Inequalities for non-moderate functions of a pair of stochastic processes. (English) Zbl 0789.60016

Jacka, S. D.; Yor, M.

Proc. Lond. Math. Soc., III. Ser. 67, No. 3, 649-672 (1993).

We find sufficient and (in the case of self-similar processes) necessary conditions for inequalities of the form \[ \mathbb{E} F(X_ L) \leq A+\mu \mathbb{E} G(Y_ L) \text{ for all } L, \] where \(F\) and \(G\) are nonmoderate increasing functions, \(X\) and \(Y\) are increasing optional processes, and \(L\) runs through the class of nonnegative random variables (random times). In particular, we show in Theorem 1 that if \(B\) is a Brownian motion, \[ \mathbb{E} \exp ((B^*_ L)^ p) \leq A+\mu \mathbb{E} \exp (\theta(\sqrt L)^{2p/(2-p)}) \text{ and } \mathbb{E} \exp ((\sqrt L)^ p) \leq A'+\mu' \mathbb{E} \exp (\theta'(B^*_ L)^{2p/(2-p)}) \] for suitable values of \(A\), \(A'\), \(\mu\), \(\mu'\), \(\theta\), and \(\theta'\), provided \(0<p<2\). We go on to apply our results to martingale sequences and Wiener chaos.

Reviewer: S.D.Jacka

MSC:

60E15	Inequalities; stochastic orderings
60G07	General theory of stochastic processes

Keywords:

nonmoderate functions; inequalities; random times; norm inequalities; Brownian motion