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Consider a linear SDE of the form:

$$dX_t = \alpha X_tdt + \beta dW_t,\\ X_t = x$$ I would like to determine an explicit solution, as well as that solution's distribution if possible.

My issue with the first task is that we are given $X_t = x$, and not, as usual, $X_0 = x$. With regards to the second task, I do believe such solutions will be .... normal? Log-normal?

EDIT: I found the solution to the first question, but only for $X_0 = x$. So I am wondering, do we just replace $t$ in the formula with $T-t$?

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    Avoid writing $dX_t$ and $X_t$ with the same subscript, that is causing confusion. And you could write it as $T-t$ sure, note that in this example the drift and diffusion coefficients do not depend on time, and therefore this problem is equivalent to solving up to time $S = T - t$ the SDE with $X_0 = x$. – 2017-02-24
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    So you would say the solution is $exp( \alpha (T-t)) + \int_0^{T-t} exp( \alpha (T - t - s)) \beta dW_s$? Seems a but clunky. – 2017-02-24
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    It does look clunky! My point was to, as you first suggested, replace any mention of $T-t $ with $S $ and solve from $s=0$ – 2017-02-24

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