Conditional hitting time estimation in a nonlinear filtering model by the Brownian bridge method

TL;DR

This paper develops a method to estimate the probability that a nonlinear diffusion process crosses a barrier after a certain time, based on discrete observations of a correlated observation process, using filtering, quantization, and Monte Carlo techniques.

Contribution

It introduces a novel approach combining nonlinear filtering, optimal quantization, and Monte Carlo simulations for barrier crossing probability estimation in diffusion models.

Findings

Effective estimation of crossing probabilities from discrete observations.

Integration of filtering and quantization techniques improves accuracy.

Applicable to correlated diffusion processes in nonlinear filtering context.

Abstract

The model consists of a signal process $X$ which is a general Brownian diffusion process and an observation process $Y$, also a diffusion process, which is supposed to be correlated to the signal process. We suppose that the process $Y$ is observed from time 0 to $s>0$ at discrete times and aim to estimate, conditionally on these observations, the probability that the non-observed process $X$ crosses a fixed barrier after a given time $t>s$. We formulate this problem as a usual nonlinear filtering problem and use optimal quantization and Monte Carlo simulations techniques to estimate the involved quantities.