Maximal inequalities for bessel processes

Abstract

It is proved that the uniform law of large numbers (over a random parameter set) for the -dimensional ( ) Bessel process started at 0 is valid: for all stopping times for . The rate obtained (on the right-hand side) is shown to be the best possible. The following inequality is gained as a consequence: for all stopping times for , where the constant satisfies as . 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.

Keywords

• Bessel process
• Uniform law of large numbers
• Stopping time
• (Geometric) Brownian motion (with drift)
• Time change
• Ito's formula
• Burkholder–Gundy's inequality
• Optimal stopping
• Doob's maximal inequality
• Infinitesimal operator
• Trap
• Reflecting boundary
• Entrance boundary