Abstract
A tight upper bound is given on the distribution of the maximum of a supermartingale. Specifically, it is shown that if $Y$ is a semimartingale with initial value zero and quadratic variation process $[Y,Y]$ such that $Y + [Y,Y]$ is a supermartingale, then the probability the maximum of $Y$ is greater than or equal to a positive constant $a$ is less than or equal to$1/(1+a).$ The proof makes use of the semimartingale calculus and is inspired by dynamic programming.
Highlights
The basic framework of the classical martingale calculus will be used, as described, for example, in [6, 7, 9, 10, 11]
All random processes are assumed to be defined on a complete probability space (Ω, F, P ) with a filtration of σ-algebras (Ft : t ≥ 0) that is assumed to satisfy the usual conditions of right-continuity and inclusion of all sets of probability zero
The following condition is stronger than the supermartingale condition, requiring that the downward drift be at least as strong as a constant γ times the rate of variation of the process, as measured by the quadratic variation process
Summary
The basic framework of the classical martingale calculus will be used, as described, for example, in [6, 7, 9, 10, 11]. The following condition is stronger than the supermartingale condition, requiring that the downward drift be at least as strong as a constant γ times the rate of variation of the process, as measured by the quadratic variation process. The necessary conditions in Proposition 1.1(b) cannot be satisfied for two distinct strictly positive values of a. This raises the question as to how close to equality the bound (1.1) can be for all values of a, for a single choice of Y not depending on a.
Sign-in/Register to view highlights & summary Sign-in/Register to access full text options 
Free) 
Talk to us
Join us for a 30 min session where you can share your feedback and ask us any queries you have
Schedule a call
