Sensitivity analysis for ruin probabilities: canonical risk model

The surplus process of an insurance portfolio is defined as the wealth obtained by the premium payments minus the reimbursements made at the time of claims. When this process becomes negative (if ever), we say that ruin has occurred. The general setting is the Gambler's Ruin Problem. In this paper we address the problem of estimating derivatives (sensitivities) of ruin probabilities with respect to the rate of accidents. Estimating probabilities of rare events is a challenging problem, since naïve estimation is not applicable. Solution approaches are very recent, mostly through the use of importance sampling techniques. Sensitivity estimation is an even harder problem for these situations. We shall study three methods for estimating ruin probabilities: one via importance sampling (IS), and two others via indirect simulation: the storage process (SP), which restates the problems in terms of a queuing system, and the convolution formula (CF). To estimate the sensitivities, we apply the Rare Perturbation Analysis (RPA) method to IS, the Infinitesimal Perturbation Analysis (IPA) method to SP and the score function method to CF. Simulation methods are compared in terms of their efficiency, a criterion that appropriately weighs precision and CPU time. As well, we indicate how other criteria such as set-up time and prior formulae development may actually be problem-dependent.     

Smoothed perturbation analysis algorithms for estimating the derivatives of occupancy-related functions in serial queueing networks

We present unbiased Smoothed Perturbation Analysis (SPA) estimators for the derivatives of occupancy-related performance functions in serial networks ofG/G/1 queues with respect to parameters of the distributions of service times at the queues. The sample functions for these performance measures are piecewise constant, and established Infinitesimal Perturbation Analysis (IPA) methods typically fail to provide unbiased estimators in this case. The performance measures considered in this paper are: the average network occupancy as seen by an arrival, the average occupancy of a specific queue as seen by an arrival to it, the probability that a customer is blocked at a specific queue, and the probability that a customer leaves a queue idle. The SPA estimators derived are quite simple and flexible, and they lend themselves to straightforward analysis. Unlike most of the established SPA algorithms, ours are not based on the comparison of hazard rates, and the proofs of their unbiasedness do not require the boundedness of such hazard rates.Supported in part by the National Science Foundation under Grant ECS-8801912, by the Office of Naval Research under Contract N00014-87-K-0304, and by NASA under Contract NAG 2-595.     

Variance comparison between infinitesimal perturbation analysis and likelihood ratio estimators to stochastic gradient

We theoretically compare variances between the Infinitesimal Perturbation Analysis (IPA) estimator and the Likelihood Ratio (LR) estimator to Monte Carlo gradient for stochastic systems. The results presented in Cui et al. (2020) [2] on variance comparison between these two estimators are substantially improved. We also prove a practically interesting result that the IPA estimators to European vanilla and arithmetic Asian options' Delta, respectively, have smaller variance when the underlying asset's return process is independent with the initial price and square integrable.     

Some results on quadratic hedging with insider trading

We consider the hedging problem in an arbitrage-free incomplete financial market, where there are two kinds of investors with different levels of information about the future price evolution, described by two filtrations F and G=F∨σ(G) where G is a given r.v. representing the additional information. We focus on two types of quadratic approaches to hedge a given square-integrable contingent claim: local risk minimization (LRM) and mean-variance hedging (MVH). By using initial enlargement of filtrations techniques, we solve the hedging problem for both investors and compare their optimal strategies under both approaches.

In particular, for LRM, we show that for a large class of additional non trivial r.v.s G both investors will pursue the same locally risk minimizing portfolio strategy and the cost process of the ordinary agent is just the projection on F of that of the insider. For the MVH approach, we study also some general stochastic volatility model, including Hull and White, Heston and Stein and Stein models. In this more specific setting and for r.v.s G which are measurable with respect to the filtration generated by the volatility process, we obtain an expression for the insider optimal strategy in terms of the ordinary agent optimal strategy plus a process admitting a simple feedback-type representation.     

Approximate traveling wave solution for shallow water wave equation

Reconstruction of Variational Iteration Method (RVIM) is used for computing the coupled Whitham-Broer-Kaup shallow water. Then RVIM solution is verified against exact one and is compared with powerful approximate solutions, the Homotopy Perturbation Method (HPM) and Homotopy Analysis Method (HAM). The existent error of the methods is computed and convergence of the RVIM solution has been presented. Results obtained expose effectiveness and capability of this method to solve the nonlinear systems in mechanics, analytically.     

Scheduling arrivals to a stochastic service mechanism

We consider the problem of scheduling the arrivals of a fixed number of customers to a stochastic service mechanism to minimize an expected cost associated with operating the system. We consider the special case of exponentially distributed service times and the problems in general associated with obtaining exact analytic solutions. For general service time distributions we obtain approximate numerical solutions using a stochastic version of gradient search employing Infinitesimal Perturbation Analysis estimates of the objective function gradient obtained via simulation.     

