Suppose that, for all $n$, $B_{j,n}$ for $j=1,...,n$ is a triangular array of Bernoulli variates. Define
$$ p_{j,n} = \mathbb{P}\left[B_{j,n}=1\mid B_{j-1,n}\right] $$
and assume that
$$ p_{j,n} = p_{j-1,n}+\alpha\,(\psi-p_{j-1,n})+\nu\,B_{j-1,n}=p_{j-1,n}\,(1-\alpha)+\alpha\,\psi_n+\nu\,B_{j-1,n} $$
with $p_{0,n}=\psi>0$, $\alpha>0$ and $\nu>0$. In essence, the Bernoulli process is self-exciting in the sense that a $B_{j-1,n}=1$ increases the probability of having $B_{j,n}=1$.
My purpose is to compute $\mathbb{E}\left[B_{j,n}\right]$ recursively. I can find a recursive formula for $p_{j,n}$
whence, by induction,
Nevertheless I cannot find a similar formula for $\mathbb{E}\left[B_{j,n}\right]$, any idea?
By law of total expectation, you readily have a recursive formula for the expectation:
$$ \mathbb E[B_{j, n}] = \mathbb E[\mathbb E[B_{j, n} \mid B_{j-1, n}]] = \phi + \frac{\nu}{1 - \alpha} \sum^{j - 1}_{l = 0} \mathbb E[B_{l, n}](1-\alpha)^{j-l} $$