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?
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 $$ dY_t=\frac12g''(X_t)\sigma(X_t)^2dt+g'(X_t)dX_t =dt+dB_t. $$