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I came across this SDE and since I am not an expert I am wondering if this SDE is known to have an closed form solution for first passage times.

The SDE is

$$dY_t=(a+be^{ct}) \, dt+\sigma \, dB_t$$

How does one go about finding an explicit distribution for first passage times in this case?

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No need for a SDE, $Y$ is simply $Y_t=Y_0+at+(b/c)(e^{ct}-1)+\sigma B_t$. "Name"? No. "Explicit distribution for first passage times"? No. – Did Apr 16 '14 at 15:32
If I wanted to try and get the explicit distributions of First Passage Times how would I do it? – Nuno Calaim Apr 16 '14 at 16:55
Did you read my first comment? Otherwise, this can go on forever... – Did Apr 16 '14 at 18:00
I read your comment: but I interpreted it as: "no one yet has done all the math in order to come up with an explicit distribution for first passage times" but your second comment implies: "no one will ever be able to do it" – Nuno Calaim May 5 '14 at 12:52

Your SDE is an example of a linear SDE $$dY_t = (\alpha(t)+\beta(t)Y_t)dt+(\gamma(t)+\delta(t)Y_t)dW_t,$$ where $\alpha$, $\beta$, $\gamma$ and $\delta$ denote deterministic processes and $W$ denotes a one-dimensional Wiener process. In your case, processes $\beta$ and $\delta$ are zero processes and $\gamma$ is a constant process and $\alpha(t) = a+be^{ct}.$ There exists a closed form for the (strong) solution of this SDE, which can be found on Wikipedia. Since your SDE is $$Y_t = A(t) + \sigma W_t,$$ where $A(t) = \int_0^t\alpha(s)ds$ is an exponential curve, studying first passage times of constant levels for $Y$ is equivalent to studying first passage times of exponential curves for the Wiener process.

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