Timeline for Using the optional stopping theorem on a stochastic process
Current License: CC BY-SA 3.0
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| Mar 23, 2017 at 16:44 | comment | added | user83457 | Suppose you manage to find a martingale Z with $Z_T = 0$ for your stopping time T (e.g. $X-c$). Then you can be pretty sure that the optional stopping theorem does not hold for $Z$ up to time T. If it did, by the martingal property Z is identically 0. You may want to browse the appropriate sections in karlin & taylor volume 2. You have options. I think K&T discuss the use of solutions of $GF = 1 $, G the generator, to find $E(T)$, and the passage time density itself satisfies a PDE (I think) though I am not sure it is discussed in K & T | |
| Mar 23, 2017 at 15:06 | comment | added | Ian | One way would be to artificially stop the process at a boundary to the left of $x(0)$, solve that PDE, and then send that boundary to $-\infty$. You can get some intuition for this trick from gambler's ruin. Consider studying the time to hit zero in a simple (possibly asymmetric) random walk on $\mathbb{N}_0$. You can understand this quantity by artificially deciding that the gambler will quit if he earns some amount $n$ (which gives two boundary conditions for a second order recurrence, as you should have) and then make the gambler "infinitely greedy" by sending $n \to \infty$. | |
| Mar 23, 2017 at 14:41 | review | First posts | |||
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| Mar 23, 2017 at 14:39 | history | asked | Circonflexe | CC BY-SA 3.0 |