For the purpose of economical considerations, I consider a discrete period of $n \in {1, ..., 40}$, during which a negative event (e.g. fire) can occur - it follows Bernoulli distribution $P$ (yes-no distribution) with a success probability of $\alpha$. I want to study the value of a certain good (a house in this example) $W_n$ throughout the periods $n$ (the underlying values $V_1$, ..., $V_n$ which do not account for the negative events are known; $W_1$, ... $W_n$ are the prices with penalities due to fires) - the problem is: each time event occurs the value decreases by the parameter $p \in (0,1)$ and then $p$ linearly reverts by $m$ periods to 1 and still affects such periods during which it reverts. E.g. fire happening to the house in $n = 1$ would set $W_1$ to $(1-p)V_1$ and consequently, assuming $m = 3$ and no further fires happening, $W_2 = (1- \frac{2}{3}p) V_2$, ..., $W_4$ = $V_4$, ..., $W_{40}$ = $V_{40}$. Two other examples:
- No fire for $n = 1$ would translate to $W_1$ = $V_1$.
- Assuming $m = 3$. Two consequtive sucessful Bernoulli trials for $n = 1,2$ would result in $W_1 = (1-p) V1$ and $W_2 = (1-p)*(1- \frac{2}{3}p)V_2$.
Question: I see this as a recurrence problem depending on $m$. However, is the close form formula for the expected value of $W_n$ possible to obtain?