Uniqueness of adapted solutions to scalar BSDEs with Peano-type generators
Résumé
It is well known that a Backward Stochastic Differential Equation (BSDE) with a Peano-type generator has infinite many solutions when the terminal value is zero. In this paper, we study the uniqueness of adapted solutions of such a BSDE when the terminal value is positive. Two methods are developed. The first one is to connect the BSDE to an optimal stochastic control problem: under suitable integrability of the terminal value, with a verification argument, we prove that the first component of the adapted solution pair is the value process for some given stochastic optimal control problem. The second one appeals to a change of variables, with a flavor of more analyses: by a change of variables, the original BSDE is reduced to a convex quadratic BSDE, and then using the θ-difference method, we give a sharp result in some special case which can be applied to the BSDE defined by Kreps-Porteus utility.
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