/tjm.19.2015.2551

Mathematical Reviews number (MathSciNet)
MR3332315

Zentralblatt MATH identifier
1357.60044

Subjects
Primary: 60G40: Stopping times; optimal stopping problems; gambling theory [See also 62L15, 91A60] 62L15: Optimal stopping [See also 60G40, 91A60] 35F20: Nonlinear first-order equations

Keywords
maximal inequality nonlinear differential equation optimal stopping problem

Citation

Makasu, Cloud. A SHARP MAXIMAL INEQUALITY FOR A GEOMETRIC BROWNIAN MOTION. Taiwanese J. Math. 19 (2015), no. 2, 585--591. doi:10.11650/tjm.19.2015.2551. https://projecteuclid.org/euclid.twjm/1499133648


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References

  • S. E. Graversen and G. Peskir, Optimal stopping and maximal inequalities for geometric Brownian motion, J. Appl. Prob., 35 (1998), 856-872.
  • G. Peskir, Optimal stopping of the maximum process: the maximality principle, Ann. Prob., 26 (1998), 1614-1640.
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  • L. A. Shepp and A. N. Shiryaev, A new look at the “Russian option”, Theory Prob. Appl., 39 (1994), 103-119.