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A continuous locally defined process is a pair $(X,T)$ specified on the stochastic interval $[0,T)$, with a lifetime stopping time $T$ approached by an announcing sequence for a stopping time $T_n$. Each stopped process is an ordinary continuous adapted process. This is the convention used for continuous local differential equations; no value at $T$ is required. Restricting a globally defined process gives examples, but finite lifetimes also allow approach to a domain boundary or explosion.

A locally defined stochastic process is a pair $(X,T)$, where $T$ is a lifetime stopping time and $X_t(\omega )$ is defined for $0\le t<T(\omega )$, that is, on a stochastic interval. For continuous local problems one uses stopping times $T_n\uparrow T$ with $T_n<T$ on $\{T>0\}$, so the process stopped at each $T_n$ has an ordinary continuous adapted process version. This is an announcing sequence for a stopping time; no value at $T$ or after $T$ is implicit in the pair.

A local solution of a stochastic differential equation on an open domain $U$ takes its values in $U$ before $T$ and, on every such stopped interval, satisfieswith the local integrability needed for the ordinary and Itô integrals. A maximal local solution of a stochastic differential equation has no extension to a strictly later lifetime that agrees with it before $T$. For locally Lipschitz continuous coefficients on $U$, take a nested compact exhaustion $K_n$ of $U$ and the successive exit times: their increasing limit is the maximal lifetime, and on $\{T<\infty \}$ the solution eventually leaves every compact subset of $U$. To obtain the announcing sequence with finite stops even when the path never exits a compact set, cap these exit times by $n$; the capped times still increase to the maximal lifetime. Thus a finite maximal lifetime means exit from the domain or explosion; it need not mean divergence to infinity. For $U=(0,\infty )$, reaching the boundary zero terminates the local solution even if an absorbing extension could be defined for a different domain.

Integrating the stopped stochastic differential equation for $Y$ and then taking a common full-probability event for a localizing sequence gives, simultaneously for every $t<T$,The integral is finite on each compact time interval before $T$: the continuous positive path has a positive minimum there. Its integrand is nonnegative, soThis is a pathwise comparison using the same Brownian motion, not a comparison of independent distributions. Continuity lets the identities established initially at rational stopped times hold on the entire stochastic interval.

For $0<\alpha <1$, use the power transformation $Y=X^{1-\alpha }$ on the positive stochastic interval. The Itô formula, with $d[X]=X^{2\alpha }dt$, givesThus $Y_t\le x_0^{1-\alpha }+(1-\alpha )B_t$ simultaneously for $t<T$. The right side reaches zero almost surely by recurrence of one-dimensional Brownian motion, and the same positive-path contradiction proves $T$ cannot exceed that Brownian first-passage time. This establishes the finite lifetime threshold for a power diffusion throughout $0<\alpha <1$.

At $\alpha =1$, the solution is the geometric Brownian motionIt is finite and positive at every finite time. On every compact time interval it has a positive minimum, so the hitting times of $1/n$ tend to infinity. Although the strong law for Brownian motion implies $X_t\to 0$ as $t\to \infty$, this is not a finite lifetime. ThereforeThe power transformation is a rescaled Lamperti transform; at $\alpha =1$ the corresponding transformation is the logarithm. No claim about pathwise uniqueness after adjoining the boundary zero is needed: the coefficients are locally Lipschitz continuous inside the positive domain, which is the domain of the given maximal local solution of a stochastic differential equation.