Optimal decision under ambiguity for diffusion processes

In this paper we consider stochastic optimization problems for an ambiguity averse decision maker who is uncertain about the parameters of the underlying process. In a first part we consider problems of optimal stopping under drift ambiguity for one-dimensional diffusion processes. Analogously to the case of ordinary optimal stopping problems for one-dimensional Brownian motions we reduce the problem to the geometric problem of finding the smallest majorant of the reward function in a two-parameter function space. In a second part we solve optimal stopping problems when the underlying process may crash down. These problems are reduced to one optimal stopping problem and one Dynkin game. Examples are discussed.     

Optimal Stopping Problem with a Vector-Valued Reward Function

In this note, using the well-known method of scalarization, we give an explicit characterization of the Pareto optimal stopping time for a vector-valued optimal stopping problem with only two reward functions. The present problem is a natural generalization of the classical McDonald-Siegel optimal stopping problem.     

有限时区上的最优转换和停止问题

讨论了有限时区上的最优转换和停止问题,它是一类同时具备脉冲控制和最优停止特征的最优控制问题.问题的最优值以及最优转换和停止决策可以由具有混合障碍的多维反射倒向随机微分方程的解来刻画.接着考虑了形式更一般的反射倒向随机微分方程并证明了方程解的存在唯一性.     

有限时区上的最优转换和停止问题

讨论了有限时区上的最优转换和停止问题,它是一类同时具备脉冲控制和最优停止特征的最优控制问题.问题的最优值以及最优转换和停止决策可以由具有混合障碍的多维反射倒向随机微分方程的解来刻画.接着考虑了形式更一般的反射倒向随机微分方程并证明了方程解的存在唯一性.     

Explicit solutions in one-sided optimal stopping problems for one-dimensional diffusions

Consider the optimal stopping problem of a one-dimensional diffusion with positive discount. Based on Dynkin's characterization of the value as the minimal excessive majorant of the reward and considering its Riesz representation, we give an explicit equation to find the optimal stopping threshold for problems with one-sided stopping regions, and an explicit formula for the value function of the problem. This representation also gives light on the validity of the smooth-fit (SF) principle. The results are illustrated by solving some classical problems, and also through the solution of: optimal stopping of the skew Brownian motion and optimal stopping of the sticky Brownian motion, including cases in which the SF principle fails.     

A model for optimal stopping in advertisement

In this work we propose a model for optimal advertisement in new product diffusion based on the Bass model and assuming that the effect of the environmental pressure in the diffusion of the product is subject to a stochastic dependence. The optimal stopping problem is reduced to a free boundary problem which is analyzed and solved numerically, in order to determine an optimal stopping rule for the advertisement campaign. The numerical solution is obtained through a policy iteration like contraction scheme, the convergence properties of which are studied in detail. Furthermore, the expected time until the optimal stopping of the campaign is estimated. Finally, a combined optimal stopping and control problem for the optimization of the advertisement effectiveness is also proposed and solved numerically. Our results are expected to provide useful guidelines for campaign managers, for the choice of effectiveness and duration of an advertisement campaign.     

The Convergence Rate from Discrete to Continuous Optimal Investment Stopping Problem

The author studies the optimal investment stopping problem in both continuous and discrete cases, where the investor needs to choose the optimal trading strategy and optimal stopping time concurrently to maximize the expected utility of terminal wealth.Based on the work of Hu et al. (2018) with an additional stochastic payoff function,the author characterizes the value function for the continuous problem via the theory of quadratic reflected backward stochastic differential equations (BSDEs for short) with unbounded terminal condition. In regard to the discrete problem, she gets the discretization form composed of piecewise quadratic BSDEs recursively under Markovian framework and the assumption of bounded obstacle, and provides some useful a priori estimates about the solutions with the help of an auxiliary forward-backward SDE system and Malliavin calculus. Finally, she obtains the uniform convergence and relevant rate from discretely to continuously quadratic reflected BSDE, which arise from corresponding optimal investment stopping problem through above characterization.     

Lower semicontinuity property of multiparameter optimal stopping value and its application to multiparameter prophet inequalities

This paper is concerned with the optimal stopping problem for discrete time multiparameter stochastic processes with the index set Nd. The optimal stopping value of a discrete time multiparameter integrable stochastic process whose negative part is uniformly integrable, is lower semicontinuous for the topology of convergence in distribution. The multiparameter version of prophet inequality for the one-parameter optimal stopping problem is formulated and the lower semicontinuity property of the optimal stopping value is applied to the multiparameter prophet inequality.     

An optimal stopping problem for a geometric Brownian motion with poissonian jumps

This paper examines an optimal stopping problem for a geometric Brownian motion with random jumps. It is assumed that jumps occur according to a time-homogeneous Poisson process and the proportions of these sizes are independent and identically distributed nonpositive random variables. The objective is to find an optimal stopping time of maximizing the expected discounted terminal reward which is defined as a nondecreasing power function of the stopped state. By applying the “smooth pasting technique” [1,2], we derive almost explicitly an optimal stopping rule of a threshold type and the optimal value function of the initial state. That is, we express the critical state of the optimal stopping region and the optimal value function by formulae which include only given problem parameters except an unknown to be uniquely determined by a nonlinear equation.     

