Stochastic bridges of linear systems
This provides a theoretical framework for modeling constrained stochastic trajectories in physics and engineering, but the contribution is incremental as it extends known techniques.
The paper derives a stochastic differential equation that generates a bridge process for inertial particles with random acceleration, conditioned on position and velocity at endpoints, generalizing the Brownian bridge to higher-order linear diffusions.
We study a generalization of the Brownian bridge as a stochastic process that models the position and velocity of inertial particles between the two end-points of a time interval. The particles experience random acceleration and are assumed to have known states at the boundary. Thus, the movement of the particles can be modeled as an Ornstein-Uhlenbeck process conditioned on position and velocity measurements at the two end-points. It is shown that optimal stochastic control provides a stochastic differential equation (SDE) that generates such a bridge as a degenerate diffusion process. Generalizations to higher order linear diffusions are considered.