We consider the holder of an individual tontine retirement account, with maximum and minimum withdrawal amounts (per year) specified. The tontine account holder initiates the account at age 65 and earns mortality credits while alive, but forfeits all wealth in the account upon death. The holder wants to maximize total withdrawals and minimize expected shortfall at the end of the retirement horizon of 30 years (i.e., it is assumed that the holder survives to age 95). The holder controls the amount withdrawn each year and the fraction of the retirement portfolio invested in stocks and bonds. The optimal controls are determined based on a parametric model fitted to almost a century of market data. The optimal control algorithm is based on dynamic programming and the solution of a partial integro differential equation (PIDE) using Fourier methods. The optimal strategy (based on the parametric model) is tested out of sample using stationary block bootstrap resampling of the historical data. In terms of an expected total withdrawal, expected shortfall (EW-ES) efficient frontier, the tontine overlay dramatically outperforms an optimal strategy (without the tontine overlay), which in turn outperforms a constant weight strategy with withdrawals based on the ubiquitous four per cent rule.

A controlled heterogeneous collection of identical items is presented. According to their level of wear and tear, they are divided into a finite number of classes and the partition of the collection is allowed to change over time. A suitable exchangeability assumption is made to preserve the property that the items be identical. The role of the occupation numbers is investigated and a filtering problem is set up, where the observation is the cardinality of a particular class. A control on the dynamics of the items is introduced, and the existence of an optimal control is proved. A discrete-time approximation for the separated problem, which is a finite-dimensional one, is performed. As a consequence, an approximation for the value function is given.

In this paper we develop policies for scheduling dynamically arriving jobs to a broad class of parallel-processing queueing systems. We show that in heavy traffic the policies asymptotically minimize a measure of the expected system backlog, which we call system work. Our results yield succinct, closed-form expressions for optimal system work in heavy traffic.

A player starts at x in (-G, G) and attempts to leave the interval in a limited playing time. In the discrete-time problem, G is a positive integer and the position is described by a random walk starting at integer x, with mean increments zero, and variance increment chosen by the player from [0, 1] at each integer playing time. In the continuous-time problem, the player's position is described by an Ito diffusion process with infinitesimal mean parameter zero and infinitesimal diffusion parameter chosen by the player from [0, 1] at each time instant of play. To maximize the probability of leaving the interval (–G, G) in a limited playing time, the player should play boldly by always choosing largest possible variance increment in the discrete-time setting and largest possible diffusion parameter in the continuous-time setting, until the player leaves the interval. In the discrete-time setting, this result affirms a conjecture of Spencer. In the continuous-time setting, the value function of play is identified.

The distribution function for the negative binomial distribution is known to be an incomplete beta function. Here, some of the properties of the family of distribution functions for multivariate negative binomial distributions are explored. These properties are then used in deriving the expected cost per unit time for a multi-line joint-reordering system with Poisson demands. Policies are considered for which the quantity of any particular line in stock is the same at the beginning of every cycle. A method which gives good approximations to the optimal values of these quantities is described.