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I have a Brownian motion $X_t = \mu t+\sigma W_t$, where $W_t$ standard Brownian motion. I know that the first passage time $\tau = \min\{t|X_t\leq\alpha\}$, is Inverse Gaussian distributed i.e., $\tau\sim IG(\alpha/\mu,\alpha^2/\sigma^2)$ which I can generate easily using software such as R. I want to generate first passage times in the interval $[0,T]$. The parameters for $IG(\alpha/\mu,\alpha^2/\sigma^2)$ are constants with no dependency on T. My problem is that the values I am getting a too big for an interval such as $[0,1]$.

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To generate values in $[0,1]$ according to the distribution of $\tau$ is impossible since $\tau\gt1$ with positive probability. To generate values in $[0,1]$ according to the distribution of $\tau$ *conditionally on $\tau\leqslant1$* is easy: generate values of $\tau$ and keep only those falling in $[0,1]$. –  Did Jun 23 '12 at 6:29
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