I am trying to solve the following problem:
Let $X$ be the strong solution of the following Stochastic Differential Equation: $\mathrm dX_t = sign(X_t)dt + \mathrm dW_t, X_0 = 0$, where $W_t$ is a standard Brownian Motion. Let $\tau = inf\{t \geq 0: |X_t| \geq 1\}$. Compute $E[\tau]$.
Note that a strong solution does exist (see Shreve Stochastic Calculus and Brownian Motion pg 341).
I have tried looking for a solution with absolute values but couldn't get anywhere. Does anyone have any ideas for this? Thanks.