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 $n\ge 1$.