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$S_{n}$ 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: $$ P(S_{n} \geq 0 , \forall n=1,2,\dots, N) = (1-\epsilon)^N$$

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  • $\begingroup$ In your definition of $\epsilon$ you have $S_n=s$. What is $s$? $\endgroup$ Sep 5, 2014 at 6:35
  • $\begingroup$ $s$ is a positive number. Basically, $\epsilon$ is the probability that $S_{n+1}$ is negative given that $S_{n}$ is some positive number. $\endgroup$
    – kagami
    Sep 5, 2014 at 6:38
  • $\begingroup$ And that probability is the same for all $s$? Note that $P(S_1>0)=1-P(S_1\leq 0)$ and not $1-P(S_1<0)$ which it seems that you are using. $\endgroup$ Sep 5, 2014 at 6:42
  • $\begingroup$ Thanks for your remark. Yes, you can say that the probability is the same for all s. $\endgroup$
    – kagami
    Sep 5, 2014 at 6:58
  • $\begingroup$ Related (and probably making the present question obsolete): math.stackexchange.com/q/922285 $\endgroup$
    – Did
    Sep 9, 2014 at 8:46

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