ПРОИЗВОДНЫЕ В СРЕДНЕМ СЛУЧАЙНЫХ ПРОЦЕССОВ И ДИФФУЗИОННЫЕ МОДЕЛИ В ЭКОНОМИКЕ

Abstract

The article is devoted to diffusion models. The authors discuss the theoretical and methodological foundations of diffusion models in financial mathematics. Like the economic system, the modern world is developing rapidly. It seems impossible to predict what will happen tomorrow, how the emergence of new technologies will affect the market, and how changes in random factors will affect the product and the market as a whole. Diffusion models are one of the main methods for studying economic objects and processes. This is why it is so important to develop a diffusion model. The authors propose extending the applicability of the models by passing from Itô type stochastic equations to equations with so-called derivatives in the mean. For this, following E. Nelson, the authors introduce the concept of derivatives in the mean on the right and on the left. The equation with the derivative in the mean does not involve the Wiener process, therefore, it is not assumed in advance that the solution is diffusional. The article describes some well-known diffusion models, in which the transition from equations like an Itô type stochastic differential equation to equations satisfying a system of equations with derivatives in the mean leads to an expansion of the set of possible solutions. The authors also consider a generalization of geometric Brownian motion that satisfies a system of stochastic equations with derivatives in the mean and can cover a wider class of problems.