Fleet replacement under technological shocks

The paper analyzes a fleet replacement problem under general assumptions about technological improvement. The corresponding optimization model considers the variable lifetime of assets in deterministic infinite-horizon settings. The optimal dynamics of the asset lifetime and investment is investigated analytically and numerically under continuous technological change and technological shocks that affect the operating and new asset costs. The obtained results contribute to the OR replacement theory and suggest simple rules for the efficient fleet replacement under improving technology.     

Optimal equipment replacement without paradoxes: A continuous analysis

Combining known continuous- and discrete-time models of equipment replacement, we show that the optimal equipment lifetime is shorter when the embodied technological change is more intense. The paper has been inspired by a paradox in the equipment replacement raised by Cheevaprawatdomrong and Smith in Oper. Res. Lett. 31 (2003).     

Un modéle d'lnspections optimales

We consider the following model: we inspect the motion of a Markov process with which an “evolution cost” is associated. We inspect the process at times T ₁…, T _n ,…. If when we inspect, its value is in a given set A, it continues its evolution, otherwise we kill it. At each inspection we associate an "inspection cost" and a "killing cost". The problem consists of finding a sequence of optimal inspections. After the modelization we construct the value function by an iterative procedure as in impulse control theory, by using the theory of analytic functions and theorems of section. Thanks to the criteria of optimality we get a sequence of optimal inspections under very general hypotheses.     

On minimizing drawdown risks of lifetime investments

Drawdown measures the decline of portfolio value from its historic high-water mark. In this paper, we study a lifetime investment problem aiming at minimizing the risk of drawdown occurrences. Under the Black–Scholes framework, we examine two financial market models: a market with two risky assets, and a market with a risk-free asset and a risky asset. Closed-form optimal trading strategies are derived under both models by utilizing a decomposition technique on the associated Hamilton–Jacobi–Bellman (HJB) equation. We show that it is optimal to minimize the portfolio variance when the fund value is at its historic high-water mark. Moreover, when the fund value drops, the proportion of wealth invested in the asset with a higher instantaneous rate of return should be increased. We find that the instantaneous return rate of the minimum lifetime drawdown probability (MLDP) portfolio is never less than the return rate of the minimum variance (MV) portfolio. This supports the practical use of drawdown-based performance measures in which the role of volatility is replaced by drawdown.     

Static portfolio choice under Cumulative Prospect Theory

We derive the optimal portfolio choice for an investor who behaves according to Cumulative Prospect Theory (CPT). The study is done in a one-period economy with one risk-free asset and one risky asset, and the reference point corresponds to the terminal wealth arising when the entire initial wealth is invested into the risk-free asset. When it exists, the optimal holding is a function of a generalized Omega measure of the distribution of the excess return on the risky asset over the risk-free rate. It conceptually resembles Merton’s optimal holding for a CRRA expected-utility maximizer. We derive some properties of the optimal holding and illustrate our results using a simple example where the excess return has a skew-normal distribution. In particular, we show how a CPT investor is highly sensitive to the skewness of the excess return on the risky asset. In the model we adopt, with a piecewise-power value function with different shape parameters, loss aversion might be violated for reasons that are now well-understood in the literature. Nevertheless, we argue that this violation is acceptable.     

Household Lifetime Strategies under a Self-Contagious Market

In this paper, we consider the optimal strategies in asset allocation, consumption, and life insurance for a household with an exogenous stochastic income under a self-contagious market which is modeled by bivariate self-exciting Hawkes jump processes. By using the Hawkes process, jump intensities of the risky asset depend on the history path of that asset. In addition to the financial risk, the household is also subject to an uncertain lifetime and a fixed retirement date. A lump-sum payment will be paid as a heritage, if the wage earner dies before the retirement date. Under the dynamic programming principle, explicit solutions of the optimal controls are obtained when asset prices follow special jump distributions. For more general cases, we apply the Feynman–Kac formula and develop an iterative numerical scheme to derive the optimal strategies. We also prove the existence and uniqueness of the solution to the fixed point equation and the convergence of an iterative numerical algorithm. Numerical examples are presented to show the effect of jump intensities on the optimal controls.     

