Estimating the hitting time for a SDE solution

Question

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 $f$

Estimate as what goes to what? I assume it's as $x$ goes to infinity, but in that case I'm not sure how to make sense of your second question. For the first, I would for convenience find $a$ and $b$ such that $Y_t=a(X_{by}-x_0)$ satisfies $dY_t=-Y_tdt+dB'_t$. — Pierre PC, Commented Nov 2, 2020 at 19:27
Then you can play with the fact that $F(Y_t)^2-t$ is a martingale for $F(y)=\int_0^y\exp(w^2)dw$, as well as the explicit representation of $Y_t$. — Pierre PC, Commented Nov 2, 2020 at 19:29
Do you have a reference for such a calculation? I couldnt find anything like this estimation done anywhere! — gradstudent, Commented Nov 2, 2020 at 21:28
I don't. I'm not even saying I could do the exercise, I'm just throwing ideas. — Pierre PC, Commented Nov 3, 2020 at 0:03

Thomas Kojar · Accepted Answer · 2023-01-27 05:02:32Z

$$c^{-1}\frac{e^{\lambda r^{2}}}{r^{d}}\leq E_{0}(\tau_{B_{0}(r)})\leq c\frac{e^{\lambda r^{2}}}{r^{d}},$$

In terms of the other process, if you have any comparison results between the two drifts $b_{1}:=-\theta(x-x_0)$ and $b_{2}(x):=-\theta(x-x_0)+g'(x)$ then you could try using a comparison theorem for their solutions $X_{b_{1}},X_{b_{2}}$ such as the one by Nakao "On pathwise uniqueness and comparison of solutions of one-dimensional stochastic differential equations"

1 Answer 1