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Let $W_t$ is standard Brownian motion under probability measure $P$. Consider stochastic differential equation $$ dY_t = dt + Y_t dW_t, \ Y_0 = 0.$$ Note that, the above SDE has a strong non-negative solution.

Define stopping times $$\tau_0 = 0; \tau_{n+1} = \inf [ t>\tau_n: Y_t = 0 ].$$

[Q] Can one show that $\lim_{n\to \infty} \tau_n >1$ almost surely in $P$?

The above question is not true if the underlying SDE is $$ dY_t = dt + dW_t, \ Y_0 = 0,$$ since $Y_t$ is standard BM under some equivalent probability measure, and $\tau_n = 0$ for all $n\ge 1$.

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The process will never hit zero at positive times, by comparing to geometric Brownian motion. So $\tau_n=\infty$ for all $n > 0$. – George Lowther Jul 27 '12 at 12:15
@George, oops, you are right. – kenneth Jul 27 '12 at 13:30

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