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Integral of the OU (Ornstein Uhlenbeck) process conditioned on hitting a threshold value for...

Let say I have a zero-mean OU process as follows: $dX_t = -\alpha X_t + dW_t$ The process starts at $x_0 = 0%$ and I'm interested in the event in which the process hits the value $x_{\tau} = a$ for …
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