Optimal Stopping of Stochastic Transport Minimizing Submartingale Costs

On some universal sigma finite measures and some extensions of Doob's optional stopping theorem

In this paper, we associate, to any submartingale of class $(Σ)$, defined on a filtered probability space $(Ω, \mathcal{F}, \mathbb{P}, (\mathcal{F}_t)_{t \geq 0})$, which satisfies some technical conditions, a $σ$-finite measure $\mathcal{Q}$ on $(Ω, \mathcal{F})$, such that for all $t \geq 0$, and for all events $Λ_t \in \mathcal{F}_t$: $$ \mathcal{Q} [Λ_t, g\leq t] = \mathbb{E}_{\mathbb{P}} [\mathds{1}_{Λ_t} X_t]$$ where $g$ is the last hitting time of zero of the process $X$. This measure $\mathcal{Q}$ has already been defined in several particular cases, some of them are involved in the s…

On the Robust Dynkin Game

We study a robust Dynkin game over a set of mutually singular probabilities. We first prove that for the conservative player of the game, her lower and upper value processes coincide (i.e. She has a value process $V $ in the game). Such a result helps people connect the robust Dynkin game with second-order doubly reflected backward stochastic differential equations. Also, we show that the value process $V$ is a submartingale under an appropriately defined nonlinear expectations up to the first time $τ_*$ when $V$ meets the lower payoff process $L$. If the probability set is weakly compact, one…

Optimal Sequential Joint Detection and Estimation

This paper has been withdrawn by the authors. Please see arXiv:1302.6058. We consider the sequential joint detection and estimation problem. Minimizing the average stopping time subject to a combination of detection and estimation constraints we obtain the optimal triplet of stopping time, detector and estimator. In the joint detection and estimation problem the primary goal is to detect and estimate together, as opposed to the conventional testing of composite hypotheses where the primary goal is to detect only. For the first time in the literature we develop optimal solution to the sequentia…

Optimal stopping and a non-zero-sum Dynkin game in discrete time with risk measures induced by BSDEs

We first study an optimal stopping problem in which a player (an agent) uses a discrete stopping time in order to stop optimally a payoff process whose risk is evaluated by a (non-linear) $g$-expectation. We then consider a non-zero-sum game on discrete stopping times with two agents who aim at minimizing their respective risks. The payoffs of the agents are assessed by g-expectations (with possibly different drivers for the different players). By using the results of the first part, combined with some ideas of S. Hamad{è}ne and J. Zhang, we construct a Nash equilibrium point of this game by a…

Optimal Boundary Surface for Irreversible Investment with Stochastic Costs

This paper examines a Markovian model for the optimal irreversible investment problem of a firm aiming at minimizing total expected costs of production. We model market uncertainty and the cost of investment per unit of production capacity as two independent one-dimensional regular diffusions, and we consider a general convex running cost function. The optimization problem is set as a three-dimensional degenerate singular stochastic control problem. We provide the optimal control as the solution of a Skorohod reflection problem at a suitable boundary surface. Such boundary arises from the anal…

多维ergodic奇异随机控制问题的一类最优策略特征

Optimal Policy Characterization for a Class of Multi-Dimensional Ergodic Singular Stochastic Control Problems

本文研究多维ergodic奇异随机控制问题，通过Skorokhod反射在Y依赖自由边界中确定最优策略，并解决两个二维最优库存管理问题，首次将多维ergodic奇异随机控制与最优停止联系起来。