Volume 16, Issue 5

We consider various stochastic models of stock prices, financial futures, and rates of exchange of currencies both in the form of stochastic differential equations and in recurrent form. Price models are used to construct the stock portfolio, organize futures and option trading. We study in detail the problem of estimating unknown coefficients in the models and give examples of statistical modelling.

For certain quasisingular classes of elliptic problems (QCEP) we have obtained condensed finite element grids to get an optimal approximate solution in the sense of the Kolmogorov N-width. But the computational cost of any method of solving the system of finite element equations is high. That is why in this paper we undertook to invent a new fast multigrid algorithm (FMA) in order to get the optimal approximate solution for QCEP. We show that the computational cost of the FMA is asymptotically proportional to N(lnN) 3 .

We describe a successive assimilation procedure in the complete nonlinear model of climatic temperature and salinity fields. It allows us to calculate the continuous evolution of hydrophysical Black Sea fields as new observational data becomes available. We use sources of the Newton type for data assimilation in heat and salt transport equations. We analyse maps of the intra-annual variability of reconstructed sea level, currents, vertical velocity, and temperature fields.

In this paper we study the stability of various classes of semi-Lagrangian schemes such as the scheme based on the ‘pseudoparticle’ method or the scheme with ordinary cubic interpolation. We prove the theorem about the sufficient stability conditions of conventional semi-Lagrangian schemes. It is shown that the pseudoparticle method in the form proposed in [9-11] is conditionally stable. We propose the simple algorithm modification ensuring the absolute scheme stability, which is confirmed by the numerical calculation of a transition modulus for this scheme in the multidimensional case and some analytical estimates in the long-wave approximation. We give the results of test numerical experiments.

Using the variational principle, we propose and numerically estimate the method of the numerical statistical modelling of the influence domain for pollutant concentration fields against the background of climatic implementations of wind velocity fields. It is shown that when numerically modelling a pollutant spreading process as the ensemble of the corresponding climatic implementations with their subsequent averaging the estimate of the influence domain differs essentially from the estimates which are derived by using average velocity fields or separate climatic scenarios.