Bayesian stopping rules for multistart global optimization methods

By far the most efficient methods for global optimization are based on starting a local optimization routine from an appropriate subset of uniformly distributed starting points. As the number of local optima is frequently unknown in advance, it is a crucial problem when to stop the sequence of sampling and searching. By viewing a set of observed minima as a sample from a generalized multinomial distribution whose cells correspond to the local optima of the objective function, we obtain the posterior distribution of the number of local optima and of the relative size of their regions of attraction. This information is used to construct sequential Bayesian stopping rules which find the optimal trade off between reliability and computational effort.     

A stopping rule for the Robbins-Monro method

A stopping rule for the multidimensional Robbins-Monro stochastic approximation method is developed in this paper. Both moving average and stationary -mixing type of correlated noise processes are treated. Sequentially determined confidence ellipsoids are constructed to fulfill the goal for the determination of the stopping rule. The limit behavior of the algorithm is investigated. It is shown that the stopped Robbins-Monro process is asymptotically normal. Such asymptotic normality is established by means of weak convergence methods.Communicated by Y. C. Ho     

Markov stopping games with random priority

In the paper a construction of Nash equilibria for a random priority finite horizon two-person non-zero sum game with stopping of Markov process is given. The method is used to solve the two-person non-zero-sum game version of the secretary problem. Each player can choose only one applicant. If both players would like to select the same one, then the lottery chooses the player. The aim of the players is to choose the best candidate. An analysis of the solutions for different lotteries is given. Some lotteries admit equilibria with equal Nash values for the players.The research was supported in part by Committee of Scientific Research under Grant KBN 211639101.     

A note on optimal stopping of diffusions with a two-sided optimal rule

We consider a class of optimal stopping problems of diffusions with a two-sided optimal rule. We propose an approach for finding and characterizing the solution. We establish that the optimal stopping rule can be associated with the unique fixed point of an auxiliary function. The results are illustrated with an explicit example.     

The monotone case approach for the solution of certain multidimensional optimal stopping problems

This paper studies explicitly solvable multidimensional optimal stopping problems of sum- and product-type in discrete and continuous time using the monotone case approach. It gives a review on monotone case stopping using the Doob decomposition, resp. Doob–Meyer decomposition in continuous time, also in its multiplicative versions. The approach via these decompositions leads to explicit solutions for a variety of examples, including multidimensional versions of the house-selling and burglar’s problem, the Poisson disorder problem, and an optimal investment problem.     

On when to stop sampling for the maximum

Suppose a sequential sample is taken from an unknown discrete probability distribution on an unknown range of integers, in an effort to sample its maximum. A crucial issue is an appropriate stopping rude determining when to terminate the sampling process. We approach this problem from a Bayesian perspective, and derive stopping rules that minimize loss functions which assign a loss to the sample size and to the deviation between the maximum in the sample and the true (unknown) maximum. We will show that our rules offer an extremely simple approximate solution to the well-known problem to terminate the Multistart method for continuous global optimization.     

A stopping rule for choosing the best of three coins

A recently proposed stopping rule for choosing the best of three coins is presented as the first exit time from a certain polygonal region by a two-dimensional random walk. A diffusion approximation to the expected value of this exit time is developed, using techniques of conformal mapping. An algebraic method for calculating the exact value of this expectation is also described.     

Error bounds for the compound poisson approximation

Explicit error bounds in terms of probabilities and stop-loss premiums are given for two kinds of compound Poisson approximations: the first concerns the difference between the individual and the collective model; the second is about the difference of the compound negative binomial and the compound Poisson distribution.     

Sequential multi-hypothesis testing for compound Poisson processes

Suppose that there are finitely many simple hypotheses about the unknown arrival rate and mark distribution of a compound Poisson process, and that exactly one of them is correct. The objective is to determine the correct hypothesis with minimal error probability and as soon as possible after the observation of the process starts. This problem is formulated in a Bayesian framework, and its solution is presented. Provably convergent numerical methods and practical near-optimal strategies are described and illustrated on various examples.     

A random review replacement model for a system subject to compound poisson shocks

We present a transform–free analysis of the following model. The state of the system is initially 0 and thereafter increases jumpwise due to compound Poisson shocks. Each shock increases the state by a random amount. The system is inspected at random points in time. If the state is above a threshold at an inspection, the system is replaced, otherwise no action is taken. Each replacement instantaneously brings the state back to 0. (Existing models assume either exponential interinspection times or discrete shock magnitudes.) This model can be applied to reliability, inventory, and queueing problems.Interpretations are given throughout to make the results easier to understand and to apply     

Approximation of optimal stopping problems

We consider optimal stopping of independent sequences. Assuming that the corresponding imbedded planar point processes converge to a Poisson process we introduce some additional conditions which allow to approximate the optimal stopping problem of the discrete time sequence by the optimal stopping of the limiting Poisson process. The optimal stopping of the involved Poisson processes is reduced to a differential equation for the critical curve which can be solved in several examples. We apply this method to obtain approximations for the stopping of iid sequences in the domain of max-stable laws with observation costs and with discount factors.     