Three analytical methods applied to Jeffery–Hamel flow

Three new analytical approximate techniques for addressing nonlinear problems are applied to Jeffery–Hamel flow. Homotopy Analysis Method (HAM), Homotopy Perturbation Method (HPM) and Differential Transformation Method (DTM) are proposed and used in this research. These methods are very useful and applicable for solving nonlinear problems. Then, the results are compared with numerical results and the validity of these methods is shown. Comparison between obtained results showed that HAM is more acceptable and accurate than two other methods. Ultimately, the effects of Reynolds number and divergent and convergent model of the channel on features of the flow are discussed.     

Interactive error correction in implicative theories

Errors in implicative theories coming from binary data are studied. First, two classes of errors that may affect implicative theories are singled out. Two approaches for finding errors of these classes are proposed, both of them based on methods of Formal Concept Analysis. The first approach uses the cardinality minimal (canonical or Duquenne–Guigues) implication base. The construction of such a base is computationally intractable. Using an alternative approach one checks possible errors on the fly in polynomial time via computing closures of subsets of attributes. Both approaches are interactive, based on questions about the validity of certain implications. Results of computer experiments are presented and discussed.     

Optimal design of personalized prostate cancer therapy using Infinitesimal Perturbation Analysis

The standard treatment for advanced prostate cancer is hormone therapy in the form of continuous androgen suppression (CAS), which unfortunately frequently leads to resistance and relapse. An alternative scheme is intermittent androgen suppression (IAS), in which patients are submitted to cycles of treatment (in the form of androgen deprivation) and off-treatment periods in an alternating manner. In spite of extensive recent clinical experience with IAS, the design of ideal protocols for any given patient remains a challenge. The level of prostate specific antigen (PSA) is frequently monitored to determine when patients will be taken off therapy and when therapy will resume. In this work, we propose a threshold-based policy for optimal IAS therapy design that is parameterized by lower and upper PSA threshold values and is associated with a cost metric that combines clinically relevant measures of therapy success. We use a Stochastic Hybrid Automaton (SHA) model of prostate cancer evolution under IAS and perform Infinitesimal Perturbation Analysis (IPA) to adaptively adjust PSA threshold values so as to improve therapy outcomes. We also apply this methodology to clinical data from real patients, and obtain promising results and valuable insights for personalized IAS therapy design.     

On the coupling of the homotopy perturbation method and Laplace transformation

In this paper, a Laplace homotopy perturbation method is employed for solving one-dimensional non-homogeneous partial differential equations with a variable coefficient. This method is a combination of the Laplace transform and the Homotopy Perturbation Method (LHPM). LHPM presents an accurate methodology to solve non-homogeneous partial differential equations with a variable coefficient. The aim of using the Laplace transform is to overcome the deficiency that is mainly caused by unsatisfied conditions in other semi-analytical methods such as HPM, VIM, and ADM. The approximate solutions obtained by means of LHPM in a wide range of the problem’s domain were compared with those results obtained from the actual solutions, the Homotopy Perturbation Method (HPM) and the finite element method. The comparison shows a precise agreement between the results, and introduces this new method as an applicable one which it needs fewer computations and is much easier and more convenient than others, so it can be widely used in engineering too.     

Trajectories of efficiency measurement: A bibliometric analysis of DEA and SFA

This study surveys the increasing research field of performance measurement by making use of a bibliometric literature analysis. We concentrate on two approaches, namely Data Envelopment Analysis (DEA) and Stochastic Frontier Analysis (SFA) as the most important methods to evaluate the efficiency of individual and organizational performance. It is the first literature survey that analyses DEA and SFA publications jointly, covering contributions published in journals, indexed by the Web of Science database from 1978 to 2012. Our aim is to identify seminal papers, playing a major role in DEA and SFA development and to determine areas of adoption. We recognized a constant growth of publications during the years identifying DEA as a standard technique in Operations Research, whereas SFA is mainly adopted in Economic research fields. Making use of document co-citation analysis we identify Airports and Supplier Selection (DEA) as well as Banking and Agriculture (SFA) as most influential application areas. Furthermore, Sensitivity and Fuzzy Set Theory (DEA) as well as Bayesian Analysis and Heterogeneity (SFA) are found to be most influential research areas and seem to be methodological trends. By developing an adoption rate of knowledge we identify that research, in terms of citations, is more focusing on relatively old and recent research at the expenses of middle-aged contributions, which is a typical phenomenon of a fast developing discipline.     

Valuation of urban industrial land: An analytic network process approach

In this paper an application of the Analytic Network Process (ANP) to asset valuation is presented. It has two purposes: solving some of the drawbacks found in classical asset valuation methods and broadening the scope of current approaches. The ANP is a method based on Multiple Criteria Decision Analysis (MCDA) that accurately models complex environments. This approach is particularly useful in problems which work with partially available data, qualitative variables and influences among the variables, which are very common situations in the valuation context. As an illustration, the new approach has been applied to a real case study of an industrial park located in Valencia (Spain) using three different models. The results confirm the validity of the methodology and show that the more information is incorporated into the model, the more accurate the solution will be, so the presented methodology stands out as a good alternative to current valuation approaches.     

