BSDEs with weak terminal condition

Abstract

We introduce a new class of backward stochastic differential equations in which the $T$-terminal value $Y_{T}$ of the solution $(Y,Z)$ is not fixed as a random variable, but only satisfies a weak constraint of the form $E[\Psi(Y_{T})]\ge m$, for some (possibly random) nondecreasing map $\Psi$ and some threshold $m$. We name them BSDEs with weak terminal condition and obtain a representation of the minimal time $t$-values $Y_{t}$ such that $(Y,Z)$ is a supersolution of the BSDE with weak terminal condition. It provides a non-Markovian BSDE formulation of the PDE characterization obtained for Markovian stochastic target problems under controlled loss in Bouchard, Elie and Touzi [SIAM J. Control Optim. 48 (2009/10) 3123–3150]. We then study the main properties of this minimal value. In particular, we analyze its continuity and convexity with respect to the $m$-parameter appearing in the weak terminal condition, and show how it can be related to a dual optimal control problem in Meyer form. These last properties generalize to a non-Markovian framework previous results on quantile hedging and hedging under loss constraints obtained in Föllmer and Leukert [Finance Stoch. 3 (1999) 251–273; Finance Stoch. 4 (2000) 117–146], and in Bouchard, Elie and Touzi (2009/10).