Summary
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 Nualart and Pardoux [16] to analyse them. We give analytical necessary and sufficient conditions for existence and uniqueness of a solution, we establish sufficient conditions for the existence of probability densities using both the Malliavin calculus and the co-aera formula, and give sufficient conditions that the solution be either a Markov process or a Markov field.
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Supported in part by NSF Grant No. MCS-8301880
The research was carried out while this author was visiting the Institute for Advanced Study, Princeton NJ, and was supported by a grant from the RCA corporation
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Ocone, D., Pardoux, E. Linear stochastic differential equations with boundary conditions. Probab. Th. Rel. Fields 82, 489–526 (1989). https://doi.org/10.1007/BF00341281
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DOI: https://doi.org/10.1007/BF00341281