Problem: Approximate the expectation function of the solution of SDE $$dX_t=\exp\bigl(\sqrt{t}\sin(2t)\,B_t\bigr)\Bigl(-\frac{2t}{(1+t^2)^2}+\frac{B_t}{1+t^2}\Bigl(\frac{\sin(2t)}{2\sqrt{t}}+2\sqrt{t}\cos(2t)\Bigr)+\frac{(\sqrt{t}\sin(2t))^2}{2(1+t^2)}\Bigr)\,dt+\frac{\sqrt{t}\sin(2t)}{1+t^2}\exp\bigl(\sqrt{t}\sin(2t)\,B_t\bigr)\,dB_t$$ in the limit $t \to \infty$ Please provide your answer in LaTeX \boxed{} environment. Final answer should be in closed form and parsable. I don't want any self-defined constants other than e and \pi. I don't want to have any short-cut function you used throughout the solution in the final solution either. The solution should solely be a function of t.
Solution: $\boxed{u(x,t) = \frac{1}{1 + t^2} e^{t^2 \sin^2(2t)}}$