Simulating Stochastic Diffusion Processes and Processes with “Market” Time

Abstract

In financial mathematics and engineering, randomness is the dominant criterion that determines the inner character of markets. In this case, stochasticity, like Brownian motion, is not just a negligible correction, but a major approximation to the real process. That is, we can say that our world is not deterministic, its real nature is stochastic. The usual differential equation is only the first approximation to the description of real processes. The next step is stochastic equations and computer modeling of the stochastic processes [1]. Computer modeling of the behavior of complex stochastic systems and processes is a must-have tool for any financial analyst, and sometimes the only way to explore these systems. The model of continuous stochastic processes is used quite effectively when calculating financial formulas. For all its elegance, a model of continuous stochastic processes is rather limited model which only tries to describe a real process. In fact, the market has a disruptive dynamic, as there are periods of time when it is closed. The assumption of trade continuity over very short periods of time is also artificial. Modeling and simulation techniques of the stochastic diffusion processes have been a matter of active research in recent decades. Some of them can be found in papers of Kozachenko U.V., G. Deodatis G. and other. In context of our research we use ideas from [1; 2]. But in most publications dealing with simulation of stochastic processes the problem of computer modeling for processes with given marginal probability density and with new market time didn’t studied. The aim of the work was to construct the iterative scheme for modeling market time as a diffusion processes with a given marginal inverse gamma dis-