First Passage and First Exit Times for diffusion processes related to a general growth curve

Abstract

Recently a general growth curve including the well known growth equations, such as Malthus, logistic, Bertallanfy, Gompertz, has been studied. We now propose two stochastic formulations of this growth equation. They are obtained starting from a suitable parametrization of the deterministic model, by adding an additive and multiplicative noise respectively. For these processes we focus attention on the First Passage Time from a barrier and on the First Exit Time from a region delimited by two barriers. We consider thresholds, generally time dependent, for which there exist closed-forms of the probability densities of the first passage time and of the first exit time.

Figures (7)

• Figure 1: Sample paths of $X_L(t)$ and $X_G(t)$ for $n=1$, $k=20$, $\gamma=0.5$, $x_0=1$, $t_0=0$ and $\sigma=0.02$.
• Figure 2: For $n=1, \gamma=0.5, k=20, x_0=1, t_0=0$, the FPT density of $X_L(t)$ for different values of the proportion $\nu$ (on the left) and for different values of $\sigma$ (on the right).
• Figure 3: For $n=1, \gamma=0.5, k=20, x_0=1, t_0=0, \sigma=0.02$, the FET pdf of $X_L(t)$ for different values of $\nu_1$ (on the left) and for different values of $\nu_2$ (on the right).
• Figure 4: For $n=1, \gamma=0.5, k=20, x_0=1, t_0=0$, the FET pdf of $X_L(t)$ for different values of $\sigma$.
• Figure 5: For $n=1, \gamma=0.5, k=20, x_0=1, t_0=0$, the FPT density of $X_G(t)$ for different values of the proportion $\nu$ (on the left) and for different values of $\sigma$ (on the right).

Theorems & Definitions (3)

• Remark 2.1
• Remark 3.1
• Remark 3.2