Solve $dX_t=(a+bW_t)dt+cdW_t.$

Question

Question: If $a,b,c$ are constants and $W_t$ is a Brownian motion, solve the following SDE $$dX_t=(a+bW_t)dt+cdW_t.$$

First of all, this is not a linear SDE due to presence of $W_t$.

I attempted any form of integration factor but to no avail.

Any hint is appreciated.

Answer 1

Indeed as mentioned in the comments the process $$X_{t}=aB_{t}+b\int^t B_{s}ds=\int_{0}^{t}a+b(t-s)dB_{s}.$$

is as far as one can go. If you are interested in its law (Gaussian) see here Expectation of time integral of Wiener process and Integral of Brownian motion w.r.t. time.