Methodology and Computing in Applied Probability最新文献

In this paper, the objective is to provide sequences of improved non-increasing (non-decreasing) upper (lower) bounds for the solution of (defective) renewal equations in terms of the right-tail probability of a compound geometric distribution. Exponential (Lundberg type) and non-exponential type bounds are also derived. Also, under several reliability classifications, some new as well as improvements of well-known bounds are given. The results are then applied to obtain refinements of the bounds for several ruin related quantities, (such as the deficit at ruin, the joint distribution of the surplus prior to and at ruin, the mean deficit at ruin and the stop-loss premium, and the compound geometric densities). Bounds for the renewal function are also given.

In this paper, the Bahadur representation of the empirical and Bernstein polynomials estimators of the quantile function based on associated sequences are investigated. The rate of approximation depends on the rate of decay in covariances, and it converges to the optimal rate observed under independence when the covariances quickly approach zero. As an application, we establish a Berry-Esseen bound with the rate (O(n^{-1/3})) assuming polynomial decay of covariances. All these results are established under fairly general conditions on the underlying distributions. Additionally, we perform Monte Carlo simulations to evaluate the finite sample performance of the suggested estimators, utilizing an associated and non-mixing model.

We consider the random motion of a particle that moves with constant velocity in (mathbb {R}^3). The particle can move along four different directions that are attained cyclically. It follows that the support of the stochastic process describing the particle’s position at a fixed time is a tetrahedron. We assume that the sequence of sojourn times along each direction follows a Geometric Counting Process (GCP). When the initial condition is fixed, we obtain the explicit form of the probability law of the process, for the particle’s position. We also investigate the limiting behavior of the related probability density when the intensities of the four GCPs tend to infinity. Furthermore, we show that the process does not admit a stationary density. Finally, we introduce the first-passage-time problem for the first component of the process through a constant positive boundary providing the bases for future developments.

This paper extends the stratified approximation method using lognormal and gamma distributions - first introduced to price Asian options - to derive a close formula for pricing and hedging of periodic-premium variable annuities. We used the moment matching method to fit the lognormal and gamma distributions to the conditional distribution of the integral of the underlying asset on a time interval, given the terminal value of the underlying asset. The highly oscillating double integrals for computing an expectation about the integral of the underlying assets are simplified down to a single integral, which greatly reduces the computation time for pricing periodic-premium variable annuities. This method allowed us to construct a different delta hedging strategy, other than the one used in the existing literature for embedded option of periodic-premium variable annuities. Compared with the existing research on pricing periodic-premium variable annuities, we obtained more accurate results using the stratified approximation method than the numerical method of partial differential equations, and found that the underpricing problem with periodic-premium variable annuities is even more severe than previously stated in existing literature. We further investigated the price gap between single-premium and periodic-premium variable annuities in a variety of settings, and examined the impact that the model and product parameters had on the price gap. The robustness and accuracy of the proposed method is tested by numerical examples.

We investigate the valuation problem of a mortgage contract with full prepayment and default risks using the reduced-form model with regime switching. The hazard rates of full prepayment and default are specified as linear functions of the risk-free interest rate and house prices, respectively, which are characterized by Ornstein-Uhlenbeck processes with regime switching. To derive the explicit valuation formula, we derive the distribution of the number of transitions for two-state Markov processes in finite time and the conditional joint probability density function of transition times using the uniformization technique. Finally, we analyze the effect of parameters on the valuation of the mortgage.

This paper focuses on an unreliable M/M/1 retrial queue with delayed repair, in which a novel breakdown mechanism is considered, i.e., a normal breakdown may deteriorate into a major breakdown. Arriving customers are not provided with the system’s information, but must decide whether or not to join it. First, the steady state of the system is analyzed. Then, based on the practical requirements of the cloud computing system, we construct an optimization model to minimize the response time of requesting information with proportions of detection, and repair of the normal and major breakdown as the decision variable. Furthermore, equilibrium joining strategies and the socially optimal pricing strategy are studied from the perspectives of the customer and the social planner, respectively. Finally, numerical examples are used to illustrate the impact of different parameters on strategies.

Abstract

In a context of illiquidity, the reservation price is a well-accepted alternative to the usual martingale approach which does not apply. However, this price is not available in closed form and requires numerical methods such as Monte Carlo or polynomial approximations to evaluate it. We show that these methods can be inaccurate and propose a deterministic decomposition of the reservation price using the Lambert function. This decomposition allows us to perform an improved Monte Carlo method, which we name Lambert Monte Carlo (LMC) and to give deterministic approximations of the reservation price and of the optimal strategies based on the Lambert function. We also give an answer to the problem of selecting a hedging asset that minimizes the reservation price and also the cash invested. Our theoretical results are illustrated by numerical simulations.

In the world of connected automated objects, increasingly rich and structured data are collected daily (positions, environmental variables, etc.). In this work, we are interested in the characterization of the variability of the trajectories of one of these objects (robot, drone, or delivery droid for example) along a particular path from irregularly sampled data in time and space. To do so, we model the position of the considered object by a random field indexed in time, whose distribution we try to estimate (for risk analysis for example). This distribution being by construction concentrated on an unknown curve, two phases are proposed for its reconstruction: a phase of identification of this curve, by clustering and polynomial smoothing techniques, then a phase of statistical inference of the random field orthogonal to this curve, by spectral methods and kernel reconstructions. The efficiency of the proposed approach, both in terms of computation time and reconstruction quality, is illustrated on several numerical applications.