216 research outputs found
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
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