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Constrained stochastic differential equations Storm, Andrew


This work uses techniques from convex analysis to study constrained solutions (u, ƞ) to stochastic differential equations in Hilbert spaces. The process u must stay in the domain of a given convex function φ, and ƞ is a bounded variation process. The constraint is expressed by a variational inequality involving u and ƞ, and is equivalent to ƞ ∈∂Փ(u), where Փ(u) = ∫oT(ut)dt. Both ordinary and partial stochastic differential equations are considered. For ordinary equations there are minimal restrictions on the constraint function φ. By choosing φ to be the indicator of a closed convex set, previous results on reflected diffusion processes in finite dimensions are reproduced. For stochastic partial differential equations there are severe restrictions on the constraint functions. Results are obtained if φ is the indicator of a sphere or a halfspace. Other constraint functions may be possible, subject to a technical condition.

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