 Power diffusion

A power diffusion has a power of its positive state as noise coefficient. Here the normalized zero-drift family is considered on $U=(0,\infty )$ with $X_0>0$. Its coefficients are locally Lipschitz continuous on that domain, so a maximal local solution of a stochastic differential equation is pathwise unique before its lifetime. The Lamperti transform reduces its noise coefficient to a constant; the case $\alpha =1$ is geometric Brownian motion. This normalized family is part of the constant-elasticity-of-variance models, but is not the whole model with arbitrary drift and scale.