Band 28, Heft 2

The paper introduces a novel high order discretization scheme for expectation of jump-diffusion processes by using a Malliavin calculus approach and an operator splitting method. The test function of the target expectation is assumed to be only Lipschitz continuous in order to apply the method to financial problems. Then Kusuoka’s estimate is employed to justify the proposed discretization scheme. The algorithm with a numerical example is shown for implementation.

The Black–Karasinski model is a one-factor non-affine interest rate model as it describes interest rate movements driven by a single source of randomness and the drift function is a nonlinear function of the interest rate. The drift parameters represent the level and the speed of mean reversion of the interest rate. It belongs to the class of no-arbitrage models. The paper introduces some new approximate minimum contrast estimators of the mean reversion speed parameter in the model based on discretely sampled data which are efficient and studies their asymptotic distributional properties with precise rates of convergence.

Randomized scalable vector algorithms for calculation of matrix iterations and solving extremely large linear algebraic equations are developed. Among applications presented in this paper are randomized iterative methods for large linear systems of algebraic equations governed by M-matrices. The crucial idea of the randomized method is that the iterations are performed by sampling random columns only, thus avoiding not only matrix-matrix but also matrix-vector multiplications. The suggested vector randomized methods are highly efficient for solving linear equations of high dimension, the computational cost depends only linearly on the dimension.

The paper is devoted to one possible way of the model construction for the stationary Gaussian process with given accuracy and reliability in functional space C ⁢ ( [ 0 , T ] ) {C([0,T])} .

This paper considers integral approximation via importance sampling where the importance sampler is chosen from a family of skew-Student distributions. This is an alternative class of distributions than is typically considered in importance sampling applications. We describe variate generation and propose adaptive methods for fitting a member of the skew-Student family to a particular integral. We also demonstrate the utility of the approach in several examples.

In this paper, we investigate the estimation of the unknown parameters of a competing risk model based on a Weibull distributed decreasing failure rate and an exponentially distributed constant failure rate, under right censored data. The Bayes estimators and the corresponding risks are derived using various loss functions. Since the posterior analysis involves analytically intractable integrals, we propose a Monte Carlo method to compute these estimators. Given initial values of the model parameters, the maximum likelihood estimators are computed using the expectation-maximization algorithm. Finally, we use Pitman’s closeness criterion and integrated mean-square error to compare the performance of the Bayesian and the maximum likelihood estimators.

The simulated annealing (SA) algorithm is a popular intelligent optimization algorithm which has been successfully applied in many fields. In this paper, we propose a software component under the Windows environment called goRDS which implements a refined descriptive sampling (RDS) number generator of high quality in the MATLAB programming language. The aim of this generator is to sample random inputs through the RDS method to be used in the Simple SA algorithm with swap operator. In this way, the new probabilistic meta-heuristic algorithm called RDS-SA algorithm will enhance the simple SA algorithm with swap operator, the SA algorithm and possibly its variants with solutions of better quality and precision. Towards this goal, the goRDS generator was highly tested by adequate statistical tests and compared statistically to the random number generator (RNG) of MATLAB, and it was proved that goRDS has passed all tests better. Simulation experiments were carried out on the benchmark traveling salesman problem (TSP) and the results show that the solutions obtained with the RDS-SA algorithm are of better quality and precision than those of the simple SA algorithm with swap operator, since the software component goRDS represents the probability behavior of the SA input random variables better than the usual RNG.

In this paper, we study the Carathéodory approximate solution for a class of stochastic differential equations involving the local time at point zero. Based on the Carathéodory approximation procedure, we prove that stochastic differential equations involving the local time at point zero have a unique solution, and we show that the Carathéodory approximate solution converges to the solution of stochastic differential equations involving the local time at point zero with one-sided Lipschitz drift coefficient.