The Annals of Probability

BSDEs with weak terminal condition

Bruno Bouchard, Romuald Elie, and Antony Réveillac

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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).

Article information

Ann. Probab., Volume 43, Number 2 (2015), 572-604.

First available in Project Euclid: 2 February 2015

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Zentralblatt MATH identifier

Primary: 60H10: Stochastic ordinary differential equations [See also 34F05] 93E20: Optimal stochastic control
Secondary: 49L20: Dynamic programming method 91G80: Financial applications of other theories (stochastic control, calculus of variations, PDE, SPDE, dynamical systems)

Backward stochastic differential equations optimal control stochastic target


Bouchard, Bruno; Elie, Romuald; Réveillac, Antony. BSDEs with weak terminal condition. Ann. Probab. 43 (2015), no. 2, 572--604. doi:10.1214/14-AOP913.

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