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.


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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[1] ANDERSON, R. et OREY, S. 1976. Small perturbations of dynamicals with reflecting boundary. Nagoya Math. J. 60 189 216. · Zbl 0324.60063
[2] BENSOUSSAN, A. et RASCANU, A. 1994. d-Dimensional stochastic differential equation with a multivalued subdifferential operator in drift. A paraitre.
[3] BREZIS, H. 1973. Operateurs monotones et semi-groupes de contractions dans les espaces \' de Hilbert. North-Holland, Amsterdam. \'
[4] CEPA, E. 1994. Equations differentielles stochastiques multivoques. These. \' \' \'
[5] CEPA, E. 1995. Equations differentielles stochastiques multivoques. Seminaire Probabilites \' \' \' \' XXIX 86 107. Springer, Berlin. · Zbl 0833.60079
[6] CEPA, E. et LEPINGLE, D. 1997. Diffusive particles with electrostatic repulsion. Probab. \' \' Theory Related Fields 107 429 449.
[7] DUPUIS, P. et ISHII, H. 1991. On Lipschitz continuity of the solution mapping to the Skorohod problem with applications. Stochastics 35 31 62. · Zbl 0721.60062
[8] FEYEL, D. 1987. Sur la methode de Picard EDO et EDS. Seminaire Probabilites XXI \' \' \' 515 520. Springer, Berlin. · Zbl 0629.60065
[9] FILLIPOV, A. N. 1960. Differential equations with discontinuous right-hand side. Mat. Sb.N.S. 51 99 128. · Zbl 0138.32204
[10] FREIDLIN, M. I. et WENTZELL, A. D. 1984. Random Perturbations of Dynamicals Systems. Springer, New York. \' · Zbl 0522.60055
[11] LEPINGLE, D. et MAROIS, C. 1987. Equations differentielles stochastiques multivoques \' únidimensionnelles. Seminaire Probabilites XXI 520 533. Springer, Berlin. \' \' · Zbl 0616.60059
[12] LIONS, P. L. et SZNITMAN, A. S. 1984. Stochastic differential equations with reflecting boundary conditions. Comm. Pure Appl. Math. 37 511 537. · Zbl 0598.60060
[13] SAISHO, Y. 1987. Stochastic differential equations for multidimensional domains with reflecting boundary. Probab. Theory Related Fields 74 455 477. · Zbl 0591.60049
[14] SKOROHOD, A. V. 1961. Stochastic equations for diffusions in a bounded region. Theory Probab. Appl. 6 264 274. · Zbl 0215.53501
[15] TANAKA, H. 1979. Stochastic differential equations with reflecting boundary conditions in convex regions. Hiroshima Math. J. 9 163 177. · Zbl 0423.60055
[16] ORLEANS, CEDEX 2 \' FRANCE E-MAIL: cepa@labomath.univ-orleans.fr
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