A general replacement model

Summary Replacement models deal with the problem of finding the optimal random time for a preventive replacement of a technical system. A large class of such models has been considered in the literature recently. In this paper a unifying approach to the replacement problem is given regarding it as an optimal stopping problem. The generalization allows to weaken a necessary monotonicity condition and to consider different information levels. Measuring information in this context leads to discussing the problem in general terms of stochastic process theory. An example shows how to get explicit solutions and how the information level influences the optimal replacement policy.

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Zusammenfassung Ersetzungsmodelle beschreiben im Rahmen der Zuverlässigkeitstheorie das Problem, einen optimalen Zeitpunkt für die vorbeugende Ersetzung eines technischen Systems festzulegen. Eine große Vielfalt solcher Modelle ist in letzter Zeit in der Literatur zu finden. Ziel dieser Arbeit ist es, einen einheitlichen Zugang zu dem Ersetzungsproblem anzugeben, indem dieses Problem als Problem des optimalen Stoppens aufgefaßt wird. Die Verallgemeinerung vorhandener Modelle erlaubt es, eine notwendige Monotoniebedingung abzuschwächen und unterschiedliche Informationsniveaus zu betrachten. An einem Beispiel wird gezeigt, wie sich explizite Lösungen des Ersetzungsproblems gewinnen lassen und wie sich das Informationsniveau auf die optimale Ersetzungsstrategie auswirkt.

     

Existence and explicit determination of optimal stopping times

We consider large classes of continuous time optimal stopping problems for which we establish the existence and form of the optimal stopping times. These optimal times are then used to find approximate optimal solutions for a class of discrete time problems.     

离散模型下的美式期权定价

本文考虑离散时间金融市场模型中由效用函数U(x)所产生的报酬序列(U1 (Srn))n的最优停止问题.其中U(x)是由股票价格产生的效用.     

Estimates for the diameter of a martingale

We establish sharp weak type and logarithmic estimates for the diameter of the stopped Brownian motion. Then, using standard embedding theorems, we extend the results to the case of general real-valued continuous-path martingales. The proof rests on finding of the solutions to the corresponding three-dimensional optimal stopping problems.     

A generalization of Lyapunov's convexity theorem with applications in optimal stopping

Lyapunov proved that the range of finite measures defined on the same -algebra is compact, and if each measure also is atomless, then the range is convex. Although both conclusions may fail for measures on different -algebras of the same set, they do hold if the -algebras are nested, which is exactly the setting of classical optimal stopping theory.

     

Explicit solutions in one-sided optimal stopping problems for one-dimensional diffusions

Consider the optimal stopping problem of a one-dimensional diffusion with positive discount. Based on Dynkin's characterization of the value as the minimal excessive majorant of the reward and considering its Riesz representation, we give an explicit equation to find the optimal stopping threshold for problems with one-sided stopping regions, and an explicit formula for the value function of the problem. This representation also gives light on the validity of the smooth-fit (SF) principle. The results are illustrated by solving some classical problems, and also through the solution of: optimal stopping of the skew Brownian motion and optimal stopping of the sticky Brownian motion, including cases in which the SF principle fails.     

A general method for finding the optimal threshold in discrete time

We develop an approach for solving one-sided optimal stopping problems in discrete time for general underlying Markov processes on the real line. The main idea is to transform the problem into an auxiliary problem for the ladder height variables. In case that the original problem has a one-sided solution and the auxiliary problem has a monotone structure, the corresponding myopic stopping time is optimal for the original problem as well. This elementary line of argument directly leads to a characterization of the optimal boundary in the original problem. The optimal threshold is given by the threshold of the myopic stopping time in the auxiliary problem. Supplying also a sufficient condition for our approach to work, we obtain solutions for many prominent examples in the literature, among others the problems of Novikov-Shiryaev, Shepp-Shiryaev, and the American put in option pricing under general conditions. As a further application we show that for underlying random walks (and Lévy processes in continuous time), general monotone and log-concave reward functions g lead to one-sided stopping problems.     

Bounds for a constrained optimal stopping problem

In this short letter, we present an explicit upper bound for the optimal value of a bidimensional optimal stopping problem over stopping times τ subject to a constraint , where x(.) is a geometric Brownian motion coupled with an arbitrary diffusion process y(.), θ(., .) and c(.) are given positive, continuous functions and β > 0 is a fixed constant. The present result is derived from a corresponding Lagrangian dual problem, and using a recent result of Makasu (Seq Anal 27:435–440, 2008). Examples are given to illustrate our main result. Partial results of this note were obtained when the author was holding a postdoc grant PRO12/1003 at the Mathematics Institute, University of Oslo, Norway.     

随机利率下期权定价

本文在随机利率的基础上,考虑股票价格过程和利率过程分别为扩散过程和Ito过程,并且在相关的假设下,运用鞅方法推导出欧式期权价值过程所满足的微分方程;以及利率满足一种特殊方程时,运用最优停止的鞅方法,得到了随机利率下美式期权的价格和最优停时.     