Optimal variance stopping with linear diffusions

We study the optimal stopping problem of maximizing the variance of an unkilled linear diffusion. Especially, we demonstrate how the problem can be solved as a convex two-player zero-sum game, and reveal quite surprising application of game theory by doing so. Our main result shows that an optimal solution can, in a general case, be found among stopping times that are mixtures of two hitting times. This and other revealed phenomena together with suggested solution methods could be helpful when facing more complex non-linear optimal stopping problems. The results are illustrated by a few examples.     

An optimal stopping problem for fragmentation processes

In this article we consider a toy example of an optimal stopping problem driven by fragmentation processes. We show that one can work with the concept of stopping lines to formulate the notion of an optimal stopping problem and moreover, to reduce it to a classical optimal stopping problem for a generalized Ornstein–Uhlenbeck process associated with Bertoin’s tagged fragment. We go on to solve the latter using a classical verification technique thanks to the application of aspects of the modern theory of integrated exponential Lévy processes.     

Continuity Properties of Optimal Multiple Stopping Value

This article is concerned with the optimal multiple stopping problem for discrete time finite stage stochastic processes. We study lower semicontinuity and continuity properties of optimal stopping values with respect to the topology of convergence in distribution. Also, we formulate the multiple stopping version of the prophet inequality for the optimal stopping problem and apply the lower semicontinuity property of optimal stopping values to the prophet inequality for the optimal multiple stopping problem.     

Reflected BSDEs and robust optimal stopping for dynamic risk measures with jumps

We study the optimal stopping problem for dynamic risk measures represented by Backward Stochastic Differential Equations (BSDEs) with jumps and its relation with reflected BSDEs (RBSDEs). The financial position is given by an RCLL adapted process. We first state some properties of RBSDEs with jumps when the obstacle process is RCLL only. We then prove that the value function of the optimal stopping problem is characterized as the solution of an RBSDE. The existence of optimal stopping times is obtained when the obstacle is left-upper semi-continuous along stopping times. Finally, we investigate robust optimal stopping problems related to the case with model ambiguity and their links with mixed control/optimal stopping game problems. We prove that, under some hypothesis, the value function is equal to the solution of an RBSDE. We then study the existence of saddle points when the obstacle is left-upper semi-continuous along stopping times.     

Monotone stopping rules forstochastic processes in a semimartingale representation with applications

A monotone stopping problem is considered for stochastic processes in a semimartingale representation. Such a representation allows a direct infinitesimal characterization of the optimal stopping time. Transformations of such processes are investigated, which leave the semimartingale property unchanged. One of these transformations is a change of tiltration which leads to the stopping problem with partial information. Findly an application is discussed.     

On value functions for impulsive control of piecewise-deterministic processes

The paper deals with value functions for optimal stopping and impulsive control for piecewise-deterministic processes with discounted cost. The associated dynamic programming equations are variational and quasi-variational inequalities with integral and first-order differential terms The technique used is to approximate the value functions for an optimal stopping (impulsive control. switching control) problem for a piecewise-deterministic process by value functions for optimal stopping (impulsive control, switching control) problems for Feller piecewise-deterministic processes     

A general method for finding the optimal threshold in discrete time

We develop an approach for solving one-sided optimal stopping problems in discrete time for general underlying Markov processes on the real line. The main idea is to transform the problem into an auxiliary problem for the ladder height variables. In case that the original problem has a one-sided solution and the auxiliary problem has a monotone structure, the corresponding myopic stopping time is optimal for the original problem as well. This elementary line of argument directly leads to a characterization of the optimal boundary in the original problem. The optimal threshold is given by the threshold of the myopic stopping time in the auxiliary problem. Supplying also a sufficient condition for our approach to work, we obtain solutions for many prominent examples in the literature, among others the problems of Novikov-Shiryaev, Shepp-Shiryaev, and the American put in option pricing under general conditions. As a further application we show that for underlying random walks (and Lévy processes in continuous time), general monotone and log-concave reward functions g lead to one-sided stopping problems.     

The Elimination algorithm for the problem of optimal stopping

We present a new algorithm for solving the optimal stopping problem. The algorithm is based on the idea of elimination of states where stopping is nonoptimal and the corresponding correction of transition probabilities. The formal justification of this method is given by one of two presented theorems. The other theorem describes the situation when an aggregation of states is possible in the optimal stopping problem.     

Optimal stopping of a Brownian bridge with an unknown pinning point

The problem of stopping a Brownian bridge with an unknown pinning point to maximise the expected value at the stopping time is studied. A few general properties, such as continuity and various bounds of the value function, are established. However, structural properties of the optimal stopping region are shown to crucially depend on the prior, and we provide a general condition for a one-sided stopping region. Moreover, a detailed analysis is conducted in the cases of the two-point and the mixed Gaussian priors, revealing a rich structure present in the problem.     

On infinite horizon optimal stopping of general random walk

The objective of this study is to provide an alternative characterization of the optimal value function of a certain Black–Scholes-type optimal stopping problem where the underlying stochastic process is a general random walk, i.e. the process constituted by partial sums of an IID sequence of random variables. Furthermore, the pasting principle of this optimal stopping problem is studied.      

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We generalize the framework of [18] for optimal stopping time problem to allow a certain restricted class of stopping times. By using classical results in probability theory on families of random variables indexed by a restricted family of stopping times, we prove the existence of an optimal time, givecharacterizations of the minimal and maximal optimal stopping times, and provide some local properties of the value function family, in concert with all special cases studied previously.