Stochastic differential equation and stopping time

Question

Consider the following SDE: $dX_t=\frac{1}{1+X_t^2}dW_t$, $X_0=1$. For some $a<1<b$ define stopping times $\tau_a=\inf\{t\geqslant 0 : X_t\leqslant a\}$ and $\tau_b=\inf\{t\geqslant 0 : X_t\geqslant b\}$. Find $P(\tau_a<\tau_b)$.

My attempts:

Maybe it is possible to find a solution of this equation. My idea was to use Ito formula for $Z_t=X_t+\frac{1}{3}X_t^3$, but it didn't help.

Thank you in advance.

Answer 1

Hints:

1. $(X_t)_{ \geq 0}$ is a martingale.
2. By the optional stopping theorem, $(X_{t \wedge \tau})_{t \geq 0}$ is a martingale for any stopping time $\tau$. For $\tau := \tau_a \wedge \tau_b$ this implies $$\mathbb{E}(X_{t \wedge \tau})=0. \tag{1}$$
3. Show that $a \leq X_{t \wedge \tau} \leq b$ and $X_{\tau} \in \{a,b\}$. Conclude from the dominated convergence theorem and Step 2 that $$\mathbb{E}(X_{\tau}) = 0$$ i.e. $$a \mathbb{P}(X_{\tau}=a) + b \mathbb{P}(X_{\tau}=b). \tag{2}$$
4. We have $$\mathbb{P}(X_{\tau}=a) + \mathbb{P}(X_{\tau}=b) = 1. \tag{3}$$
5. $(2)$ and $(3)$ is a system of linear equations for $\mathbb{P}(X_{\tau}=a) = \mathbb{P}(\tau_a<\tau_b)$ and $\mathbb{P}(X_{\tau}=b) = \mathbb{P}(\tau_b<\tau_a)$. Solve it.