Bruno Bassan

Positive value of information in games

Abstract.We exhibit a general class of interactive decision situations in which all the agents benefit from more information. This class includes as a special case the classical comparison of statistical experiments à la Blackwell.

Mixed Optimal Stopping and Stochastic Control Problems with Semicontinuous Final Reward for Diffusion Processes

We consider mixed control problems for diffusion processes, i.e. problems which involve both optimal control and stopping. The running reward is assumed to be smooth, but the stopping reward need only be semicontinuous. We show that, under suitable conditions, the value function w has the same regularity as the final reward g, i.e. w is lower or upper semicontinuous if g is. Furthermore, when g is l.s.c., we prove that the value function is a viscosity solution of the associated variational inequality.

Optimal stopping problems with discontinous reward: Regularity of the value function and viscosity solutions

We study optimal stopping problems for diffusion processes with discontinuous reward function. We give some results about the regularity of the value function and we show that, under suitable mild conditions on the underlying process, it has the same regularity of the reward function, namely, it is lower (respectively: upper) semicontinuous if the reward function is. The proofs for the two cases are quite different, and the upper semicontinuous case requires stronger conditions. Finally, we show that, in the case of lower semicontinuous reward, under suitable conditions the value function is a (discontinuous) viscosity solution of the associated variational inequalities.

Variability orders and mean differences

Several well-known stochastic orderings are defined in terms of iterated integrals of distribution or survival functions. In this note we will provide necessary conditions for some variability orderings of the above type. These conditions will be based on the comparison of mean differences, which will be written by using the iterated integrals of survival and distribution functions. An interesting by-product of this idea is a curious formula for the variance. A bivariate version of the above results will be provided, as well.

On a 2n-valued telegraph signal and the related integrated process

We consider the stochastic process describing the position of a particle whose velocity changes in sign and magnitude at the occurrences of two independent Poisson processes We decompose the probability law of the process into n components, which jointly yield a solution of a system of n telegraph equations. In the particular case n = 2, the fourth-order equation governing the probability law is presented and its explicit expression is obtained when the Poisson rates are suitably connected

Regularity of the value function and viscosity solutions in optimal stopping problems for general markov processes

We consider optimal stopping problems for Markov processes with a semicontinuous reward function g , and we show that under suitable conditions the value function w = w [ g ] is itself semicontinuous and is a viscosity solution of the associated variational inequality.

Parameter estimation in differential equations, using random time transformations

Differential equations with measurements subject to errors are usually handled by Least Squares methods or by Likelihood methods based on diffusion-type stochastic modifications of the differential equation. We study the performance of likelihood methods based on substituting a Gaussian random time transformation as argument in the solution of the original deterministic differential equation. This method may be applied to the simultaneous estimation of parameters describing a number of differential equations, based on data with dependent measurement errors. The model is fitted to disease progress curves derived from a real data set consisting of disease assessments of melon plants infected by Zucchini Yellow Mosaic Virus (ZYMV).

Stochastic comparisons of Itô processes

Stochastic comparisons of Markov processes have mostly been in terms of transition functions or infinitesimal generators. For Ito processes, that is, solutions of stochastic differential equations, it is possible to obtain very intuitive comparisons in terms of three deterministic functions that govern the drift, diffusion, and jumps. Some further results on semimartingale Hunt processes show the detrimental effect of time changes upon such comparisons.

An Optimal Stopping Problem Arising from a Decision Model with Many Agents

We study an optimal stopping problem for a nonhomogeneous Markov process, with a reward function that is lower semicontinuous everywhere and smooth in certain regions. We prove that the payoff (value function) is lower semicontinuous as well and solves a so-called generalized Stefan problem in each of these regions. We provide some results for the geometry of the “stopping observations” set. Our results generalize those in Bassan, Brezzi, and Scarsini (1996). The problem we consider stems from an economic model in which several self-interested agents desire information, whereas a social planner, although benevolent toward the agents, might decide to withhold information in order to induce diversification in their behavior.

Information in continuous time decision models with many agents

Several agents with different subjective probabilities make a binary decision at a time determined by a planner. Each agent chooses the action that has the highest probability of success. Given that their probabilities differ, so will their choices. From time 0 until decision time, all the agents are entitled to access the same increasing flow of information. The planner, who gains from having as many agents as possible making the right choice, faces the following tradeoff: the more information she feeds to the agents, the better off they will be in making their decisions, but the less likely they will be to diversify their actions, so the more difficult it will be for her to hedge her positions. The model gives rise to a continuous time optimal stopping problem.