Regularity of the American Put option in the Black–Scholes model with general discrete dividends

We analyze the regularity of the value function and of the optimal exercise boundary of the American Put option when the underlying asset pays a discrete dividend at known times during the lifetime of the option. The ex-dividend asset price process is assumed to follow the Black–Scholes dynamics and the dividend amount is a deterministic function of the ex-dividend asset price just before the dividend date. This function is assumed to be non-negative, non-decreasing and with growth rate not greater than 1. We prove that the exercise boundary is continuous and that the smooth contact property holds for the value function at any time but the dividend dates. We thus extend and generalize the results obtained in Jourdain and Vellekoop (2011) [10] when the dividend function is also positive and concave. Lastly, we give conditions on the dividend function ensuring that the exercise boundary is locally monotonic in a neighborhood of the corresponding dividend date.     

The effect of tax depreciation on the stochastic replacement policy

The optimal replacement policy for an asset subject to a stochastic deteriorating operating cost is determined for three different tax depreciation schedules and a known re-investment cost, as the solution to a two-factor model using a quasi-analytical method. We find that tax depreciation exerts a critical influence over the replacement policy by lowering the operating cost thresholds. Although typically a decline in the corporate tax rate, increase in any initial capital allowance, or decrease in the depreciation lifetime (increase in depreciation rate) results in a lower operating cost threshold which justifies replacing older equipment, these results are not universal, and indeed for younger age assets the result may be the opposite. An accelerating depreciation schedule may incentivize early replacement in a deterministic context, but not necessarily for an environment of uncertainty.     

Optimal impulsive control of compartment models,I: Qualitative aspects

Optimal impulsive control of systems arising from linear compartment models for drug distribution in the human body is considered. A system of linear, time-invariant, homogeneous differential equations is given along with a set of continuous constraints on state and control. The object is to develop a constructive algorithm for the computation of the optimal control relative to a convex cost functional. It is first shown that under suitable hypotheses, satisfying the continuous constraints is equivalent to satisfying the constraints at a finite set of abstractly definedcritical points. Once these critical points have been determined, the solution of the optimal control problem is found as the solution of a finite-dimensional convex programming problem. The set of critical points can often be determineda priori solely from the qualitative behavior of the solutions of the system. A class of such problems, generalizing the so-calledplateau effect, is considered in detail. It is shown that the solution achieving the plateau effect is indeed optimal in certain cases. In a subsequent paper, an iterative algorithm will be given for the solution of these problems when the critical points cannot all be determineda priori.This work was supported in part by the National Science Foundation under Grant No. GP-20130.     

Network economics and optimal replacement of age-structured IT capital

The constructed hierarchical optimization model of vintage capital replacement takes into account network effects and the age-dependent technological structure of capital equipment. It involves the control of a network coalition choice, endogenous investments, capital structure, and capital lifetime. The qualitative analysis of the model shows how the lifetime and financial structure of the IT capital depends on technological change. These results are relevant for strategic management on a firm level. Provided numeric examples simulate the optimal lifetime of personal computers. This research is partially supported by the NATO grant CLG 982209.     

Optimization of financial and energy structure of productive capital

^** Email: nahritonenko{at}pvamu.edu^*** Email: yyatsenko{at}hbu.edu Optimal control of special non-linear Volterra integral equationsis used to optimize the structure and lifetime of age-dependentproductive capital at an individual enterprise level. The equationsdescribe a multifactor vintage capital model. The optimizationproblem is to select investment, specific capital cost, specificenergy consumption and capital lifetime that maximize net profiton finite and infinite horizons. The structure of optimal trajectoriesis investigated and relevant relations among technological andfinancial parameters of the productive capital are established.     

