Cash Management and Control Band Policies for Spectrally One-sided Levy Processes

We study the control band policy arising in the context of cash balance management. A policy is specified by four parameters (d,D,U,u). The controller pushes the process up to D as soon as it goes below d and pushes down to U as soon as it goes above u, while he does not intervene whenever it is within the set (d, u). We focus on the case when the underlying process is a spectrally one-sided Levy process and obtain the expected fixed and proportional controlling costs as well as the holding costs under the band policy

Contraction options and optimal multiple-stopping in spectrally negative Levy models

This paper studies the optimal multiple-stopping problem arising in the context of the timing option to withdraw from a project in stages. The profits are driven by a general spectrally negative Levy process. This allows the model to incorporate sudden declines of the project values, generalizing greatly the classical geometric Brownian motion model. We solve the one-stage case as well as the extension to the multiple-stage case. The optimal stopping times are of threshold-type and the value function admits an expression in terms of the scale function. A series of numerical experiments are conducted to verify the optimality and to evaluate the efficiency of the algorithm.Comment: 32 page

Inventory Control for Spectrally Positive Levy Demand Processes

A new approach to solve the continuous-time stochastic inventory problem using the fluctuation theory of Levy processes is developed. This approach involves the recent developments of the scale function that is capable of expressing many fluctuation identities of spectrally one-sided Levy processes. For the case with a fixed cost and a general spectrally positive Levy demand process, we show the optimality of an (s,S)-policy. The optimal policy and the value function are concisely expressed via the scale function. Numerical examples under a Levy process in the beta-family with jumps of infinite activity are provided to confirm the analytical results. Furthermore, the case with no fixed ordering costs is studied.Comment: Final version. To appear in Mathematics of Operations Researc

Optimality of two-parameter strategies in stochastic control

In this note, we study a class of stochastic control problems where the optimal strategies are described by two parameters. These include a subset of singular control, impulse control, and two-player stochastic games. The parameters are first chosen by the two continuous/smooth fit conditions, and then the optimality of the corresponding strategy is shown by verification arguments. Under the setting driven by a spectrally one-sided Levy process, these procedures can be efficiently done thanks to the recent developments of scale functions. In this note, we illustrate these techniques using several examples where the optimal strategy as well as the value function can be concisely expressed via scale functions

On the continuous and smooth fit principle for optimal stopping problems in spectrally negative Levy models

We consider a class of infinite-time horizon optimal stopping problems for spectrally negative Levy processes. Focusing on strategies of threshold type, we write explicit expressions for the corresponding expected payoff via the scale function, and further pursue optimal candidate threshold levels. We obtain and show the equivalence of the continuous/smooth fit condition and the first-order condition for maximization over threshold levels. As examples of its applications, we give a short proof of the McKean optimal stopping problem (perpetual American put option) and solve an extension to Egami and Yamazaki (2013).Comment: 26 page

Solving Optimal Dividend Problems via Phase-type Fitting Approximation of Scale Functions

The optimal dividend problem by De Finetti (1957) has been recently generalized to the spectrally negative L\'evy model where the implementation of optimal strategies draws upon the computation of scale functions and their derivatives. This paper proposes a phase-type fitting approximation of the optimal strategy. We consider spectrally negative L\'evy processes with phase-type jumps as well as meromorphic L\'evy processes (Kuznetsov et al., 2010a), and use their scale functions to approximate the scale function for a general spectrally negative L\'evy process. We obtain analytically the convergence results and illustrate numerically the effectiveness of the approximation methods using examples with the spectrally negative L\'evy process with i.i.d. Weibull-distributed jumps, the \beta-family and CGMY process.Comment: 33 pages, 8 figure

Asymptotic theory of sequential detection and identification in the hidden Markov models

We consider a unified framework of sequential change-point detection and hypothesis testing modeled by means of hidden Markov chains. One observes a sequence of random variables whose distributions are functionals of a hidden Markov chain. The objective is to detect quickly the event that the hidden Markov chain leaves a certain set of states, and to identify accurately the class of states into which it is absorbed. We propose computationally tractable sequential detection and identification strategies and obtain sufficient conditions for the asymptotic optimality in two Bayesian formulations. Numerical examples are provided to confirm the asymptotic optimality and to examine the rate of convergence

Optimality of doubly reflected Levy processes in singular control

We consider a class of two-sided singular control problems. A controller either increases or decreases a given spectrally negative Levy process so as to minimize the total costs comprising of the running and control costs where the latter is proportional to the size of control. We provide a sufficient condition for the optimality of a double barrier strategy, and in particular show that it holds when the running cost function is convex. Using the fluctuation theory of doubly reflected Levy processes, we express concisely the optimal strategy as well as the value function using the scale function. Numerical examples are provided to confirm the analytical results

On the Refracted-Reflected Spectrally Negative L\'evy Processes

We study a combination of the refracted and reflected L\'evy processes. Given a spectrally negative L\'evy process and two boundaries, it is reflected at the lower boundary while, whenever it is above the upper boundary, a linear drift at a constant rate is subtracted from the increments of the process. Using the scale functions, we compute the resolvent measure, the Laplace transform of the occupation times as well as other fluctuation identities that will be useful in applied probability including insurance, queues, and inventory management.Comment: 28 pages, forthcoming in Stochastic Processes and their Application

Games of singular control and stopping driven by spectrally one-sided Levy processes

We study a zero-sum game where the evolution of a spectrally one-sided Levy process is modified by a singular controller and is terminated by the stopper. The singular controller minimizes the expected values of running, controlling and terminal costs while the stopper maximizes them. Using fluctuation theory and scale functions, we derive a saddle point and the value function of the game. Numerical examples under phase-type Levy processes are also given.Comment: To appear in Stochastic Processes and Their Application