Timeline for probability question regarding brownian motion
Current License: CC BY-SA 2.5
10 events
| when toggle format | what | by | license | comment | |
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| Feb 14, 2011 at 15:55 | vote | accept | Qiang Li | ||
| Feb 10, 2011 at 22:44 | comment | added | Apollo | A useful fact for simplifying the resulting integral is that $\int_{-\infty}^\lam \phi(u)\Phi(\alpha+\beta u)du = \Psi[\lambda,\frac{\alpha}{\sqrt{1+\beta^2}};\frac{-\beta}{\sqrt{1+\beta^2}}]$ where $\Psi[x,y;\rho]$ is the bivariate normal cumulative distribution function. This is a straightforward bit of algebra (expand the inner cumulative normal as an integral and play with it until you get the standard bivariate). | |
| Feb 10, 2011 at 22:34 | comment | added | Apollo | We're integrating along the distribution of $X$ at time $a$ and multiplying by the conditional probability (given our location at time $a$) that we make it further to time $T$ without hitting zero. If we're already below zero then this probability is $0$ (so the lower bound of the integral starts at $0$) if we're above zero then we just need to keep the minimum of the remaining path above zero. | |
| Feb 10, 2011 at 22:31 | comment | added | Qiang Li | I mean: how to understand it is the required probability. | |
| Feb 10, 2011 at 22:31 | comment | added | Qiang Li | @Apollo, looks great! Can I ask you: how to understand the integral? | |
| Feb 10, 2011 at 22:24 | history | edited | Apollo | CC BY-SA 2.5 |
added more details; added 14 characters in body
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| Feb 10, 2011 at 22:18 | comment | added | Apollo | Added a little more detail. | |
| Feb 10, 2011 at 22:17 | history | edited | Apollo | CC BY-SA 2.5 |
added 658 characters in body
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| Feb 10, 2011 at 22:08 | comment | added | Qiang Li | @Apollo: could you please write down the integral? I am not sure I understand what you meant here. Does this way guarantee for all time in the interval $[a, T]$, no 0 is hit? | |
| Feb 10, 2011 at 22:00 | history | answered | Apollo | CC BY-SA 2.5 |