Generalized Gaussian Bridges

TL;DR

This paper introduces two representations of generalized Gaussian bridges—orthogonal and canonical—highlighting their constructions, properties, and applications, especially in insider trading scenarios, with a focus on prediction-invertible processes like fractional Brownian motion.

Contribution

It develops explicit orthogonal and canonical representations for generalized Gaussian bridges, including non-semimartingale cases, and explores their applications in financial modeling.

Findings

Canonical representation avoids future path knowledge.

Orthogonal representation requires full path information.

Application to insider trading models using fractional Brownian motion.

Abstract

A generalized bridge is the law of a stochastic process that is conditioned on N linear functionals of its path. We consider two types of representations of such bridges: orthogonal and canonical. The orthogonal representation is constructed from the entire path of the underlying process. Thus, future knowledge of the path is needed. The orthogonal representation is provided for any continuous Gaussian process. In the canonical representation the filtrations and the linear spaces generated by the bridge process and the underlying process coincide. Thus, no future information of the underlying process is needed. Also, in the semimartingale case the canonical bridge representation is related to the enlargement of filtration and semimartingale decompositions. The canonical representation is provided for the so-called prediction-invertible Gaussian processes. All martingales are trivially prediction-invertible. A typical non-semimartingale example of a prediction-invertible Gaussian process is the fractional Brownian motion. We apply the canonical bridges to insider trading.