We consider the problem of finding the minimal initial data of a controlled process which guarantees to reach a controlled target with a given probability of success or, more generally, with a given level of expected loss. By suitably increasing the state space and the controls, we show that this problem can be converted into a stochastic target problem, i.e. find the minimal initial data of a controlled process which guarantees to reach a controlled target with probability one. Unlike the existing literature on stochastic target problems, our increased controls are valued in an unbounded set. In this paper, we provide a new derivation of the dynamic programming equation for general stochastic target problems with unbounded controls, together with the appropriate boundary conditions. These results are applied to the problem of quantile hedging in financial mathematics, and are shown to recover the explicit solution of Föllmer and Leukert [5].

We study a discrete-time approximation for solutions of systems of decoupled forward-backward stochastic differential equations with jumps. Assuming that the coefficients are Lipschitz-continuous, we prove the convergence of the scheme when the number of time steps n goes to infinity. When the jump coefficient of the first variation process of the forward component satisfies a non-degeneracy condition which ensures its inversibility, we obtain the optimal convergence rate n −1/2. The proof is based on a generalization of a remarkable result on the path-regularity of the solution of the backward equation derived by Zhang [28, 29] in the nojump case. A similar result is obtained without the non-degeneracy assumption whenever the coefficients are C 1 b with Lipschitz derivatives. Several extensions of these results are discussed. In particular, we propose a convergent scheme for the resolution of systems of coupled semilinear parabolic PDE's.

In this paper, we investigate a moral hazard problem in finite time with lump-sum and continuous payments, involving infinitely many Agents with mean field type interactions, hired by one Principal. By reinterpreting the mean-field game faced by each Agent in terms of a mean field forward backward stochastic differential equation (FBSDE for short), we are able to rewrite the Principal's problem as a control problem of McKean-Vlasov SDEs. We review one general approaches to tackle it, introduced recently in [1,43,44,45,46] using dynamic programming and Hamilton-Jacobi-Bellman (HJB for short) equations, and mention a second one based on the stochastic Pontryagin maximum principle, which follows [10]. We solve completely and explicitly the problem in special cases, going beyond the usual linear-quadratic framework. We finally show in our examples that the optimal contract in the N −players' model converges to the mean-field optimal contract when the number of agents goes to +∞, thus illustrating in our specific setting the general results of [8].

In this note, we provide an innovative and simple approach for proving the existence of a unique solution for multidimensional reflected BSDEs associated to switching problems. Getting rid of a monotonicity assumption on the driver function, this approach simplifies and extends the recent results of Hu & Tang [4] or Hamadene & Zhang [3].

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We consider the infinite horizon optimal consumption-investment problem under the drawdown constraint, i.e. the wealth process never falls below a fixed fraction of its running maximum. We assume that the risky asset is driven by the constant coefficients Black and Scholes model. For a general class of utility functions, we provide the value function in explicit form, and we derive closed-form expressions for the optimal consumption and investment strategy.

In this paper, we study a new type of BSDE, where the distribution of the Y -component of the solution is required to satisfy an additional constraint, written in terms of the expectation of a loss function. This constraint is imposed at any deterministic time t and is typically weaker than the classical pointwise one associated to reflected BSDEs. Focusing on solutions (Y, Z, K) with deterministic K, we obtain the well-posedness of such equation, in the presence of a natural Skorokhod type condition. Such condition indeed ensures the minimality of the enhanced solution, under an additional structural condition on the driver. Our results extend to the more general framework where the constraint is written in terms of a static risk measure on Y . In particular, we provide an application to the super hedging of claims under running risk management constraint. Introduction.Backward Stochastic Differential Equations (BSDEs) have been introduced by Pardoux and Peng [15] and share a strong connection with stochastic control problems. Solving a BSDE typically consists in the obtention of an adapted couple process (Y, Z) with the following dynamics:In their seminal paper, Pardoux and Peng provide the existence of a unique solution (Y, Z) to this equation for a given square integrable terminal condition ξ and a Lipschitz random driver f . Since then, many extensions have been derived in several directions. The regularity of the driver can for example be weakened. The underlying dynamics can be fairly more complex, via for example the addition of jumps. These extensions allow in particular to provide representation of solutions to a large class of stochastic control problems, and to tackle several meaningful applications in mathematical finance.More interestingly, the consideration of additional conditions on the stochastic control problems of interest naturally led to the consideration of constrained BSDEs. In such a case, the solution of a constrained BSDE contains an additional adapted increasing process K, such that (Y, Z, K) satisfiestogether with a chosen constraint on the solution. The process K interprets as the extra cost on the value process Y , due to the additional constraint. In such a framework, this equation admits an infinite ), partially supported by Lebesgue center of mathematics ("Investissements d'avenir" program -ANR-11-LABX-0020-01 and by ANR-15-CE05-0024-02.number of solutions, as the roles of Y and K are too closely connected. The underlying stochastic control problem of interest typically indicates that one should look for the minimal solution (in terms of Y ) of such equation. Motivated by optimal stopping or related obstacle problems, El Karoui et al. [10] introduced the notion of reflected BSDE, where the constraint is of the formThe obstacle process L is a lower bound on the solution Y and interprets as the reward payoff, if one chooses to stop immediately. It is worth noticing that the minimal solution (Y, Z, K) is fully characterized by the following so-called Skorokhod conditionThis conditio...

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[Ψ(Y T )] ≥ m, for some (possibly random) non-decreasing map Ψ 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 [2]. 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 [6,7], and in Bouchard, Elie and Touzi [2].