Diffusion bridge with misspecification: theory construction and application to high-resolution fish count data

TL;DR

This paper develops a theoretical framework for diffusion bridges under model misspecification, providing bounds on performance indices and applying it to high-resolution fish count data to assess uncertainty impacts.

Contribution

It introduces exactly-solvable optimization problems based on Girsanov transformation to quantify model uncertainty in diffusion bridges, with practical application to environmental data.

Findings

Bounds on diffusion bridge performance index under misspecification

Model uncertainty significantly affects fish count estimation

Method provides a way to incorporate ambiguity aversion in stochastic models

Abstract

Stochastic processes of bridge types having pinned initial and terminal conditions have been widely used in applied research areas, but they all have a common drawback in that the model at hand is possibly misspecified owing to its stochastic nature; namely, parameter values and coefficients are distorted compared to the ground truth. We consider a pair of novel exactly-solvable optimization problems that provide both the lower and upper bounds of the performance index of a diffusion bridge. Our formulation is based on the Girsanov transformation, in which the model uncertainty is measured through relative entropy. We provide a sufficient condition under which these optimization problems are well-posed, and hence admit the corresponding maximizer/minimizer that achieves the worst-case lower and upper bounds given the ambiguity aversion or uncertainty size. We apply the proposed method to the latest 10-min, high-resolution fish count data of a migratory fish in a river and discuss the influence of model uncertainty on the estimation of the total fish count, which is an important problem in resource and environmental management.