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Multivalued Skorohod problem. (Problème de Skorohod multivoque.) (French) Zbl 0937.34046

Summary: An existence and uniqueness result is proven for a generalization (by introduction of a multivalued maximal monotone operator) of the deterministic Skorohod problem (with normal reflection) associated with a closed convex \(D\) in \(\mathbb{R}^d\). The maximal monotone operator formulation allows for drifts that blow up as one gets near the boundary. This “multivalued approach” clarifies the connection between nonlinear semigroup theory and the Skorohod problem. As a consequence, the author discusses then the stochastic case: multivalued stochastic differential equations are thus revisited. Therefore, he gives an alternative way to construct diffusions with normal reflecting boundary conditions and discontinuous, exploding drift.

MSC:

34F05 Ordinary differential equations and systems with randomness
60H10 Stochastic ordinary differential equations (aspects of stochastic analysis)
47N20 Applications of operator theory to differential and integral equations
47N30 Applications of operator theory in probability theory and statistics
60J60 Diffusion processes
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