nonlinear stochastic differential equation

Question

I have problem with solving this nonlinear SDE $$dX_t=X_t(1+X^{2}_t)dt + (1+X^{2}_t)dW_t$$ So far I've tried using Ito's formula but without any success. Could you give me an advice on how to start with this equation?

Answer 1

Apply the Ito's lemma to the function $\arctan(X_t)$: $$\begin{align} d\arctan(X_t) &= \frac{1}{1+X_t^2}dX_t + \frac{1}{2}\cdot \left(-\frac{2X_t}{(1+X_t^2)^2} \right) \cdot d\langle X,X \rangle _t \\ &=\frac{1}{1+X_t^2}\left( X_t(1+X^{2}_t)dt + (1+X^{2}_t)dW_t \right) + \frac{1}{2}\cdot \left(-\frac{2X_t}{(1+X_t^2)^2} \right) \cdot (1+X^{2}_t)^2 dt \\ d\arctan(X_t)&=dW_t \end{align}$$ Then $$\arctan(X_t) -\arctan(X_0) = W_t$$ that is $$\color{red}{X_t =\tan\left( W_t + C\right)} \hspace{1cm} \text{for } C\in \mathbb{R}$$