(Please feel free to suggest a better title -- I'm still not sure what to call this in the first place.)
I'm having trouble getting my head wrapped around a time-series stochastics problem I've run into. I think the main issue I'm having is that I don't know what area of statistics I need to be digging into to solve it. I've tried my best to break it down into the basic statistics terminology that I do remember -- feel free to let me know if I need to add or correct anything:
I'm trying to solve for z:
- x, y, and z are probabilities
- S is a time-series variable following a stochastic process that is close to normal, but with skew and kurtosis
- K is an upper limit for S that we pick at time 0; K > S at time 0
- x is the probability that S will be above K at time t; we don't calculate x -- it's provided by a third party; they don't publish skew or kurtosis, we might be able to get a stdev estimate
- x is recalculated and republished by the third party at each time tick
- y is a lower limit for x that we pick at time 0; y < x at time 0
- z is the probability of x crossing below y before time t
Given this information, can anyone see a way to calculate z?
It seems like, given that the inputs are already expressed in terms of probabilities, we should not need to know stdev, skew, or kurtosis... or am I completely confused?
Simplified Example:
Hydrologists estimate that there is an x=40% probability of a particular property being underwater exactly 10 years from now. An insurer writes a 10-year insurance contract for that property, specifying that no premium is due, and no coverage will start, unless and until the hydrologists' estimate first drops below y=30%.
What is the probability that coverage will start before the contract expires in 10 years?
I keep wanting to do simple things like z = 1 - (x - y), but that doesn't look right.
Answering my own question, after thinking about it a while... I think the reason I'm having trouble figuring this out is that, in order to get z, I would need to know the stdev of x. Alternatively, I'd need the stdev of the stdev of S.
I'd welcome any disagreement with that conclusion. ;-)