The logistic S.D.E.

Jean-Sébastien Giet (1) , Pierre Vallois (2) , Sophie Wantz-Mézières (2)

1 IECL - Institut Élie Cartan de Lorraine
2 BIGS - Biology, genetics and statistics

We consider the logistic S.D.E which is obtained by addition of a diffusion coefficient of the type β√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 Vt 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 Tε, where Tε is the first hitting of level ε > 0 by (Vt). We calculate their Laplace transforms and also the first moment of Tε. Finally we determine the asymptotic behavior of Tε, as ε → 0.

Mots clés: Laplace transforms; birth and death process; diffusion processes; Logistic equation; tumor; first hitting time; radiotherapy treatment

Mots clés: diffusion; équation logistique; transformée de Laplace; processus de naissance et mort

Domaines: Probabilités [math.PR]

Jean-Sébastien Giet, Pierre Vallois, Sophie Wantz-Mézières. The logistic S.D.E.. Theory of Stochastic Processes, 2015, 20(36) (1), pp.28 à 62. ⟨hal-01096323⟩