Journal of Inequalities and Applications 
Volume 2 (1998), Issue 2, Pages 99-119
doi:10.1155/S102558349800006X

Maximal inequalities for bessel processes

S. E. Graversen1 and G. Peškir1,2

 1Institute of Mathematics, University of Aarhus, Ny Munkegade, Aarhus 8000, Denmark
2Department of Mathematics, University of Zagreb, Bijenička 30, Zagreb 10000, Croatia
Received 18 February 1997

Abstract

It is proved that the uniform law of large numbers (over a random parameter set) for the α-dimensional (α≥1) Bessel process Z=(Zt)t≥0 started at 0 is valid: E(max0≤t≤T|Zt2α−t|)≤12αE(T) for all stopping times T for Z. The rate obtained (on the right-hand side) is shown to be the best possible. The following inequality is gained as a consequence: E(max0≤t≤TZt2)≤G(α)E(T) for all stopping times T for Z, where the constant G(α) satisfies G(α)α=1+O(1α) as α→∞. This answers a question raised in [4]. The method of proof relies upon representing the Bessel process as a time changed geometric Brownian motion. The main emphasis of the paper is on the method of proof and on the simplicity of solution.