Derivation of the exponential family by stochastic differential equations

The probability distributions included in the exponential family have useful algebraic properties, and so have been used in probability theory and statistics. However, with the exception of the normal distribution, it has been thought that they cannot be derived from stochastic differential equations. For example, the Poisson distribution, which is included in the exponential family, plays an important role in Lévy processes, but the derivation of Poisson processes is based on the assumption that the probability process at each time follows a Poisson distribution, which is not logical. Therefore, we derived a stochastic differential equation that can be used to derive an exponential family. First, we derived a relationship between the sum of the time when an event occurred and the time when it did not occur, and the number of times an event occurred. Based on this relationship, we gave a stochastic differential equation that can be used to derive an exponential family. Finally, we derived the exponential distribution, Poisson distribution, and binomial distribution from the stochastic differential equation.