The Annals of Probability

Problàme de Skorohod multivoque

Emmanuel Cépa

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Résumé

On prouve un résultat d’existence et d’unicité pour une généralisation (par introduction d’un opérateur maximal monotone multivoque) du problème de Skorohod (avec réflexion normale) déterministe associé àun convexe fermé $D$ de $\mathbb{R}^d$. La formulation Í partir des opérateurs maximaux monotones multivoques permet de considérer des drifts singuliers explosant au bord du domaine. Cette ‘‘approche multivoque’’ clarifie la connexion entre problème de Skorohod et semigroupes nonlinéaires. En guise d’application, on considère ensuite le cas stochastique: les équations différentielles stochastiques multivoques sont ainsi revisitées. Par conséquent, cette contribution fournit une nouvelle méthode de construction de diffusions réfléchies normalement possédant un coefficient de dérive discontinu, explosif.

Abstract

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, we discuss then the stochastic case: multivalued stochastic differential equations are thus revisited. Therefore, we give an alternative way to construct diffusions with normal reflecting boundary conditions and discontinuous, exploding drift.

Article information

Source
Ann. Probab. Volume 26, Number 2 (1998), 500-532.

Dates
First available in Project Euclid: 31 May 2002

Permanent link to this document
http://projecteuclid.org/euclid.aop/1022855642

Mathematical Reviews number (MathSciNet)
MR1626174

Digital Object Identifier
doi:10.1214/aop/1022855642

Subjects
Primary: 35F05: Linear first-order equations 47N20: Applications to differential and integral equations 47N30: Applications in probability theory and statistics 60H10: Stochastic ordinary differential equations [See also 34F05] 60J60: Diffusion processes [See also 58J65]

Keywords
Multivalued stochastic differential equations Skorohod problem nonlinear semigroup reflected diffusion with exploding drift diffusive particle with electrostatic repulsion

Citation

Cépa, Emmanuel. Problàme de Skorohod multivoque. The Annals of Probability 26 (1998), no. 2, 500--532. doi:10.1214/aop/1022855642. http://projecteuclid.org/euclid.aop/1022855642.


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  • 45067 ORLEANS, CEDEX 2 ´ FRANCE E-MAIL: cepa@labomath.univ-orleans.fr