IDEAL PERTURBATION LEMMA

We explain the essence of perturbation problems. The key to understanding is the structure of chain homotopy equivalence – the standard one must be replaced by a finer notion which we call a strong chain homotopy equivalence.

We formulate an Ideal Perturbation Lemma and show how both new and classical (including the Basic Perturbation Lemma) results follow from this ideal statement.     

Perturbation analysis of communication networks with feedback control using stochastic hybrid models

Communication networks may be abstracted through Stochastic Fluid Models (SFM) with the node dynamics described by switched flow equations as various events take place, thus giving rise to hybrid automaton models with stochastic transitions. The inclusion of feedback mechanisms complicates these dynamics. In a tandem setting, a typical feedback mechanism is the control of a node processing rate as a threshold-based function of the downstream node’s buffer level. We consider the problem of controlling the threshold parameters so as to optimize performance metrics involving average workload and packet loss and show how Infinitesimal Perturbation Analysis (IPA) can be used to analyze congestion propagation through a network and develop gradient estimators of such metrics.     

Condensation film on an inclined rotating disk

The three dimensional problem of steady fluid deposition on an inclined rotating disk is solved by similarity transform. For a given spraying rate there may be one, two or no steady state solution. The inclination causes a downward draining flow and a lateral flow. Perturbation solutions compare well with exact similarity solutions when the fluid film is thin.     

Resource contention games in multiclass stochastic flow models

We consider resource contention games in a stochastic hybrid system setting using Stochastic Flow Models (SFM) with multiple classes and class-dependent objectives. We present a general modeling framework for such games, where Infinitesimal Perturbation Analysis (IPA) estimators are derived for the derivatives of various class-dependent objectives. This allows us to study these games from the point of view of system-centric optimization of a performance metric and compare it to the user-centric approach where each user optimizes its own performance metric. We derive explicit solutions for a specific model in which the competing user classes employ threshold control policies and service is provided on a First Come First Serve (FCFS) basis. The unbiasedness of the IPA estimators is established in this case and it is shown that under certain conditions the system-centric and user-centric optimization solutions coincide.     

Application of homotopy-perturbation and variational iteration methods to nonlinear heat transfer and porous media equations

Perturbation methods depend on a small parameter which is difficult to be found for real-life nonlinear problems. To overcome this shortcoming, two new but powerful analytical methods are introduced to solve nonlinear heat transfer problems in this article; one is He's variational iteration method (VIM) and the other is the homotopy-perturbation method (HPM). The VIM is to construct correction functionals using general Lagrange multipliers identified optimally via the variational theory, and the initial approximations can be freely chosen with unknown constants. The HPM deforms a difficult problem into a simple problem which can be easily solved. Nonlinear convective–radiative cooling equation, nonlinear heat equation (porous media equation) and nonlinear heat equation with cubic nonlinearity are used as examples to illustrate the simple solution procedures. Comparison of the applied methods with exact solutions reveals that both methods are tremendously effective.     

A probabilistic analysis of the set covering problem

A probabilistic analysis of the minimum cardinality set covering problem (SCP) is developed, considering a stochastic model of the (SCP), withn variables andm constraints, in which the entries of the corresponding (m, n) incidence matrix are independent Bernoulli distributed random variables, each with constant probabilityp of success. The behaviour of the optimal solution of the (SCP) is then investigated as bothm andn grow asymptotically large, assuming either an incremental model for the evolution of the matrix (for each size, the matrixA is obtained bordering a matrix of smaller size by new columns and rows) or an independent one (for each size, an entirely new set of entries forA are considered). Two functions ofm are identified, which represent a lower and an upper bound onn in order the (SCP) to be a.e. feasible and not trivial. Then, forn lying within these bounds, an asymptotic formula for the optimum value of the (SCP) is derived and shown to hold a.e.The performance of two simple randomized algorithms is then analyzed. It is shown that one of them produces a solution value whose ratio to the optimum value asymptotically approaches 1 a.e. in the incremental model, but not in the independent one, in which case the ratio is proved to be tightly bounded by 2 a.e. Thus, in order to improve the above result, a second randomized algorithm is proposed, for which it is proved that the ratio between the approximate solution value and the optimum approaches 1 a.e. also in the independent model.     

Intelligent multicriteria decision support: Overview and perspectives

This paper presents a survey of the history and the recent status of the Multicriteria Decision Support Systems (MCDSSs). The last generation of MCDSSs is based on the synergistic operation of Artificial Intelligence and Multicriteria Analysis (MA) approaches. The contribution of these two different approaches into an integrated MCDSS is analysed through this paper, which concludes providing the recent software developments in this field.     

Ratio-based RTS determination in weight-restricted DEA models

This paper provides computationally efficient approaches for determining to which returns to scale (RTS) class a unit belongs in weight-restricted Data Envelopment Analysis (DEA) models. A non-traditional computational algorithm is introduced. The suggested approach is based on the calculation of certain ratios within the data set and offers obvious computational advantages over the traditional approaches involving the solution of standard DEA models. Some theorems and algorithms are given. Computational advantages of the provided results are discussed and one of the algorithms is illustrated using real world data.