Parameter estimation for diffusion processes: unknown facts in the well-known theory

The probabilistic distribution of local time of a homogeneous transient dif- fusion process is found. To this end, a second order differential equation corre- sponding to the process generator is considered, and properties of its monotone solutions as functions of a parameter are established with the help of analytic tools. At the same time a probabilistic representation of monotone solutions are used. Combining the techniques of differential equations theory and stochastic processes theory allowed to identify the parameter of exponential distribution of the local time