BSDEs with nonlinear weak terminal condition

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In a recent paper, Bouchard, Elie and Reveillac BER have studied a new class of Backward Stochastic Differential Equations with weak terminal condition, for which the T-terminal value Yₜ of the solution (Y,Z) is not fixed as a random variable, but only satisfies a constraint of the form E[Ψ(Yₜ)] ≥ m. The aim of this paper is to introduce a more general class of BSDEs with nonlinear weak terminal condition. More precisely, the constraint takes the form Eᶠ₀,ₜ[Ψ(Yₜ)] ≥ m, where Eᶠ represents the f-conditional expectation associated to a nonlinear driver f. We carry out a similar analysis as in BER of the value function corresponding to the minimal solution Y of the BSDE with nonlinear weak terminal condition: we study the regularity, establish the main properties, in particular continuity and convexity with respect to the parameter m, and finally provide a dual representation and the existence of an optimal control in the case of concave constraints. From a financial point of view, our study is closely related to the approximative hedging of an European option under dynamic risk measures constraints. The nonlinearity f raises subtle difficulties, highlighted throughout the paper, which cannot be handled by the arguments used in the case of classical expectations constraints studied in BER.