Tamás Szántai

Improved Bounds and Simulation Procedures on the Value of the Multivariate Normal Probability Distribution Function

Improved bounds and simulation procedures on the value of the multivariate normal probability distribution function value are given in the paper. The authors variance reduction technique was based on the Bonferroni bounds involving the first two binomial moments only. The new variance reduction technique is adapted to the most refined new bounds developed in the last decade for the estimation the probability of union respectively intersection of events. Numerical test results prove the efficiency of the simulation procedures described in the paper.

Probability bounds given by hypercherry trees

In this article new lower and upper bounds are given for the probability of the union of events. For this purpose the new concept of hypercherry trees has been introduced. Earlier the concept of cherry tree and its application for bounding the probability of union of events was introduced by Bukszár and Prékopa. This, based on the cherry tree bound, is always an upper bound, and it can be regarded as a generalisation of the upper bound introduced by Hunter by means of maximum weight spanning trees. Later the Hunter bound was generalised by Tomescu. He used the concept of hypertrees in the framework of uniform hypergraphs and on the basis of these new hypergraph structures it became possible to define not only upper but also lower bounds on the probability of union of events. The new bounds of the paper are generalisations of Tomescus bounds in the same sense as the upper bound by Bukszár and Prékopa was a generalisation of the Hunter bound. The efficiency of the new bounds is illustrated on some test problems according to multivariate normal probability distribution function calculations.

On the Approximation of a Discrete Multivariate Probability Distribution Using the New Concept of t-Cherry Junction Tree

Most everyday reasoning and decision making is based on uncertain premises. The premises or attributes, which we must take into consideration, are random variables, so that we often have to deal with a high dimensional discrete multivariate random vector. We are going to construct an approximation of a high dimensional probability distribution that is based on the dependence structure between the random variables and on a special clustering of the graph describing this structure. Our method uses just one-, two- and three-dimensional marginal probability distributions. We give a formula that expresses how well the constructed approximation fits to the real probability distribution. We then prove that every time there exists a probability distribution constructed this way, that fits to reality at least as well as the approximation constructed from the Chow–Liu dependence tree. In the last part we give some examples that show how efficient is our approximation in application areas like pattern recognition and feature selection.

On numerical calculation of probabilities according to Dirichlet distribution

The main difficulty in numerical solution of probabilistic constrained stochastic programming problems is the calculation of the probability values according to the underlying multivariate probability distribution. In addition, when we are using a nonlinear programming algorithm for the solution of the problem, the calculation of the first and second order partial derivatives may also be necessary.In this paper we give solutions to the above problems in the case of Dirichlet distribution. For the calculation of the cumulative distribution function values, the Lauricella function series expansions will be applied up to 7 dimensions. For higher dimensions we propose the hypermultitree bound calculations and a variance reduction simulation procedure based on these bounds. There will be given formulae for the calculation of the first and second order partial derivatives, too. The common property of these formulae is that they involve only lower dimensional cumulative distribution function calculations. Numerical test results will also be presented.

Optimization techniques for planning highway pavement improvements

In this paper, two different decision models for the planning of highway pavement improvements are presented. In the first model, we want to get a prescribed improvement in the state of the highway network with minimal agency cost. In the second model, a given amount of money is distributed between the highway sections in different states in such a way that the achieved improvements should be the best in some sense. The first model helps the administration in the estimation of the necessary cost for the annual highway improvements. The second model in addition gives an objective tool to the administration for the distribution of the total amount of money between the different regions of the country. We present the construction of the models in detail. Both of them use Markov transition probabilities according to the states of the highway sections and produce a special structure, large-scale, linear programming problem. Some numerical results are presented on the data from Hungarian highways.

New Bounds and Approximations for the Probability Distribution of the Length of the Critical Path

A project is defined as the collection of activities (or events) {a, b,…} among which a precedence relation a ≺ b is defined. It is supposed to be transitive, i.e., if a ≺ b and b ≺c then a ≺c. Any project can be depicted as a directed network, where the directed arcs represent the activities. Without restricting generality, we may assume that there is exactly one node such that no arc leads into it and there is exactly one node such that no arc goes out of it. These two nodes will be called original and terminal nodes, respectively.

Computing bounds for the probability of the union of events by different methods

Let A1,…,An be arbitrary events. The underlying problem is to give lower and upper bounds on the probability P(A1∪⋯∪An) based on

On the analytical–numerical valuation of the Bermudan and American options

P(A_{i_{1}}\cap\cdots\cap A_{i_{k}})

On the connection between cherry-tree copulas and truncated R-vine copulas

, 1≤i1<⋯<ik≤n, where k=1,…,d, and d≤n (usually d≪n) is a certain integer, called the order of the problem or the bound. Most bounding techniques fall in one of the following two main categories: those that use (hyper)graph structures and the ones based on binomial moment problems. In this paper we compare bounds from the two categories with each other, in particular the bounds yielded by univariate and multivariate moment problems are compared with Bukszár’s hypermultitree bounds. In the comparison we considered several numerical examples, most of which have important practical applications, e.g., the approximation of the values of multivariate cumulative distribution functions or the calculation of network reliability. We compare the bounds based on how close they are to the real value and the time required to compute them, however, the problems arising in the implementations of the methods as well as the limitations of the usability of the bounds are also illustrated.

On the binomial tree method and other issues in connection with pricing Bermudan and American options

The paper further develops, both from the theoretical and numerical points of view the analytical valuation of the American options, initiated by Geske and Johnson (1984) for the American put with no dividend. We present and prove closed form formulas for the value of the Bermudan put and call, with dividend, paid continuously at a constant rate, where a general number and not necessarily equal length intervals subdivide the time. Based on the obtained formulas and recent, efficient numerical integration techniques, to obtain values of the multivariate normal c.d.f., the Bermudan put and call option values are calculated for up to twenty subdividing intervals. The sequences of option values are smoothed by sums of exponential functions and the latters are used to predict the values of the American options. Numerical results are presented and compared with those, published in the literature. It is shown that the binomial method systematically overestimates the option price, and, according to our numerical results, so do many other methods. Some properties of Richardson extrapolation are explored.