Power diffusion ($d{X}_t=X_t^{\alpha }d{B}_t$)

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.

For the positive-domain power diffusion with $0<\alpha \le 1$, assuming the lifetime is the limit of the hitting times of $1/n$, it is finite almost surely exactly when $\alpha <1$. For $\alpha <1$, the Itô formula givesThe negative drift bounds this positive process above by $X_0^{1-\alpha }+(1-\alpha )B$, whose first hit of zero is finite by recurrence of one-dimensional Brownian motion. The lifetime must precede that hit. At $\alpha =1$, $X_t=X_0{e}^{B_t-t/2}$ is positive and finite on every compact time interval, so the lifetime is infinite. Approaching zero only as $t\to \infty$ is not a finite boundary lifetime.