Geodesic
The logistic S.D.E.
Teoriâ slučajnyh processov, Tome 20 (2015) no. 1, pp. 28-62

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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 $\varepsilon\rightarrow 0$.
Keywords: Logistic equation, tumor, radiotherapy treatment, Laplace transforms, birth and death process, first hitting time.
Mots-clés : diffusion processes 
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     title = {The logistic {S.D.E.}},
     journal = {Teori\^a slu\v{c}ajnyh processov},
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     number = {1},
     year = {2015},
     language = {en},
     url = {http://geodesic.mathdoc.fr/item/THSP_2015_20_1_a2/}
}
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Jean-Sébastien Giet; Pierre Vallois; Sophie Wantz-Mézières. The logistic S.D.E.. Teoriâ slučajnyh processov, Tome 20 (2015) no. 1, pp. 28-62. http://geodesic.mathdoc.fr/item/THSP_2015_20_1_a2/