Joint replacement in an operational planning phase

We consider the problem of combining replacements of multiple components in an operational planning phase. Within an infinite or finite time horizon, decisions concerning replacement of components are made at discrete time epochs. The optimal solution of this problem is limited to only a small number of components. We present a heuristic rolling horizon approach that decomposes the problem; at each decision epoch an initial plan is made that addresses components separately, and subsequently a deviation from this plan is allowed to enable joint replacement. This approach provides insight into why certain actions are taken. The time needed to determine an action at a certain epoch is only quadratic in the number of components. After dealing with harmonisation and horizon effects, our approach yields average costs less than 1% above the minimum value.     

Optimal Replacement Planning by Geometric Programming

Geometric programming (GP) is suggested as an analytical toolfor solving replacement problems with infinite time horizon.The GP solution method is described and explained through theformulation and solution of a typical replacement problem. Asimple example is worked out to demonstrate the pint that GPhas potential as an appropriate mathematical tool for the analysisof certain types of replacement problems.     

A stochastic programming approach for planning horizons of infinite horizon capacity planning problems

Planning horizon is a key issue in production planning. Different from previous approaches based on Markov Decision Processes, we study the planning horizon of capacity planning problems within the framework of stochastic programming. We first consider an infinite horizon stochastic capacity planning model involving a single resource, linear cost structure, and discrete distributions for general stochastic cost and demand data (non-Markovian and non-stationary). We give sufficient conditions for the existence of an optimal solution. Furthermore, we study the monotonicity property of the finite horizon approximation of the original problem. We show that, the optimal objective value and solution of the finite horizon approximation problem will converge to the optimal objective value and solution of the infinite horizon problem, when the time horizon goes to infinity. These convergence results, together with the integrality of decision variables, imply the existence of a planning horizon. We also develop a useful formula to calculate an upper bound on the planning horizon. Then by decomposition, we show the existence of a planning horizon for a class of very general stochastic capacity planning problems, which have complicated decision structure.     

Convex infinite horizon programs

We establish conditions under which a sequence of finite horizon convex programs monotonically increases in value to the value of the infinite program; a subsequence of optimal solutions converges to the optimal solution of the infinite problem. If the conditions we impose fail, then (roughtly) the optimal value of the infinite horizon problem is an improper convex function. Under more restrictive conditions we establish the necessary and sufficient conditions for optimality. This constructive procedure gives us a way to solve the infinite (long range) problem by solving a finite (short range) problem. It appears to work well in practice.     

Uniqueness of unbounded viscosity solutions for impulse control problem

We study here the impulse control problem in infinite as well as finite horizon. We allow the cost functionals and dynamics to be unbounded and hence the value function can possibly be unbounded. We prove that the value function is the unique viscosity solution in a suitable subclass of continuous functions, of the associated quasivariational inequality. Our uniqueness proof for the infinite horizon problem uses stopping time problem and for the finite horizon problem, comparison method. However, we assume proper growth conditions on the cost functionals and the dynamics.     

A unified treatment of single component replacement models

In this paper we discuss a general framework for single component replacement models. This framework is based on the regenerative structure of these models and by using results from renewal theory a unified presentation of the discounted and average finite and infinite horizon cost models is given. Finally, some well-known replacement models are discussed, and making use of the previous results an easy derivation of their cost functions is presented.     

Finite Time Merton Strategy under Drawdown Constraint: A Viscosity Solution Approach

We consider the optimal consumption-investment problem under the drawdown constraint, i.e. the wealth process never falls below a fixed fraction of its running maximum. We assume that the risky asset is driven by the constant coefficients Black and Scholes model and we consider a general class of utility functions. On an infinite time horizon, Elie and Touzi (Preprint, [2006]) provided the value function as well as the optimal consumption and investment strategy in explicit form. In a more realistic setting, we consider here an agent optimizing its consumption-investment strategy on a finite time horizon. The value function interprets as the unique discontinuous viscosity solution of its corresponding Hamilton-Jacobi-Bellman equation. This leads to a numerical approximation of the value function and allows for a comparison with the explicit solution in infinite horizon.     

Infinite horizon programs; convergence of approximate solutions

This paper deals with infinite horizon, dynamic programs, stated in discrete time, and afflicted by no uncertainty. The essential objective, to be minimized, is the accumulated value of all discounted future costs, and it is assumed to satisfy the crucial condition that every lower level set is bounded with respect to a certain norm. That norm, as well as the natural space of trajectories, is problem intrinsic.In contrast to standard Markov decision processes (MDP) we admit unbounded singleperiod cost functions and exponential growth within an unlimited state space. Also, no assumption about stationarity in problem data is made.We show, under broad hypotheses, that any minimizing sequence accumulates to points which solve the dynamic program optimally. This result is important for the study of approximation schemes.Supported by grants from Total Marine via NTNF, and Wilhelm Keilhau's Minnefond.     

Asymptotic behavior of the finite-time expected time-integrated negative part of some risk processes and optimal reserve allocation

In the renewal risk model, we study the asymptotic behavior of the expected time-integrated negative part of the process. This risk measure has been introduced by Loisel (2005) [1]. Both heavy-tailed and light-tailed claim amount distributions are investigated. The time horizon may be finite or infinite. We apply the results to an optimal allocation problem with two lines of business of an insurance company. The asymptotic behavior of the two optimal initial reserves is computed.     

On the solvability of Bellman's functional equations for Markov renewal programming

The functional equations of undiscounted, stationary, infinite horizon Markov renewal programming are shown to possess a solution, by an elementary application of the Leray-Schauder fixed point theorem.     

