Paul Snell

Posted on Friday, 09 May, 2003 - 01:29 am: I need to think of an example of a simple random walk with 2 absorbing barriers, other than the gambler's ruin. Can anyone help me?

Andre Rzym

Posted on Friday, 09 May, 2003 - 09:52 am: This is a rather strange (albeit interesting) thing to ask. The first example would be in finance. There are transactions where a client will buy an option (paying money upfront for it) and will get a payment back at some predetermined point in the future provided some index (gold, some FX rate, interest rates or whatever) have neither gone above or below a certain level in the meantime. They are known as double-knockouts. There is a better example, but to find it, it is probably best to derive the equations surrounding gamblers ruin: Let p(s,t) be the probability that the asset value of the player is s at time t. We discretise time into steps dt, and assume that in dt, the asset value can move by ds = ±\s Ö   dt   . Assuming probabilities of up and down moves are equal, we have p(s,t+dt) = [(p(s+ds,t) + (p(s-ds,t)]/2 p(s,t)+ ¶p ¶t . dt = p(s,t) + 1 2 . ¶2p ¶s2 .(ds)2 ¶p ¶t .d = 1 2 . ¶2 p ¶s2 . s2. dt ¶p ¶t = s2 2 . ¶2 p ¶s2 This is the forward Kolmogorov equation. But it is also the one dimensional heat conduction equation . This is the/an answer:take a metal bar, heat it to some temperature (distribution), then fix the temperature of the endpoints of the bar (at zero to represent a knockout event). There is a 1 to 1 correspondence between the subsequent evolution of the temperature distribution and the probability distribution of the value of the gamblers assets. Andre