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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 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 at 16:55
    
Did you read my first comment? Otherwise, this can go on forever... –  Did Apr 16 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 at 12:52

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