Multivalued backward stochastic differential equations with oblique subgradients. (English) Zbl 1328.60137
Summary: We study the existence and uniqueness of the solution for the following backward stochastic variational inequality with oblique reflection (for short, \(\mathrm{BSVI}(H(t, y) \partial \varphi(y))\)), written in differential form
\[
-dY_t + H(t, Y_t) \partial \varphi(Y_t)(dt) \ni F(t, Y_t, Z_t)dt - Z_t d B_t, \;\;t \in [0, T], \;\;Y_T = \eta,
\]
where \(H\) is a bounded symmetric smooth matrix and \(\varphi\) is a proper convex lower semicontinuous function, with \(\partial \varphi\) being its subdifferential operator. The presence of the product \(H \partial \varphi\) does not permit the use of standard techniques because it conserves neither the Lipschitz property of the matrix nor the monotonicity property of the subdifferential operator. We prove that, if we consider the dependence of \(H\) only on the time, the equation admits a unique strong solution and, allowing the dependence on the state of the system, the above \(\mathrm{BSVI}(H(t, y) \partial \varphi(y))\) admits a weak solution in the sense of the Meyer-Zheng topology. However, for that purpose we must renounce at the dependence on \(Z\) for the generator function and we situate our problem in a Markovian framework.
MSC:
| 60H10 | Stochastic ordinary differential equations (aspects of stochastic analysis) |
| 60H99 | Stochastic analysis |
| 60H30 | Applications of stochastic analysis (to PDEs, etc.) |
| 93E03 | Stochastic systems in control theory (general) |
Keywords:
multivalued backward stochastic differential equations; subdifferential operators; oblique reflection; Meyer-Zheng topologyReferences:
| [1] | Billingsley, P., Convergence of Probability Measures (1999), John Wiley & Sons: John Wiley & Sons New York, NY · Zbl 0172.21201 |
| [2] | Bismut, J. M., Conjugate convex functions in optimal stochastic control, J. Math. Anal. Appl., 44, 384-404 (1973) · Zbl 0276.93060 |
| [3] | Boufoussi, B.; van Casteren, J., An approximation result for a nonlinear Neumann boundary value problem via BSDEs, Stochastic Process. Appl., 114, 331-350 (2004) · Zbl 1073.60062 |
| [4] | Brézis, H., Opérateurs Maximaux Monotones et Semigroupes de Contractions Dans les Espaces de Hilbert (1973), North-Holland: North-Holland Amsterdam · Zbl 0252.47055 |
| [5] | Buckdahn, R.; Engelbert, H.-J.; Răşcanu, A., On weak solutions of backward stochastic differential equations, Teor. Veroyatn. Primen., 49, 1, 70-108 (2004) · Zbl 1095.60019 |
| [6] | Gassous, A.; Răşcanu, A.; Rotenstein, E., Stochastic variational inequalities with oblique subgradients, Stochastic Process. Appl., 122, 7, 2668-2700 (2012) · Zbl 1255.60095 |
| [7] | Hu, Y.; Tang, S., Multi-dimensional BSDE with oblique reflection and optimal switching, Probab. Theory Related Fields, 147, 1-2, 89-121 (2010) · Zbl 1188.60029 |
| [8] | Jakubowski, A., A non-Skorokhod topology on the Skorokhod space, Electron. J. Probab., 2, 4, 1-21 (1997) · Zbl 0890.60003 |
| [9] | Karatzas, I.; Shreve, S., Brownian Motion and Stochastic Calculus (1988), Springer-Verlag · Zbl 0638.60065 |
| [10] | LeJay, A., BSDE driven by Dirichlet process and semi-linear parabolic PDE. Application to homogenization, Stochastic Process. Appl., 1, 1-39 (2002) · Zbl 1058.60045 |
| [11] | Maticiuc, L.; Pardoux, E.; Răşcanu, A.; Zălinescu, A., Viscosity solutions for systems of parabolic variational inequalities, Bernoulli, 16, 1, 258-273 (2010) · Zbl 1455.35311 |
| [12] | Maticiuc, L.; Răşcanu, A., Backward stochastic generalized variational inequality, (Applied Analysis and Differential Equations (2007), World Sci. Publ.: World Sci. Publ. Hackensack, NJ), 217-226 · Zbl 1168.60023 |
| [13] | Maticiuc, L.; Răşcanu, A., A stochastic approach to a multivalued Dirichlet-Neumann problem, Stochastic Process. Appl., 120, 777-800 (2010) · Zbl 1195.35192 |
| [14] | Pardoux, E., BSDEs, weak convergence and homogenization of semilinear PDEs, (Nonlinear Analysis, Differential Equations and Control. Nonlinear Analysis, Differential Equations and Control, NATO Science Series, vol. 528 (1999)), 503-549 · Zbl 0959.60049 |
| [15] | Pardoux, E.; Peng, S., Solution of a backward stochastic differential equation, Systems Control Lett., 14, 55-61 (1990) · Zbl 0692.93064 |
| [16] | Pardoux, E.; Peng, S., Backward SDE’s and quasilinear parabolic PDE’s, (Stochastic PDE and their Applications. Stochastic PDE and their Applications, LNCIS, vol. 176 (1992), Springer), 200-217 · Zbl 0766.60079 |
| [17] | Pardoux, E.; Răşcanu, A., Backward stochastic differential equations with subdifferential operator and related variational inequalities, Stochastic Process. Appl., 76, 2, 191-215 (1998) · Zbl 0932.60070 |
| [18] | Pardoux, E.; Răşcanu, A., Backward stochastic variational inequalities, Stoch. Stoch. Rep., 67, 3-4, 159-167 (1999) · Zbl 0948.60049 |
| [19] | Pardoux, E.; Răşcanu, A., (Stochastic Differential Equations, Backward SDEs, Partial Differential Equations. Stochastic Differential Equations, Backward SDEs, Partial Differential Equations, Stochastic Modelling and Applied Probability, vol. 69 (2014), Springer) · Zbl 1321.60005 |
| [20] | Răşcanu, A.; Rotenstein, E., The Fitzpatrick function—a bridge between convex analysis and multivalued stochastic differential equations, J. Convex Anal., 18, 1, 105-138 (2011) · Zbl 1210.60070 |
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