Thanks Alan, surely one of these options will play out.

Ok. Let's assume I have a GBM without drift, i.e. [$]dS_t = \sigma S_t dW_t[$]. Assume some level [$]B>S_{t_0}[$]. Define [$]M^+=max_{0\leq s \leq t} S_t[$]. With the help of some textbook, I could figure out that  [$]E(\tau<t) = P(M^+ \geq B) = 1 + e^{2\alpha b} N(\frac{-b-\alpha t}{\sqrt{t}}) - N(...

Cool, thank you, took me a moment to figure out. Can you guide me as to what the steps to solving the following integral are?

[$]-\frac{1}{2\sqrt{2\pi}}\int^T_0 \frac{(b-\alpha t)}{\sqrt{t}}e^{-\frac{1}{2}(\frac{b-\alpha t}{\sqrt{t}})^2}dt[$]

Appreciate your help.

Best
Ralf

What do you mean by posting it mathematically? I find it difficult to post it mathematically without that editor.

Sorry, where has the formula editor gone?

As part of the problem, I believe I must solve an integral of the form : [img=250x46]data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAAPoAAAAuCAMAAAA2nfQ2AAAAAXNSR0IArs4c6QAAAJZQTFRFAAAAAAAAAAA6AABmADo6ADpmADqQAGa2OgAAOgA6OgBmOjqQOma2OpDbZgAAZgA6ZgBmZjoAZjqQZmYAZmY6Zma2ZpCQZpC2ZpDbZrbbZrb/kDoAkDo6kDpmk...

Thank you all.

@ Alan: I was following your suggested approach but somehow failed to solve the resulting integral. I guess I will have to try harder. I assume this works with standard calculus. 

Best
Ralf

Hi, For a geometric Brownian motion (GBM), presumably one can analytically compute the expected first-passage time (tau) to a given level conditional on the timeframe under consideration being shorter equal than some fixed future time T, i.e. E(tau|tau <= T). Can anyone point me to some document whe...

thanks for that, Ralf.

thanks for that, Ralf.

Hi,Is there a closed-form solution for the expectation of the first passage time for a Brownian motion. If so, can you point me to a good reference?Thanks and regardsRalf

Suppose I have an n x p Matrix X with covariance Matrix C = X'X/(n-1). Is there an intuitive and also formal explanation why the singular value decomposition (SVD) of X, i.e. X=USV', yields the same V and, for the first p diagonal elements, same S as the SVD of (n-1)C?Many thanksRalf

[$](1/\gamma)^{1/\alpha}\delta(1+1/\alpha,\gamma x^\alpha)[$] it must be then, with [$]\delta[$] being the lower incomplete gamma fct. Thanks much for help!

yes, if there is one ...

Yep, was thinking of that, because the unconditional expectation can be expressed by a gamma-fct. Still, I can't see the way there and hoped to get some hint (I think normally you go via mgf but dont think that makes sense in the conditional case).ThanksRalf