Consider a the following OU process in one dimensions, $$dX = -\theta(X -x_0)dt + \sqrt{s}dW $$

Now one can define the time $t_x$ as the time it takes for the solution to reach the point $x$.

Then apparently the following estimate holds,

• $$\mathbb E [ t_x] \sim \sqrt{\frac{\pi s}{\theta}} \cdot \frac{e^{\frac{\theta(x-x_0^2)}{s}}}{\theta (x- x_0)} $$

  Can someone kindly reference me a derivation of this?

• In the above the point $x$ is not special in anyway from the point of view of the SDE. But suppose I construct the following possibly more interesting situation :

  Consider a function $f(x) = \frac{\theta}{2} \cdot (x - x_0)^2 + g(x)$ and suppose $x_* = {\rm argmin} f(x)$. Now we consider the SDE, $dX = -(\theta(X -x_0) + g'(X))dt + \sqrt{s}dW $ Now can similar estimates be made for $\mathbb{E}[ t_{x_*}]$ ? ( making whatever might be convenient assumptions on $g$ except to set it to a constant) If necessary we can assume that $x_0$ is a critical point or a non-trivial local minima of $f$

Consider a the following OU process in one dimension, $$dX = -\theta(X -x_0)dt + \sqrt{s}dW $$

Now one can define the time $t_x$ as the time it takes for the solution to reach the point $x$.

Then apparently the following estimate holds,

• $$\mathbb E [ t_x] \sim \sqrt{\frac{\pi s}{\theta}} \cdot \frac{e^{\frac{\theta(x-x_0^2)}{s}}}{\theta (x- x_0)} $$

  Can someone kindly reference me a derivation of this?

• In the above the point $x$ is not special in anyway from the point of view of the SDE. But suppose I construct the following possibly more interesting situation :

  Consider a function $f(x) = \frac{\theta}{2} \cdot (x - x_0)^2 + g(x)$ and suppose $x_* = {\rm argmin} f(x)$. Now we consider the SDE, $dX = -(\theta(X -x_0) + g'(X))dt + \sqrt{s}dW $ Now can similar estimates be made for $\mathbb{E}[ t_{x_*}]$ ? ( making whatever might be convenient assumptions on $g$ except to set it to a constant) If necessary we can assume that $x_0$ is a critical point or a non-trivial local minima of $f$