On singular stochastic control and optimal stopping of spectrally negative jump diffusions

We consider a broad class of singular stochastic control problems of spectrally negative jump diffusions in the presence of potentially nonlinear state-dependent exercise payoffs. We analyse these problems by relying on associated variational inequalities and state a set of sufficient conditions under which the value of the considered problems can be explicitly derived in terms of the increasing minimal r-harmonic map. We also present a set of inequalities bounding the value of the optimal policy and prove that increased policy flexibility increases both the value of the optimal strategy as well as the rate at which this value grows.     

On an optimal stopping problem of Gusein-Zade

We study the problem of selecting one of the r best of n rankable individuals arriving in random order, in which selection must be made with a stopping rule based only on the relative ranks of the successive arrivals. For each r up to r=25, we give the limiting (as n→∞) optimal risk (probability of not selecting one of the r best) and the limiting optimal proportion of individuals to let go by before being willing to stop. (The complete limiting form of the optimal stopping rule is presented for each r up to r=10, and for r=15, 20 and 25.) We show that, for large n and r, the optical risk is approximately (1−t^*)^r, where t^*≈0.2834 is obtained as the roof of a function which is the solution to a certain differential equation. The optimal stopping rule τ_r,n lets approximately t^*n arrivals go by and then stops ‘almost immediately’, in the sense that τ_r,n/n→t^* in probability as n→∞, r→∞     

Maintenance optimal control three-machine replacement model under technological breakthrough expectations

This paper addresses a finite-horizon profit maximization three-machine replacement problem. More precisely, a model is formulated allowing for preventive maintenance to slow down machine quality and profit reduction caused by obsolescence, to determine the timing of replacing an existing machine by another available machine with improved technology. This decision is considered under uncertainty regarding the introduction time of a machine with a not-yetachieved technology. Given an exponential probability distribution function of the introduction time, the optimality of a bang-bang nonincreasing preventive maintenance control is shown.Moreover, subproblems maximizing the expected discounted profit are analyzed. Closed-form solutions are provided to compare machines of different technologies and to derive an analytical sensitivity analysis concerned with many issues related to the problem. The results are not necessarily intuitive and simple. For example, different relationships between the planning horizon and the preventive maintenance switching time are presented for the three-machine problem versus the single-machine problem.The focus of this paper is on the formulation and the analytical analysis of the problem rather than on its computational aspects.     

On the martingale and free-boundary approaches in sequential detection problems with exponential penalty for delay

We study the connection between the martingale and free-boundary approaches in sequential detection problems for the drift of a Brownian motion, under the assumption of exponential penalty for the delay. By means of the solution of a suitable free-boundary problem, we show that the reward process can be decomposed into the product between a gain function of the boundary point and a positive martingale inside the continuation region.     

On value preserving and growth optimal portfolios

In a discrete-time financial market setting, the paper relates various concepts introduced for dynamic portfolios (both in discrete and in continuous time). These concepts are: value preserving portfolios, numeraire portfolios, interest oriented portfolios, and growth optimal portfolios. It will turn out that these concepts are all associated with a unique martingale measure which agrees with the minimal martingale measure only for complete markets.     

美式期权的效用最大化问题

本文考虑有限离散和连续的金融市场模型 ,且市场是有效的 ,研究不同效用函数 U(x)所产生的报酬序列 { U(Sn)(1 +r) n} ,报酬函数 U(St)ert 的最优停止问题即何时达到美式效用最大化问题 .其中 U(x)是由股票价格产生的效用 .     

Bivariate gamma distributions, sums and ratios

Exact distributions of R = X +Y and W = X/(X +Y ) and the corresponding moment properties are derived when X and Y follow five flexible bivariate gamma distributions. The expressions turn out to involve several special functions.     

Generalized Wald equations in discrete time

Let S = {S_n, n ? 1} be a martingale. Expectations of m^th order quantities associated with S are related by two forms of Wald-type identity, called Generalized Wald equations. The previously known sufficient conditions for the validity of Wald equations are shown to be of a set of three equivalent conditions, each of which is necessary as well as sufficient for the validity of both types of Generalized Wald Equation.     

双复合Poisson风险模型下盈余首次达到给定水平的时间分析

针对双复合Poisson风险模型,利用鞅论的知识,研究了盈余首次达到给定水平的时刻的拉普拉斯变换、期望、方差和3阶中心矩。     

Statistical causality,martingale problems and local uniqueness

The paper considers a statistical concept of causality in continuous time in the filtered probability spaces which is based on the Granger’s definition of causality. The given causality concept is then applied to the solution of the martingale problem (associated with the stochastic differential equation driven with semimartingales). More precisely, we show that the given causality concept is closely connected to the concept of extremality of measures for the solutions of the martingale problem, for the stopped martingale problem and for the local martingale problem. We also show the equivalence between some models of causality and local uniqueness (for the solutions of the martingale problem).