Some Properties of Density Functions on Maxima of Solutions to One-Dimensional Stochastic Differential Equations

This article proves some properties of the probability density function concerning maxima of a solution to one-dimensional stochastic differential equations. We first obtain lower and upper bounds on the density function of the discrete time maximum of the solution. We then prove that the density function of the discrete time maximum converges to that of the continuous time maximum of the solution. Finally, we prove the positivity of the density function of the continuous time maximum and a relationship between the density functions of the continuous time maximum and the solution itself.