Classes of Ordinary Differential Equations Obtained for the Probability Functions of Lévy Distribution

Levy distribution is one of few stable distributions. The absence of closed form of some of the probability functions of distribution has inspired researchers into finding alternate options such as approximations. In this paper, homogenous ordinary differential equations (ODES) of different orders were obtained for the probability density function, survival function, hazard function and reversed hazard function of Levy distribution. This is possible since the aforementioned probability functions are differentiable. However, approximation remains the only option for the quantile function and inverse survival function of the distribution. This is because those functions may not be reduced to an ODE as a result of the intractable nature of the cumulative distribution function which is used in obtaining them. . Differentiation and modified product rule were used to obtain the required ordinary differential equations, whose solutions are the respective probability functions. The different conditions necessary for the existence of the ODEs were obtained and it is in consistent with the support that defined the various probability functions considered. The parameters that defined each distribution greatly affect the nature of the ODEs obtained. This method provides new ways of classifying and approximating other probability distributions apart from one considered in this research. Algorithms for implementation can be helpful in improving the results. Index Terms— Differentiation, product