Bridge representation and modal-path approximation

TL;DR

This paper develops a bridge representation for the joint density of a system combining Brownian motion and fractional Brownian motion with drifts, enabling efficient small-time density approximations via modal-path substitution.

Contribution

It introduces a novel bridge representation for the joint density of coupled Brownian and fractional Brownian motions, facilitating practical small-time density approximations.

Findings

Derived a bridge representation for the joint density of the processes.

Provided a method for small-time density approximation using modal-paths.

Simplified the computation of joint densities in complex stochastic systems.

Abstract

The article shows a bridge representation for the joint density of a system of stochastic processes consisting of a Brownian motion with drift coupled with a correlated fractional Brownian motion with drift. As a result, a small time approximation of the joint density is readily obtained by substituting the conditional expectation under the bridge measure by a single path: the modal-path from the initial point to the terminal point.