Moderate and $$L^p$$ Maximal Inequalities for Diffusion Processes and Conformal Martingales

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 generalize the \(L^p\) maximal inequalities for diffusions. 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.