Abstract
In this paper, we study a multi-dimensional backward stochastic differential equation (BSDE) with oblique reflection, which is a BSDE reflected on the boundary of a special unbounded convex domain along an oblique direction, and which arises naturally in the study of optimal switching problem. The existence of the adapted solution is obtained by the penalization method, the monotone convergence, and the a priori estimates. The uniqueness is obtained by a verification method (the first component of any adapted solution is shown to be the vector value of a switching problem for BSDEs). As applications, we apply the above results to solve the optimal switching problem for stochastic differential equations of functional type, and we give also a probabilistic interpretation of the viscosity solution to a system of variational inequalities.
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Part of this work was completed when Y. Hu was visiting Laboratory of Mathematics for Nonlinear Sciences, Fudan University, whose hospitality is greatly appreciated. S. Tang is supported in part by NSFC Grant #10325101, Basic Research Program of China (973 Program) Grant # 2007CB814904 and Chang Jiang Scholars Program. Part of this work was completed when S. Tang was visiting IRMAR, Université Rennes 1, whose hospitality is greatly appreciated.
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Hu, Y., Tang, S. Multi-dimensional BSDE with oblique reflection and optimal switching. Probab. Theory Relat. Fields 147, 89–121 (2010). https://doi.org/10.1007/s00440-009-0202-1
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DOI: https://doi.org/10.1007/s00440-009-0202-1
Keywords
- Backward stochastic differential equations
- Oblique reflection
- Optimal switching
- Variational inequalities