Volume 11, Issue 2

Following the pioneering papers of Fournié, Lasry, Lebouchoux, Lions and Touzi, an important work concerning the applications of the Malliavin calculus in numerical methods for mathematical finance has come after. One is concerned with two problems: computation of a large number of conditional expectations on one hand and computation of Greeks (sensitivities) on the other hand. A significant test of the power of this approach is given by its application to pricing and hedging American options. The paper gives a global and simplified presentation of this topic including the reduction of variance techniques based on localization and control variables. A special interest is given to practical implementation, number of numerical tests are presented and their performances are carefully discussed.

The paper is devoted to asymptotical analysis of Linear Congruential Generators (LCGs) with increasing moduli and multiplicators. The study is performed within the scope of a stochastic model of LCGs, the technique of weak convergence of distributions is used. It is proved, f.e., that the classical spectral test can be described in terms of a special probability metric inducing a topology of weak convergence. A number of theoretical examples of LCGs with good and bad asymptotical equidistribution properties is presented.

Three-dimensional low discrepancy sequence is constructed via dynamical system. Different from one-dimensional cases, the low discrepancy sequence cannot be constructed only by expanding transformation. The transformation used in this paper is not only expanding but also shuffling.

In this paper we introduce an efficient stochastic method to solve the time evolution of a bivariate population balance equation which has been developed for modelling nano-particle dynamics. We have adapted the existing stochastic models used in the study of coagulation dynamics to solve a variant of the sintering-coagulation equation proposed by Xiong & Pratsinis. Hitherto stochastic models based on Markov jump processes have not taken into account the surface area evolution. We produce numerical results efficiently with the direct simulation and mass flow algorithms and study the convergence behaviour as the number of stochastic particles increases. We find a marked preference for using the mass flow algorithm to determine the higher order volume and area moments of the particle size distribution function. The computational efficiency of these algorithms is remarkable when compared to the sectional method that has been used previously to study this equation.