Risk-sensitive stopping problems for continuous-time Markov chains

In this paper we consider stopping problems for continuous-time Markov chains under a general risk-sensitive optimization criterion for problems with finite and infinite time horizon. More precisely our aim is to maximize the certainty equivalent of the stopping reward minus cost over the time horizon. We derive optimality equations for the value functions and prove the existence of optimal stopping times. The exponential utility is treated as a special case. In contrast to risk-neutral stopping problems it may be optimal to stop between jumps of the Markov chain. We briefly discuss the influence of the risk sensitivity on the optimal stopping time and consider a special house selling problem as an example.     

The Wiener Sequential Testing Problem with Finite Horizon

We present a solution of the Bayesian problem of sequential testing of two simple hypotheses about the mean value of an observed Wiener process on the time interval with finite horizon. The method of proof is based on reducing the initial optimal stopping problem to a parabolic free-boundary problem where the continuation region is determined by two continuous curved boundaries. By means of the change-of-variable formula containing the local time of a diffusion process on curves we show that the optimal boundaries can be characterized as a unique solution of the coupled system of two nonlinear integral equations.     

Switching control of piecewise-deterministic processes

A finite collection of piecewise-deterministic processes are controlled in order to minimize the expected value of a performance functional with continuous operating cost and discrete switching control costs. The solution of the associated dynamic programming equation is obtained by an iterative approximation using optimal stopping time problems.This research was supported in part by NSF Grant No. DMS-8508651 and by University of Tennessee Science Alliance Research Incentive Award.     

A Connection between Singular Stochastic Control and Optimal Stopping

We show that the value function of a singular stochastic control problem is equal to the integral of the value function of an associated optimal stopping problem. The connection is proved for a general class of diffusions using the method of viscosity solutions.     

A Connection between Singular Stochastic Control and Optimal Stopping

We show that the value function of a singular stochastic control problem is equal to the integral of the value function of an associated optimal stopping problem. The connection is proved for a general class of diffusions using the method of viscosity solutions.     

Solving non–linear optimal stopping problems by the method of time–change

Some non–linear optimal stopping problems can be solved explicitly by using a common method which is based on time–change.We describe this method and illustrate its use by considering several examples dealing with Brownian motion.In each of these examples we derive explicit formulas for the value function and display the optimal stopping time. The main emphasis of the paper is on the method of proof and its unifying scope     

Optimal stopping of Markov switching Lévy processes

We consider a finite time horizon optimal stopping of a regime-switching Lévy process. We prove that the value function of the optimal stopping problem can be characterized as the unique viscosity solution of the associated Hamilton–Jacobi–Bellman variational inequalities.     

Optimal stopping with irregular reward functions

We consider optimal stopping problems with finite horizon for one-dimensional diffusions. We assume that the reward function is bounded and Borel-measurable, and we prove that the value function is continuous and can be characterized as the unique solution of a variational inequality in the sense of distributions.     

On optimal stopping of multidimensional diffusions

This paper develops an approach for solving perpetual discounted optimal stopping problems for multidimensional diffusions, with special emphasis on the $d$-dimensional Wiener process. We first obtain some verification theorems for diffusions, based on the Green kernel representation of the value function. Specializing to the multidimensional Wiener process, we apply the Martin boundary theory to obtain a set of tractable integral equations involving only harmonic functions that characterize the stopping region of the problem in the bounded case. The approach is illustrated through the optimal stopping problem of a $d$-dimensional Wiener process with a positive definite quadratic form reward function.     

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.     

Stopping of functionals with discontinuity at the boundary of an open set

We explore properties of the value function and existence of optimal stopping times for functionals with discontinuities related to the boundary of an open (possibly unbounded) set O. The stopping horizon is either random, equal to the first exit from the set O, or fixed (finite or infinite). The payoff function is continuous with a possible jump at the boundary of O. Using a generalization of the penalty method, we derive a numerical algorithm for approximation of the value function for general Feller-Markov processes and show existence of optimal or ε-optimal stopping times.     

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 numerical scheme for a class of nonlinear Fredholm integral equations of the second kind

In this paper an iterative approach for obtaining approximate solutions for a class of nonlinear Fredholm integral equations of the second kind is proposed. The approach contains two steps: at the first one, we define a discretized form of the integral equation and prove that by considering some conditions on the kernel of the integral equation, solution of the discretized form converges to the exact solution of the problem. Following that, in the next step, solution of the discretized form is approximated by an iterative approach. We finally on some examples show the efficiency of the proposed approach.     

