Q: Solving a stochastic differential equationn Does anyone has ideas on how to solve this equation.
$$dX_{t} = \left(\sqrt{1+X_{t}^{2}} + \frac{1}{2}X_{t}\right)\,dt + \sqrt{1 + X_{t}^{2}} \,dBt$$ where $Bt$ is a standard Brownian Motion. 
I have tried to solve $$\frac{\partial f}{\partial x} = \sqrt{1 + f^2}$$
$$\frac{\partial f}{\partial t} + 0.5\frac{\partial^{2} f}{\partial x^{2}} = 0.5f + \sqrt{1 + f^{2}}$$
As for the 1st equation, I can find the solution $f(x) = \sinh(x) + C$, but clearly, it is not the solution for the 2nd equation. Does anyone have ideas on this?

A: You could take the inspiration from the first solution the other way around, it is easier to transform the known equation than to reverse-engineer starting from an unknown quantity. Set
$$
Y_t=g(X_t)=\arg\sinh(X_t),~~g'(x)=\frac1{\sqrt{1+x^2}},~~g''(x)=-\frac{x}{\sqrt{1+x^2}^3}.
$$
Then by the Ito theorem
$$
dY_t=\frac12g''(X_t)\sigma(X_t)^2dt+g'(X_t)dX_t
=dt+dB_t.
$$
