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 $\delta t$, and assume that in $\delta t$, the asset value can move by $\delta s=\pm \backslash s\sqrt{\delta t}$. Assuming probabilities of up and down moves are equal, we have ${\displaystyle \begin{array}{ccc}\multicolumn{1}{c}{p(s,t+\mathrm{dt})}& =& [(p(s+\mathrm{ds},t)+(p(s-\mathrm{ds},t)]/2\\ \multicolumn{1}{c}{p(s,t)+\frac{\partial p}{\partial t}.\delta t}& =& p(s,t)+\frac{1}{2}.\frac{{\partial }^{2}p}{\partial s^2}.(\delta s{)}^2\\ \multicolumn{1}{c}{\frac{\partial p}{\partial t}.\delta }& =& \frac{1}{2}.\frac{{\partial }^{2}p}{\partial s^2}.{\sigma }^2.\delta t\\ \multicolumn{1}{c}{\frac{\partial p}{\partial t}}& =& \frac{{\sigma }^2}{2}.\frac{{\partial }^{2}p}{\partial s^2}\end{array}}$ 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