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.

中文翻译：

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对数正态分布的随机过程的两种期望行为

本文根据对数正态分布和两种期望的随机过程的实现来考虑某些功能。这些功能和期望的解释是通过精算数学给出的。使用Black-Scholes公式对这两种期望类型的行为进行了比较。阐述了遵循随机扩散方程的随机过程的标准。所获得的标准在有关石油价格变化的数值示例上得到了验证，并可用于预测金融危机。