Linear stochastic differential equations with boundary conditions (Q1113195)

We study linear stochastic differential equations with affine boundary conditions. The equation is linear in the sense that both the drift and the diffusion coefficient are affine functions of the solution. The solution is not adapted to the driving Brownian motion, and we use the extended stochastic calculus of \textit{D. Nualart} and \textit{E. Pardoux} [Probab. Theory Relat. Fields 78, No.4, 535-581 (1988; Zbl 0639.60057)] to analyze them. We give analytical necessary and sufficient conditions for the existence and uniqueness of a solution, we establish sufficient conditions for the existence of probability densities using both the Malliavin calculus and the co-area formula, and give sufficient conditions that the solution be either a Markov process or a Markov field.