The infinite time horizon dynamic pricing problem with multi-unit demand

This paper studies a dynamic pricing problem for a monopolist selling multiple identical items to potential buyers arriving over time, where the time horizon is infinite, the goods are imperishable and the buyers’ arrival follows a renewal process. Each potential buyer has some private information about his purchasing will, and this private information is unknown to the seller and therefore characterized as a random variable in this paper. Thus, the buyers may have multi-unit demand. Meanwhile, the seller needs to determine the optimal posted price such that his expected discounted revenue is maximized. This problem is formulated as a stochastic dynamic programming in this paper and then how to obtain the solution is explored. A numerical study shows that the optimal posted price performs better than that of optimal fixed price, and this advantage becomes obvious as the interest rate and/or the number of initial items increases.     

Mean-field stochastic control with elephant memory in finite and infinite time horizon

ABSTRACT

Our purpose of this paper is to study stochastic control problems for systems driven by mean-field stochastic differential equations with elephant memory, in the sense that the system (like the elephants) never forgets its history. We study both the finite horizon case and the infinite time horizon case.

• In the finite horizon case, results about existence and uniqueness of solutions of such a system are given. Moreover, we prove sufficient as well as necessary stochastic maximum principles for the optimal control of such systems. We apply our results to solve a mean-field linear quadratic control problem.

• For infinite horizon, we derive sufficient and necessary maximum principles.

  As an illustration, we solve an optimal consumption problem from a cash flow modelled by an elephant memory mean-field system.

     

Duality in infinite dimensional linear programming

We consider the class of linear programs with infinitely many variables and constraints having the property that every constraint contains at most finitely many variables while every variable appears in at most finitely many constraints. Examples include production planning and equipment replacement over an infinite horizon. We form the natural dual linear programming problem and prove strong duality under a transversality condition that dual prices are asymptotically zero. That is, we show, under this transversality condition, that optimal solutions are attained in both primal and dual problems and their optimal values are equal. The transversality condition, and hence strong duality, is established for an infinite horizon production planning problem.This material is based on work supported by the National Science Foundation under Grant No. ECS-8700836.     

Risk-sensitive control of continuous time Markov chains

We study risk-sensitive control of continuous time Markov chains taking values in discrete state space. We study both finite and infinite horizon problems. In the finite horizon problem we characterize the value function via Hamilton Jacobi Bellman equation and obtain an optimal Markov control. We do the same for infinite horizon discounted cost case. In the infinite horizon average cost case we establish the existence of an optimal stationary control under certain Lyapunov condition. We also develop a policy iteration algorithm for finding an optimal control.     

Optimal policies for inventory systems with discretionary sales, random yield and lost sales

We determine replenishment and sales decisions jointly for an inventory system with random demand, lost sales and random yield. Demands in consecutive periods are independent random variables and their distributions are known. We incorporate discretionary sales, when inventory may be set aside to satisfy future demand even if some present demand may be lost. Our objective is to minimize the total discounted cost over the problem horizon by choosing an optimal replenishment and discretionary sales policy. We obtain the structure of the optimal replenishment and discretionary sales policy and show that the optimal policy for finite horizon problem converges to that of the infinite horizon problem. Moreover, we compare the optimal policy under random yield with that under certain yield, and show that the optimal order quantity （sales quantity） under random yield is more （less） than that under certain yield.     

Stationary solutions of SPDEs and infinite horizon BDSDEs with non-Lipschitz coefficients

We prove a general theorem that the -valued solution of an infinite horizon backward doubly stochastic differential equation, if exists, gives the stationary solution of the corresponding stochastic partial differential equation. We prove the existence and uniqueness of the -valued solutions for backward doubly stochastic differential equations on finite and infinite horizon with linear growth without assuming Lipschitz conditions, but under the monotonicity condition. Therefore the solution of finite horizon problem gives the solution of the initial value problem of the corresponding stochastic partial differential equations, and the solution of the infinite horizon problem gives the stationary solution of the SPDEs according to our general result.     

The gerber-shiu expected discounted penalty function for Lévy insurance risk processes

In this paper,we study a general Lévy risk process with positive and negative jumps.A renewal equation and an infinite series expression are obtained for the expected discounted penalty function of this risk model.We also examine some asymptotic behaviors for the ruin probability as the initial capital tends to infinity.     

On the Selection of One Feedback Nash Equilibrium in Discounted Linear-Quadratic Games

We study a selection method for a Nash feedback equilibrium of a one-dimensional linear-quadratic nonzero-sum game over an infinite horizon. By introducing a change in the time variable, one obtains an associated game over a finite horizon T > 0 and with free terminal state. This associated game admits a unique solution which converges to a particular Nash feedback equilibrium of the original problem as the horizon T goes to infinity.     

Applications des EDSR à horizon infini à un problème de contrôle impulsionnel

We study an impulse control problem in infinite horizon. To solve this problem, we extend to the infinite horizon case results of double barrier reflected backward stochastic differential equations. The properties of the Snell envelope reduce our problem to the existence of a pair of continuous processes.     

Robust static hedging of barrier options in stochastic volatility models

Static hedge portfolios for barrier options are extremely sensitive with respect to changes of the volatility surface. In this paper we develop a semi-infinite programming formulation of the static super-replication problem in stochastic volatility models which allows to robustify the hedge against model parameter uncertainty in the sense of a worst case design. From a financial point of view this robustness guarantees the hedge performance for an infinite number of future volatility surface scenarios including volatility shocks and changes of the skew. After proving existence of such robust hedge portfolios and presenting an algorithm to numerically solve the underlying optimization problem, we apply the approach to a detailed example. Surprisingly, the optimal robust portfolios are only marginally more expensive than the barrier option itself.