most recent 30 from http://mathoverflow.net 2013-05-24T04:00:31Z http://mathoverflow.net/feeds/question/103275 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/103275/does-this-series-stopping-times-marching-forward kenneth 2012-07-27T06:30:57Z 2012-07-27T06:30:57Z

<p>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.</p> <p>Define stopping times $$\tau_0 = 0; \tau_{n+1} = \inf [ t>\tau_n: Y_t = 0 ].$$</p> <p>[Q] Can one show that $\lim_{n\to \infty} \tau_n >1$ almost surely in $P$?</p> <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 $n\ge 1$.</p>