BSDEs with Singular Terminal Condition and a Control Problem with Constraints

Abstract

We provide a probabilistic solution of a not necessarily Markovian control problem with a state constraint by means of a Backward Stochastic Differential Equation (BSDE). The novelty of our solution approach is that the BSDE possesses a singular terminal condition. We prove that a solution of the BSDE exists, thus partly generalizing existence results obtained by Popier in [9] and [10]. We perform a verification and discuss special cases for which the control problem has explicit solutions.2010 MSC : 60H10, 91G80, 93E20.

Continuous Viscosity Solutions to Linear-Quadratic Stochastic Control Problems with Singular Terminal State Constraint

“…We also show how the uniqueness result carries over to non-Markovian models. This complements the analysis in [3,15].…”

A note on costs minimization with stochastic target constraints

“…Clearly, by choosing u 1]. Now, we are ready to state the main results which will be proved in Section 4.…”

“…Let x = 0, T = 1 and c = Φ −1 (y). Introduce the progressively measurable procesŝ 1) and the stochastic procesŝ…”

“…For a deterministic time horizon T > 0, for a random variable Ξ known at time T , for a cost function F : R + → R + and x ∈ R, find a control u with state process X t = x + t 0 u s ds which minimizes the expected costs given by E T 0 F (u t )dt under the terminal constraint X T ≥ Ξ. This question was inspired by a series of papers [1,7,8,10,11] which dealt with stochastic tracking problems under the terminal state constraint: {X T = Ξ} on a given event A. In this case, it is well established that the optimal control to such a problem is typically characterized by two coupled backward stochastic differential equations (BSDEs) with singular terminal constraints.…”

Limit behaviour of the minimal solution of a BSDE with singular terminal condition in the non Markovian setting

“…Extension have been studied in [11] and [16]. Recently singular BSDE were used to solve a particular stochastic control problem with application to portfolio management (see [1], [10], [12]). In this framework, the generator does not depend on z and has the following form:…”