Simulating Stochastic Diffusion Processes and Processes with “Market” Time

The paper focuses on modelling, simulation techniques and numerical methods concerned stochastic processes in subject such as financial mathematics and financial engineering. The main result of this work is simulation of a stochastic process with new market active time using Monte Carlo techniques.The processes with market time is a new vision of how stock price behavior can be modeled so that the nature of the process is more real. The iterative scheme for computer modelling of this process was proposed.It includes the modeling of diffusion processes with a given marginal inverse gamma distribution. Graphs of simulation of the Ornstein-Uhlenbeck random walk for different parameters, a simulation of the diffusion process with a gamma-inverse distribution and simulation of the process with market active time are presented.To simulate stochastic processes, an iterative scheme was used: xk+1 = xk + a(xk, tk) ∆t + b(xk, tk) √ (∆t) εk,, where εk each time a new generation with a normal random number distribution.Next, the tools of programming languages for generating random numbers (evenly distributed, normally distributed) are investigated. Simulation (simulation) of stochastic diffusion processes is carried out; calculation errors and acceleration of convergence are calculated, Euler and Milstein schemes. At the next stage, diffusion processes with a given distribution function, namely with an inverse gamma distribution, were modelled. The final stage was the modelling of stock prices with a new "market" time, the growth of which is a diffusion process with inverse gamma distribution. In the proposed iterative scheme of stock prices, we use the modelling of market time gains as diffusion processes with a given marginal gamma-inverse distribution.The errors of calculations are evaluated using the Milstein scheme. The programmed model can be used to predict future values of time series and for option pricing.

Local and global analysis of macromolecular Atomic Displacement Parameters

This paper describes the global and local analyses of Atomic Displacement Parameters (ADP) of macromolecules solved and refined using X-ray crystallography method. It is shown that the distribution of ADPs follows the (mixture of) Shifted Inverse Gamma distribution(s). The parameters of the mixture of SIGDs are estimated using Expectation/Maximisation methods. In addition, a method for resolution and individual ADP dependent local analysis of neighbouring atoms has been designed. This method facilitates the detection of the mismodelled atoms and indicates potential identity of heavy metal atoms. It also helps in detecting of disordered and/or wrongly modelled ligands. Both global and local analyses can be used to detect errors in atomic structures thus helping in (re)building, refinement and validation of macromolecular structures. It can also serve as an additional validation tool during data deposition to the PDB.SynopsisMacromolecular atomic B value distributions have been modelled using a mixture of Shifted Inverse Gamma Distribution. Also, B value and resolution dependent local ADP differences have been applied for validation of heavy atoms and ligands.

Weighted Analogue of Inverse Gamma Distribution: Statistical Properties, Estimation and Simulation Study

In this article we propose a new weighted version of inverse Gamma distribution known as Weighted Inverse Gamma distribution (WIGD). We examine the Length biased and Area biased versions of Weighted Inverse Gamma distribution. Basic structural properties viz moments, mode, moment generating function (mgf), characteristic function (cf), hazard rate function and measures of uncertainty. The parameters of this model are estimated from both classical (namely, maximum likelihood estimator and method of moments, and compare them by using extensive numerical simulations) and Bayesian point of view. The Bayes estimates are estimated by using non-informative Jeffrey’s prior and informative Inverse Chi square prior under different types of loss function (symmetric and asymmetric loss functions). Finally, a simulation study has been conducted for comparing weighted inverse gamma distribution with other competing distributions.