# A Hoeffding's inequality for uniformly ergodic diffusion process

## TL;DR

This paper extends Hoeffding's inequality to continuous-time uniformly ergodic diffusion processes, providing a new probabilistic bound useful in stochastic process analysis and applications.

## Contribution

It introduces a Hoeffding's inequality for diffusion processes, bridging a gap between discrete-time Markov chains and continuous-time diffusions.

## Key findings

- Derived a Hoeffding's inequality for diffusion processes.
- Illustrated the results with examples involving Jacobi diffusion and Ornstein-Uhlenbeck process.
- Provided bounds for large deviation probabilities in continuous-time settings.

## Abstract

In this note, we present a version of Hoeffding's inequality in a continuous-time setting, where the data stream comes from a uniformly ergodic diffusion process. Similar to the well-studied case of Hoeffding's inequality for discrete-time uniformly ergodic Markov chain, the proof relies on techniques ranging from martingale theory to classical Hoeffding's lemma as well as the notion of deviation kernel of diffusion process. We present two examples to illustrate our results. In the first example we consider large deviation probability on the occupation time of the Jacobi diffusion, a popular process used in modelling of exchange rates in mathematical finance, while in the second example we look at the exponential functional of a finite interval analogue of the Ornstein-Uhlenbeck process introduced by Kessler and S{\o}rensen (1999).