Dynamic Markov bridges motivated by models of insider trading

TL;DR

This paper introduces a new class of stochastic processes called dynamic Markov bridges, constructed from a given Markovian Brownian martingale, with explicit decompositions and applications to insider trading models.

Contribution

It develops explicit constructions and decompositions of dynamic Markov bridges using PDEs and filtering, and applies these to a non-Gaussian insider trading equilibrium model.

Findings

Explicit semimartingale decompositions under different filtrations.

Construction of dynamic bridges from Markovian Brownian martingales.

Solution of a non-Gaussian insider trading equilibrium model.

Abstract

Given a Markovian Brownian martingale $Z$, we build a process $X$ which is a martingale in its own filtration and satisfies $X_1 = Z_1$. We call $X$ a dynamic bridge, because its terminal value $Z_1$ is not known in advance. We compute explicitly its semimartingale decomposition under both its own filtration $\cF^X$ and the filtration $\cF^{X,Z}$ jointly generated by $X$ and $Z$. Our construction is heavily based on parabolic PDE's and filtering techniques. As an application, we explicitly solve an equilibrium model with insider trading, that can be viewed as a non-Gaussian generalization of Back and Pedersen's \cite{BP}, where insider's additional information evolves over time.