On the Behavior of Two Types of Expectations of a Random Process with Log-normal Distribution

Abstract

The paper considers some functions depending on the realizations of a random process with log-normal distribution and two types of expectations. The interpretations of these functions and expectations are given in terms of actuarial mathematics. The comparison of the behavior of these two types of expectations is given using the Black-Scholes formulas. Criteria for a random process to obey a stochastic equation of diffusion are elaborated. The obtained criteria are verified on a numerical example on the change in the price of oil, and can be used to predict financial crises.