Inventory Models with Markovian Demands and Cost Functions of Polynomial Growth

This paper studies stochastic inventory problems with unbounded Markovian demands, ordering costs that are lower semicontinuous, and inventory/backlog (or surplus) costs that are lower semicontinuous with polynomial growth. Finite-horizon problems, stationary and nonstationary discounted-cost infinite-horizon problems, and stationary long-run average-cost problems are addressed. Existence of optimal Markov or feedback policies is established. Furthermore, optimality of (s, S)-type policies is proved when, in addition, the ordering cost consists of fixed and proportional cost components and the surplus cost is convex.     

Optimality of a Threshold Policy in theM/M/ queue with repeated vacations

Consider anM/M/1 queueing system with server vacations where the server is turned off as soon as the queue gets empty. We assume that the vacation durations form a sequence of i.i.d. random variables with exponential distribution. At the end of a vacation period, the server may either be turned on if the queue is non empty or take another vacation. The following costs are incurred: a holding cost ofh per unit of time and per customer in the system and a fixed cost of each time the server is turned on. We show that there exists a threshold policy that minimizes the long-run average cost criterion. The approach we use was first proposed in Blanc et al. (1990) and enables us to determine explicitly the optimal threshold and the optimal long-run average cost in terms of the model parameters.     

Asymptotic analysis for target asset portfolio allocation with small transaction costs

In this paper we discuss the asset allocation in the presence of small proportional transaction costs. The objective is to keep the asset portfolio close to a target portfolio and at the same time to reduce the trading cost in doing so. We derive the variational inequality and prove a verification theorem. Furthermore, we apply the second order asymptotic expansion method to characterize explicitly the optimal no transaction region when the transaction cost is small and show that the boundary points are asymmetric in relation to the target portfolio position, in contrast to the symmetric relation when only the first order asymptotic expansion method is used, and the leading order is a constant proportion of the cubic root of the small transaction cost. In addition, we use the asymptotic results for the boundary points and obtain an expansion for the value function. The results are illustrated in the numerical example.     

Minimizing the probability of lifetime ruin under borrowing constraints

We determine the optimal investment strategy of an individual who targets a given rate of consumption and who seeks to minimize the probability of going bankrupt before she dies, also known as lifetime ruin. We impose two types of borrowing constraints: First, we do not allow the individual to borrow money to invest in the risky asset nor to sell the risky asset short. However, the latter is not a real restriction because in the unconstrained case, the individual does not sell the risky asset short. Second, we allow the individual to borrow money but only at a rate that is higher than the rate earned on the riskless asset.We consider two forms of the consumption function: (1) The individual consumes at a constant (real) dollar rate, and (2) the individual consumes a constant proportion of her wealth. The first is arguably more realistic, but the second is closely connected with Merton’s model of optimal consumption and investment under power utility. We demonstrate that connection in this paper, as well as include a numerical example to illustrate our results.     

Multi-asset investment-consumption model with transaction costs

In this paper, we consider the multi-asset optimal investment-consumption model: a riskless asset and d risky assets. when the initial time is t?0, for a proportional transaction costs and discount factors, we proof that the value function of the model is a unique viscosity solution of a Hamilton-Jacobi-Bellman (HJB) equations.     

Optimale inspektion bei möglicherweise fehlerbehafteten inspektionsergebnissen

This article deals with the optimal inspection of a stochastically failing system with known lifetime distribution when a failure can be detected by inspection only. At a predetermined moment a plan stops with a special action which is not necessary an inspection. The objective function is the expected loss per cycle. As the duration of checking is non-negligible there are three different kinds of errors concerning the systems state at the end of and inspection epoch. We derive several conditions for the (N-) optimality of inspection plans, and in case of a uniformly distributed lifetime of the system we calculate (N-) optimal plans, which we compare with the numerical results of Luss/Kander and Hunter.     

Growth Optimal Portfolio Selection Under Proportional Transaction Costs with Obligatory Diversification

A continuous time long run growth optimal or optimal logarithmic utility portfolio with proportional transaction costs consisting of a fixed proportional cost and a cost proportional to the volume of transaction is considered. The asset prices are modeled as exponent of diffusion with jumps whose parameters depend on a finite state Markov process of economic factors. An obligatory portfolio diversification is introduced, accordingly to which it is required to invest at least a fixed small portion of our wealth in each asset.