Randomly Stopped Nonlinear Fractional Birth Processes

We present and analyse the nonlinear classical pure birth process $\mathpzc{N} (t)$, $t>0$, and the fractional pure birth process $\mathpzc{N}^\nu (t)$, $t>0$, subordinated to various random times. We derive the state probability distribution $\hat{p}_k (t)$, $k \geq 1$ and, in some cases, we also present the corresponding governing differential equation. Various types of compositions of the fractional pure birth process $\mathpzc{N}^\nu(t)$ have been examined in the second part of the paper. In particular, the processes $\mathpzc{N}^\nu(T_t)$, $\mathpzc{N}^\nu(\mathpzc{S}^\alpha(t))$, $\mathpzc{N}^\nu(T_{2\nu}(t))$, have been analysed, where $T_{2\nu}(t)$, $t>0$, is a process related to fractional diffusion equations. As a byproduct of our analysis, some formulae relating Mittag--Leffler functions are obtained.