Solve the SDE $dX_t = \frac{1}{2 X_t} dt + dB_t$

Question

Solve the following stochastic differential equations

$ dX_t = \frac{1}{2 X_t} dt + dB_t$

or equivalently with a transformation $Y_t = X_t^2$

$ dY_t = dt + 2 \sqrt{Y_t} dB_t$ with $Y_0 = y_0 > 0$

where B is one dimentional standard Brownian motion.

The solution is Given by $Y_t = (\sqrt{y_0} + B_t)^2$. You can check this by finding the differential of the solution using Ito's formula. This solution was by guess. The way to solve this problem is by introducing a transformation of $Y_t$ that will give a linear SDE that one can solve easily. Then the solution can be found by the inverse of the used transformation.

Answer 1

Hint: For the second SDE use the substitution $$Z_t := \exp(-\sqrt{Y_t}).$$ For the general approach see e.g. (the second part of) this answer.

Remark: As @user144410 pointed out, the solution of the first SDE can be obtained from the second one using the transformation $X_t = \sqrt{Y_t}$.