Volume 23, Issue 3

In this work, a new numerical method to calculate the characteristics of the solution to stochastic differential equations is presented. This method is based on the Fokker–Planck equation for the transition probability function and the representation of the transition probability function by means of eigenfunctions of the Fokker–Planck operator. The results of the numerical experiments are presented.

In this paper we extend the results in [5] in two directions: First , we show that by bias correcting the estimated mean reversion parameter we can also have better finite sample properties of the testing procedure using a t-statistic in the near unit root situation when the mean reversion parameter is approaching its lower bound versus using the Jackknife estimator of Phillips and Yu [8]. Second , we show that although Tang and Chen [10] demonstrate that the variance of the maximum likelihood estimator of the long term mean parameter is of an order equal to the reciprocal of the sample size (the same order as that of the bias and variance of the mean reversion parameter estimator and so it does not converge very fast to its true value), the t-statistic related to that parameter does not exhibit large empirical size distortions and so does not need to be bias corrected in practice.

An algorithm is developed for generating samples of vector-valued Gaussian processes and fields. The algorithm is based on Karhunen–Loève (KL) representations of vector-valued random functions Z⁢(x){Z(x)} with finite variances and their construction involves two steps. First, truncation levels {mi}{\{m_{i}\}} are selected for the KL representations of the components {Zi⁢(x)}{\{Z_{i}(x)\}} of Z⁢(x){Z(x)} such that they meet imposed accuracies. Second, the truncation levels {mi}{\{m_{i}\}} are accepted or increased if the accuracies of resulting cross correlation functions of Z⁢(x){Z(x)} satisfy or violate preset constraints. Theoretical arguments are used to prove the validity of the proposed KL-based models of Z⁢(x){Z(x)}. The models are applied to develop an efficient Monte Carlo algorithm for generating samples of vector-valued Gaussian functions. Numerical examples illustrate the implementation of the proposed Monte Carlo algorithm and demonstrate its performance.

We suggest in this paper a Random Walk on Spheres (RWS) method for solving transient drift-diffusion-reaction problems which is an extension of our algorithm we developed recently [26] for solving steady-state drift-diffusion problems. Both two-dimensional and three-dimensional problems are solved. Survival probability, first passage time and the exit position for a sphere (disc) of the drift-diffusion-reaction process are explicitly derived from a generalized spherical integral relation we prove both for two-dimensional and three-dimensional problems. The distribution of the exit position on the sphere has the form of the von Mises–Fisher distribution which can be simulated efficiently. Rigorous expressions are derived in the case of constant velocity drift, but the algorithm is then extended to solve drift-diffusion-reaction problems with arbitrary varying drift velocity vector. The method can efficiently be applied to calculate the fluxes of the solution to any part of the boundary. This can be done by applying a reciprocity theorem which we prove here for the drift-diffusion-reaction problems with general boundary conditions. Applications of this approach to methods of cathodoluminescence (CL) and electron beam induced current (EBIC) imaging of defects and dislocations in semiconductors are presented.