A generalized Neyman–Pearson lemma for g-probabilities

Maximum principle for a stochastic delayed system involving terminal state constraints

We investigate a stochastic optimal control problem where the controlled system is depicted as a stochastic differential delayed equation; however, at the terminal time, the state is constrained in a convex set. We firstly introduce an equivalent backward delayed system depicted as a time-delayed backward stochastic differential equation. Then a stochastic maximum principle is obtained by virtue of Ekeland's variational principle. Finally, applications to a state constrained stochastic delayed linear-quadratic control model and a production-consumption choice problem are studied to illustrate the main obtained result.

Roy's safety-first portfolio principle in financial risk management of disastrous events

Roy pioneers the concept and practice of risk management of disastrous events via his safety-first principle for portfolio selection. More specifically, his safety-first principle advocates an optimal portfolio strategy generated from minimizing the disaster probability, while subject to the budget constraint and the mean constraint that the expected final wealth is not less than a preselected disaster level. This article studies the dynamic safety-first principle in continuous time and its application in asset and liability management. We reveal that the distortion resulting from dropping the mean constraint, as a common practice to approximate the original Roy's setting, either leads to a trivial case or changes the problem nature completely to a target-reaching problem, which produces a highly leveraged trading strategy. Recognizing the ill-posed nature of the corresponding Lagrangian method when retaining the mean constraint, we invoke a wisdom observed from a limited funding-level regulation of pension funds and modify the original safety-first formulation accordingly by imposing an upper bound on the funding level. This model revision enables us to solve completely the safety-first asset-liability problem by a martingale approach and to derive an optimal policy that follows faithfully the spirit of the safety-first principle and demonstrates a prominent nature of fighting for the best and preventing disaster from happening.