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