On Numerov's method for a class of strongly nonlinear two-point boundary value problems

The purpose of this paper is to give a numerical treatment for a class of strongly nonlinear two-point boundary value problems. The problems are discretized by fourth-order Numerov's method, and a linear monotone iterative algorithm is presented to compute the solutions of the resulting discrete problems. All processes avoid constructing explicitly an inverse function as is often needed in the known treatments. Consequently, the full potential of Numerov's method for strongly nonlinear two-point boundary value problems is realized. Some applications and numerical results are given to demonstrate the high efficiency of the approach.     

Generalized Reflected BSDE and an Obstacle Problem for PDEs with a Nonlinear Neumann Boundary Condition

Abstract

In this article, we derive the existence and uniqueness of the solution for a class of generalized reflected backward stochastic differential equation involving the integral with respect to a continuous process, which is the local time of the diffusion on the boundary, in using the penalization method. We also give a characterization of the solution as the value function of an optimal stopping time problem. Then we give a probabilistic formula for the viscosity solution of an obstacle problem for PDEs with a nonlinear Neumann boundary condition.     

A method for solving an exterior three-dimensional boundary value problem for the Laplace equation

We develop and experimentally study the algorithms for solving three-dimensionalmixed boundary value problems for the Laplace equation in unbounded domains. These algorithms are based on the combined use of the finite elementmethod and an integral representation of the solution in a homogeneous space. The proposed approach consists in the use of the Schwarz alternating method with consecutive solution of the interior and exterior boundary value problems in the intersecting subdomains on whose adjoining boundaries the iterated interface conditions are imposed. The convergence of the iterative method is proved. The convergence rate of the iterative process is studied analytically in the case when the subdomains are spherical layers with the known exact representations of all consecutive approximations. In this model case, the influence of the algorithm parameters on the method efficiency is analyzed. The approach under study is implemented for solving a problem with a sophisticated configuration of boundaries while using a high precision finite elementmethod to solve the interior boundary value problems. The convergence rate of the iterations and the achieved accuracy of the computations are illustrated with some numerical experiments.     

Convergence of a fitted finite volume method for the penalized Black–Scholes equation governing European and American Option pricing

In this paper we present an analysis of a numerical method for a degenerate partial differential equation, called the Black–Scholes equation, governing American and European option pricing. The method is based on a fitted finite volume spatial discretization and an implicit time stepping technique. The analysis is performed within the framework of the vertical method of lines, where the spatial discretization is formulated as a Petrov–Galerkin finite element method with each basis function of the trial space being determined by a set of two-point boundary value problems. We establish the stability and an error bound for the solutions of the fully discretized system. Numerical results are presented to validate the theoretical results.     

Optimal stopping time on discounted semi-Markov processes

This paper attempts to study the optimal stopping time for semi- Markov processes (SMPs) under the discount optimization criteria with unbounded cost rates. In our work, we introduce an explicit construction of the equivalent semi-Markov decision processes (SMDPs). The equivalence is embodied in the expected discounted cost functions of SMPs and SMDPs, that is, every stopping time of SMPs can induce a policy of SMDPs such that the value functions are equal, and vice versa. The existence of the optimal stopping time of SMPs is proved by this equivalence relation. Next, we give the optimality equation of the value function and develop an effective iterative algorithm for computing it. Moreover, we show that the optimal and ε-optimal stopping time can be characterized by the hitting time of the special sets. Finally, to illustrate the validity of our results, an example of a maintenance system is presented in the end.     

Cascadic multilevel methods for ill-posed problems

Multilevel methods are popular for the solution of well-posed problems, such as certain boundary value problems for partial differential equations and Fredholm integral equations of the second kind. However, little is known about the behavior of multilevel methods when applied to the solution of linear ill-posed problems, such as Fredholm integral equations of the first kind, with a right-hand side that is contaminated by error. This paper shows that cascadic multilevel methods with a conjugate gradient-type method as basic iterative scheme are regularization methods. The iterations are terminated by a stopping rule based on the discrepancy principle.     

A globally convergent semi-smooth Newton method for control-state constrained DAE optimal control problems

We investigate a semi-smooth Newton method for the numerical solution of optimal control problems subject to differential-algebraic equations (DAEs) and mixed control-state constraints. The necessary conditions are stated in terms of a local minimum principle. By use of the Fischer-Burmeister function the local minimum principle is transformed into an equivalent nonlinear and semi-smooth equation in appropriate Banach spaces. This nonlinear and semi-smooth equation is solved by a semi-smooth Newton method. We extend known local and global convergence results for ODE optimal control problems to the DAE optimal control problems under consideration. Special emphasis is laid on the calculation of Newton steps which are given by a linear DAE boundary value problem. Regularity conditions which ensure the existence of solutions are provided. A regularization strategy for inconsistent boundary value problems is suggested. Numerical illustrations for the optimal control of a pendulum and for the optimal control of discretized Navier-Stokes equations conclude the article.