Prob (X < A) within time frame?

Time:

T: timeframe (e.g. 1 year)

T/n = t: time increment (e.g. if n=12, then 1yr becomes one month)

Starting Value: A(0)

- Value as of now (t(0))

Intermediate Value: A(t(n))

- Value at any point between start and end

Ending Value: A(T)

- Value as of ending point (t(T))

Threshold Value: X

- Known, A(0) < X

Random Variable: R

- 0 < R < 1, behaving according to some known distribution

Variable’s Relationship:

A(n+1) = A(n)*R(n)

Question: How can I figure P(X < A) during timeframe ‘T’?

If A never exceeds X, then the probability is zero, and,

If A exceeds X halfway through the timeframe (T/2), then the probability is 50% (I think that is approximately correct)

I don’t want to generate discrete points and figure an integral for the space created where ‘A’ exists over ‘X’, nor do I want to use a monte-carlo method to run several trials and figure an average from the result, since these would be too computationally expensive (as they have to be repeated many times). If the solution is accurate to within a reasonable tolerance (e.g. 5% error) and is computationally inexpensive that would be ideal.

Please let me know if I should clarify something. Any help appreciated!

William

Re: Prob (X < A) within time frame?

My thoughts, could be wrong!:

1) have you considered that you may not need very many monte carlo simulations to get a figure that is accurate to within +/- 5%?

**or**

2) take logs of both sides:

$\displaystyle Log(A_{n+1}) = Log(A_{n}) + Log(R_n)$

so

$\displaystyle Log(A_{n}) = Log(A_{0}) + \sum Log(R_n)$

provided the Rn's are iid, and X is big enough that breaches will not cocur at short durations you may be able to mess around with the central limit theorum on $\displaystyle \sum Log(R_n)$.

Re: Prob (X < A) within time frame?

Your comment about running a few simulations is interesting, and I can see how experimenting with that may allow me to know how many to run to get within 5% error. I've simplified/abstracted this problem as much as I know how to; unfortunately one of the characteristics of the larger problem is that the time-value of the data points varies as well, which makes simulations about ^2 (?) expensive.

A Riemann Sum may be the way to go, although at this point I don't know how the computational demand (important to this problem) on of the two approaches would compare; I suppose I'd need to run many simulations with varying inputs and known input/output combinations to figure that out. It may be that one of these two approaches would be best, I'm not aware of a third option.

Thanks for the help! William