I have a stochastic variable x with this property: if it's measured at t1 and again at t2, then x(t2)-x(t1) has a normal distribution with mean 0 and standard deviation Sqrt[t2-t1].
I want to find the distribution of the maximum value this variable reaches between t1 and t2, or confirm my intuition that this is not well-defined.
My approach: break [t1,t2] into multiple intervals and integrate the following (PDF[NormalDistribution[x,y]] = normal probability distribution with mu of x and standard deviation of y):
PDF[NormalDistribution[0,Sqrt[1/4]]][x0]* PDF[NormalDistribution[x0,Sqrt[1/4]]][x1]* PDF[NormalDistribution[x1,Sqrt[1/4]]][x2]* PDF[NormalDistribution[x2,Sqrt[1/4]]][x3]
where each xi is integrated from -Infinity to m.
This specific example computes the probability that the maximum on [0,1] is less than m by breaking [0,1] into 4 parts.
Breaking [0,1] into more parts should yield more accurate results, although I slightly suspect that the limit diverges.
Mathematica slows to a crawl even breaking [0,1] into 5 or more parts.
I've tried replacing the normal distribution with others (uniform, DeltaDirac, C/(1+x^2), etc), with no better luck.
Googling yields many results (this appears to be a "Wiener Process"), but I can't find the actual distribution of the maximum anywhere (nor does it say anywhere that such a maximum doesn't exist).
Ultimate goal is to price box options: http://money.stackexchange.com/questions/4312/calculating-fair-value-of-an-oanda-com-box-option