Best bounds in Doob's maximal inequality for Bessel processes

JOURNAL OF MULTIVARIATE ANALYSIS

卷:75

Article

Pedersen, JL

关键词: Bessel process;    Doob's maximal inequality;    optimal stopping;    the principle of smooth fit;    the maximality principle;    free boundary problem;    Ito-Tanaka formula;    Burkholder-Davis-Gundy inequality;    the minimality principle;   

DOI  :  10.1006/jmva.1999.1892

来源: Elsevier

【 摘 要 】

Let ((Z(t)), P-z) be a Bessel process of dimension alpha > 0 started at z under P-z for z greater than or equal to 0. Then the maximal inequality [GRAPHICS] is shown to be satisfied for all stopping times tau for (Z(t)) with E-z(tau (p/2)) < , and all p > (2 - alpha) v 0. The constants (p/(p - (2 - alpha)))(p/(2-alpha)) and p/(p-(2 - alpha)) are the best possible. If lambda is the greater root of the equation lambda (1-(2 -alpha)/p) -lambda = (2 - alpha)/ (cp - c(2 - alpha)), the equality is attained in the limit through the stopping times [GRAPHICS] when c tends to the best constant (p/(p - (2 - alpha)))(p/(2 - alpha)) from above. Moreover we show that E-z(tau (lambda,) (q/2)(p)) < if and only if lambda > ((1 - (2 - alpha)/q) v 0)(p/(2 - alpha)). The proof of the inequality is based upon solving the optimal stopping problem [GRAPHICS] by applying the principle of smooth fit and the maximality principle. In addition, the exact formula for the expected waiting time of the optimal strategy is derived by applying the minimality principle. The main emphasis of the paper is on the explicit expressions obtained. (C) 2000 Academic Press.

【 授权许可】

Free