Martingales, Tauberian theorem, and strategies of gambling
Teoriâ veroâtnostej i ee primeneniâ, Tome 41 (1996) no. 4, pp. 810-826
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Using the Tauberian theorem, we get an asymptotic relation between the tail of the distribution of the quadratic characteristic of a martingale and the expectation of its terminal value. In case of continuous martingales the following result is proven: if $\tau $ is a stopping time for a standard Wiener process Wt with integrable terminal value $W_\tau $, then
\begin{equation}
\liminf_{t\rightarrow \infty}(\mathbb P\{\tau >t\}\sqrt{t}) \ge \sqrt{\frac 2\pi }|\mathbb E W_\tau | .
\end{equation}
Using a related result for discrete time martingales, we study asymptotic characteristics of some strategies of gambling and, in particular, Oscar's strategy.
Keywords:
optimal stopping, local martingales, Wald equation, uniform integrability, sharp inequalities, gambling strategies, boundary crossing problem.
@article{TVP_1996_41_4_a5,
author = {A. A. Novikov},
title = {Martingales, {Tauberian} theorem, and strategies of gambling},
journal = {Teori\^a vero\^atnostej i ee primeneni\^a},
pages = {810--826},
publisher = {mathdoc},
volume = {41},
number = {4},
year = {1996},
language = {ru},
url = {http://geodesic.mathdoc.fr/item/TVP_1996_41_4_a5/}
}
A. A. Novikov. Martingales, Tauberian theorem, and strategies of gambling. Teoriâ veroâtnostej i ee primeneniâ, Tome 41 (1996) no. 4, pp. 810-826. http://geodesic.mathdoc.fr/item/TVP_1996_41_4_a5/