Conditioned Diffusions which are Brownian Bridges

Abstract

Let X _t be a one-dimensional diffusion of the form dX _t=dB _t+μ(X _t)dt. Let Tbe a fixed positive number and let $\bar X_t$ be the diffusion process which is X _t conditioned so that X ₀=X _T=x. If the drift is constant, i.e., $\mu (x) \equiv k$, then the conditioned diffusion process $\bar X_t$ is a Brownian bridge. In this paper, we show the converse is false. There is a two parameter family of nonlinear drifts with this property.