The logistic S.D.E.

We consider the logistic S.D.E which is obtained by addition of a diffusion coefficient of the type $\beta \sqrt{x}$ to the usual and deterministic Verhust-Volterra differential equation. We show that this S.D.E is the limit of a sequence of birth and death Markov chains. This permits to interpret the solution $V_t$ as the size at time $t$ of a self-controlled tumor which is submitted to a radiotherapy treatment. We mainly focus on the family of stopping times $T_\varepsilon$, where $T_\varepsilon$ is the first hitting of level $\varepsilon>0$ by $(V_t)$. We calculate their Laplace transforms and also the first moment of $T_\varepsilon$. Finally we determine the asymptotic behavior of $T_\varepsilon$, as $\varepsilon\rightarrow 0$.