J.A.M. van der Weide

Gamma processes and peaks-over-threshold distributions for time-dependent reliability

In the evaluation of structural reliability, a failure is defined as the event in which stress exceeds a resistance that is liable to deterioration. This paper presents a method to combine the two stochastic processes of deteriorating resistance and fluctuating load for computing the time-dependent reliability of a structural component. The deterioration process is modelled as a gamma process, which is a stochastic process with independent non-negative increments having a gamma distribution with identical scale parameter. The stochastic process of loads is generated by a Poisson process. The variability of the random loads is modelled by a peaks-over-threshold distribution (such as the generalised Pareto distribution). These stochastic processes of deterioration and load are combined to evaluate the time-dependent reliability.

Discounted cost model for condition-based maintenance optimization

This paper presents methods to evaluate the reliability and optimize the maintenance of engineering systems that are damaged by shocks or transients arriving randomly in time and overall degradation is modeled as a cumulative stochastic point process. The paper presents a conceptually clear and comprehensive derivation of formulas for computing the discounted cost associated with a maintenance policy combining both condition-based and age-based criteria for preventive maintenance. The proposed discounted cost model provides a more realistic basis for optimizing the maintenance policies than those based on the asymptotic, non-discounted cost rate criterion.

Stochastic analysis of shock process and modeling of condition-based maintenance

The paper presents an analytical formulation for evaluating the maintenance cost of engineering systems that are damaged by shocks arriving randomly in time. The damage process is nonlinear in a sense that damage increments form an increasing sequence (i.e., accelerated damage) or a decreasing sequence (saturated damage) of random increments. Such processes are motivated from damage data collected from nuclear reactor components. To model the nonlinear nature of damage process, the paper proposes the use of non-homogeneous Poisson process for damage increments, which is in contrast with the common use of a renewal process for modeling the damage. The paper presents a conceptually clear and comprehensive derivation of formulas for computing the expected cost rate associated with a periodic inspection and preventive maintenance policy. Distinctions between the analysis of self-announced and latent failures are highlighted. The analytical model presented in this paper is quite generic and versatile, and it can be applied to optimize other types of maintenance policies.

The probability distribution of maintenance cost of a system affected by the gamma process of degradation: Finite time solution

Abstract The stochastic gamma process has been widely used to model uncertain degradation in engineering systems and structures. The optimization of the condition-based maintenance (CBM) policy is typically based on the minimization of the asymptotic cost rate. In the financial planning of a maintenance program, however, a more accurate prediction interval for the cost is needed for prudent decision making. The prediction interval cannot be estimated unless the probability distribution of cost is known. In this context, the asymptotic cost rate has a limited utility. This paper presents the derivation of the probability distribution of maintenance cost, when the system degradation is modelled as a stochastic gamma process. A renewal equation is formulated to derive the characteristic function, then the discrete Fourier transform of the characteristic function leads to the complete probability distribution of cost in a finite time setting. The proposed approach is useful for a precise estimation of prediction limits and optimization of the maintenance cost.

Finite-time maintenance cost analysis of engineering systems affected by stochastic degradation

The performance and reliability of engineering systems and structures are usually affected by uncertain degradation that occurs in service as a result of various physical and environmental processes, such as corrosion, erosion, fatigue, and creep. To maintain reliability of degrading systems, periodic inspection and preventive maintenance programmes are adopted. In the literature, the optimization of a maintenance programme is typically based on the minimization of the asymptotic cost rate. However, many engineering systems operate in a relatively short and finite time horizon in which the application of the asymptotic approximation becomes questionable. This paper presents an accurate formulation for computing the expected value and variance of the cost of a condition-based maintenance programme over a defined time horizon. A stochastic gamma process is used to model uncertain degradation. This paper emphasizes that the consideration of variance of the cost is of utmost importance in maintenance optimization, because it helps to identify a more robust (less uncertain) solution in a set of competing optimum solutions based on expected cost.

Renewal theory with exponential and hyperbolic discounting

To determine optimal investment and maintenance decisions, the total costs should be minimized over the whole life of a system or structure. In minimizing life-cycle costs, it is important to account for the time value of money by discounting and to consider the uncertainties involved. This article presents new results in renewal theory with costs that can be discounted according to any discount function that is nonincreasing and monotonic over time (such as exponential, hyperbolic, generalized hyperbolic, and no discounting). The main results include expressions for the first and second moment of the discounted costs over a bounded and unbounded time horizon as well as asymptotic expansions for nondiscounted costs.

American Basket and Spread Option Pricing by a Simple Binomial Tree

The return on a portfolio is the weighted average of the returns on the individual assets in the portfolio. But the dynamics of portfolio returns are not so simple. The standard assumption that the underlying asset for an option follows geometric Brownian motion is convenient for individual stocks, but it runs into trouble for combinations of stocks, because a linear combination of lognormal returns does not have a lognormal distribution. Luckily, the true portfolio return distribution can be closely approximated by a generalized lognormal using the technique of matching moments, even when some weights are negative, as in a spread trade. This makes it easy to price European options on baskets of stocks or spreads. In this article, the authors show how to extend the same basic idea to construct a binomial tree for the portfolio return, which allows for efficient pricing of contracts with American exercise.

A stochastic alternating renewal process model for unavailability analysis of standby safety equipment

The paper presents a stochastic approach to analyze instantaneous unavailability of standby safety equipment caused by latent failures. The problem of unavailability analysis is formulated as a stochastic alternating renewal process without any restrictions on the form of the probability distribution assigned to time to failure and repair duration. An integral equation for point unavailability is derived and numerically solved for a given maintenance policy. The paper also incorporates an age-based preventive maintenance policy with random repair time. In case of aging equipment, the asymptotic limit or average unavailability should be used with a caution, because it cannot model an increasing trend in unavailability as a result of increasing hazard rate (i.e. aging) of the time to failure distribution.

Applications to continuous-time processes of computational techniques for discrete-time renewal processes

For optimising maintenance, the total costs should be computed over a bounded or unbounded time horizon. In order to determine the expected costs of maintenance, renewal theory can be applied when we can identify renewals that bring a component back into the as-good-as-new condition. This publication presents useful computational techniques to determine the probabilistic characteristics of a renewal process. Because continuous-time renewal processes can be approximated with discrete-time renewal processes, it focusses on the latter processes. It includes methods to compute the probability distribution, expected value and variance of the number of renewals over a bounded time horizon, the asymptotic expansion for the expected value of the number of renewals over an unbounded time horizon, the approximation of a continuous renewal-time distribution with a discrete renewal-time distribution, and the extension of the discrete-time renewal model with the possibility of zero renewal times (in order to cope with an upper-bound approximation of a continuous-time renewal process).

A Benchmark Approach of Counterparty Credit Exposure of Bermudan Option under Lévy Process: The Monte Carlo-COS Method

An advanced method, which we call Monte Carlo-COS method, is proposed for computing the counterparty credit exposure profile of Bermudan options under Levy process. The different exposure profiles and exercise intensity under different mea- sures, P and Q, are discussed. Since the COS method [1] delivers accurate Bermudan prices, and no change of measure [2] needed to get the P-probability distribution, the exposure profile produced by the Monte Carlo-COS algorithm can be used as a benchmark result, E.g., to analyse the reliability of the popular American Monte Carlo method [3], [4] and [5]. The efficient calculation of expected exposure (EE) [6] can be further applied to the computation of credit value adjustment (CVA) [6].