Moderate and L^p maximal inequalities for diffusion processes and conformal martingales

2021/11/04 by Xian Chen, Chen, Xian, Yong Chen +5
Decision Sciences · Economics, Econometrics and Finance · Mathematics · #FOS: Mathematics #Probability (math.PR) #Probability and Risk Models #Stochastic processes and financial applications #Stochastic processes and statistical mechanics

paper · pdf · doi:10.48550/arxiv.2111.02641

openalex publication_date 2021/11/04 · openalex created_date 2021/11/08 · openalex updated_date 2026/07/28

Abstract

The L^p maximal inequalities for martingales are one of the classical results in the theory of stochastic processes. Here we establish the sharp moderate maximal inequalities for one-dimensional diffusion processes, which include the L^p maximal inequalities as special cases. Moreover, we apply our theory to many specific examples, including the Ornstein-Uhlenbeck (OU) process, Brownian motion with drift, reflected Brownian motion with drift, Cox-Ingersoll-Ross process, radial OU process, and Bessel process. The results are further applied to establish the moderate maximal inequalities for some high-dimensional processes, including the complex OU process and general conformal local martingales.

Citations

• On the Theory of the Brownian Motion

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