Optimal stopping in a search for a vertex with full degree in a random graph

We consider the following on-line decision problem. The vertices of a realization of the random graph G(n,p) are being observed one by one by a selector. At time m, the selector examines the mth vertex and knows the graph induced by the m vertices that have already been examined. The selector’s aim is to choose the currently examined vertex maximizing the probability that this vertex has full degree, i.e. it is connected to all other vertices in the graph. An optimal algorithm for such a choice (in other words, optimal stopping time) is given. We show that it is of a threshold type and we find the threshold and its asymptotic estimation.     

双复合Poisson风险模型

研究了保费收取过程是复合Po isson过程,索赔总额是复合Po isson过程的风险模型,给出了不破产概率的积分表示,以及在特殊情况下不破产概率的具体表达式,并用鞅方法得出了破产概率满足的Lundberg不等式和一般公式.     

A random walk phenomenon under an interesting stopping rule

In the simple one-dimensional random walk setup, a path is described as follows. Toss a coin. If the result is head, score +1 and move one step forward; otherwise score ?1 and move one step backward. One is interested to know the position after a given number of steps. In this paper, once again a coin-tossing experiment is carried out. But this time, going by the philosophy of Bhanu et al. if score ?2, the result is head, otherwise score +1. The situation studied is the case that one decides to stop at the score of a given number n. Unlike their earlier case, the paper considers a general coin with probability of head equal to θ for 0?θ?et al. when θ?=?1/2.     

Constrained maximum variance stopping for a finite horizon increasing random walk

We derive the optimal solution for the problem of choosing a non-anticipative decision rule to maximize the stopping variance of a finite horizon, increasing random walk subject to a distributional constraint, as well as an explicit upper limit on the variance of the walk’s stopping state. Problems of this caliber arise as subproblems for risk-constrained versions of standard stopping problems in areas including, for instance, market entry decision-making. A numerical example verifies the main result.     

Exchangeable claim sizes in a compound Poisson‐type process

When dealing with risk models the typical assumption of independence among claim size distributions is not always satisfied. Here we consider the case when the claim sizes are exchangeable and study the implications when constructing aggregated claims through compound Poisson‐type processes. In particular, exchangeability is achieved through conditional independence, using parametric and nonparametric measures for the conditioning distribution. Bayes' theorem is employed to ensure an arbitrary but fixed marginal distribution for the claim sizes. A full Bayesian analysis of the proposed model is illustrated with a panel‐type data set coming from a Medical Expenditure Panel Survey (MEPS). Copyright © 2009 John Wiley & Sons, Ltd.     

Approximation of aggregate claims distributions by compound poisson distributions

New error bounds are derived for the approximation of aggregate claims distributions by compound Poisson distributions. These approximations can be recommended in most cases in which the normal approximation fails.     

Optimal stopping for extremal processes

For an extremal process (Z_t)_t the optimal stopping problem for X_t = f(Z_t)?g(t) gives the continuous time analogue of the optimal stopping problem for max{Y₁,…,Y_k}?c_k where Y₁, Y₂,… are i.i.d. For the continuous time problem we derive optimal stopping times in explicit form and also show that the optimal stopping boundary is the limit of the optimal stopping boundaries for suitably standardized discrete problems.     

Functional approach to the random mean of a compound Cox process

The parametric process and counting statistics of a marked point process whose marks belong to a given subset of the mark space of a compound Cox process are derived in this paper by means of functional data analysis. They are illustrated by means of an example and simulation study with different intensity processes for the CCP. This work was partially supported by projects MTM2004-05992 of Dirección General de Investigación, and MTM2004-04230 of Plan Nacional de I+D+I, Ministerio de Ciencia y Tecnología jointly by the FEDER.     

On infinite horizon optimal stopping of general random walk

The objective of this study is to provide an alternative characterization of the optimal value function of a certain Black–Scholes-type optimal stopping problem where the underlying stochastic process is a general random walk, i.e. the process constituted by partial sums of an IID sequence of random variables. Furthermore, the pasting principle of this optimal stopping problem is studied.