I have this problem at the moment and I'm not sure what to do at all:
Let $d>0,$ N be a $\lambda-$Poisson process and ${\{z^i\}}_{i=1}^{\infty}$ be a sequence of aging particles such that:
i)Their initial ages ${\{z_0^i\}}_{i=1}^{\infty}$ are the success of $N_{[0,d)}$ or $\infty$ once the successes run out.
ii)Each particle ages as $$z_t^i=\frac{z_0^i e^{-bt}}{1-\beta z_0^i \int_0^t e^{-bs}ds}$$ until its death at age $d.$
iii)Each living particle at age $z<d$ gives birth to a new particle at rate $2a(d-z)$ with the age of the new particle being the [a,d]-uniform.
Let $$D_0={\{g\in C(\mathbb{R}):0\le g\le1,g(z)=1 \text{ when } z\ge d\}}$$ So that every g is eventually 1. Let D be the infinite product functions $$f(z)=\prod_{i=1}^{\infty}g(z^i)$$ Find the martingale for the infinite vector of ages $z^i.$ That is to find an operator $L$ such that $$f(Z_t)-f(Z_0)-\int_0^t Lf(Z_s)ds$$ is a martingale where $$Z_s=\begin{bmatrix} z_s^1\\ z_s^2\\ \vdots \end{bmatrix}$$
I understand each individual part of the question but have no clue where to even start putting them together within the martingale. Would appreciate some help on this, thanks.