$S_{n}$ model the price of a financial asset.
The recurrence relation is given by: $$ S_{n+1} = (1 + r\Delta t_{n} + \Delta W_{n})S_{n}, n = 0, \dots, N $$
where $\Delta W$ has a normal distribution $\cal{N} (\mathrm{0},\Delta t)$
Let $\epsilon = P(S_{n+1}<0 | S_{n} = s)$ where $S_{n}$ are random variables.
I want to compute $P (\min_{1 \leq n \leq N}{S_{n}} \geq 0)$.
I proceed as follow:
$P (\min_{1 \leq n \leq N}{S_{n}} \geq 0) = P(S_{n} \geq 0 , \forall n=1,2,\dots, N) $.
Assuming that $S_{0} \geq 0$, we can say that $P(S_{1} \geq 0) = 1- \epsilon$.
For N=2, $$P(S_{2} \geq 0, S_{1} \geq 0) = P(S_{2} \geq 0|S_{1} \geq 0) P(S_{1} \geq 0) = (1-\epsilon)^2$$
I was thinking if I could generalize this result so that, we have: $$ P(S_{n} \geq 0 , \forall n=1,2,\dots, N) = (1-\epsilon)^N$$