2
$\begingroup$

Let $X_t$ be an OU process and $Y_t$ be the Gaussian process defined by $$ Y_t = y+\int_0^t X_s ds + W_t, $$ for some Brownian motion independent of $X_t$. Let $y,a>0$; is there a large deviation result, describing the hitting time of $Y_t$ to $a$?

I've seen similar results in other papers, for example here but I can't seem to find this type of result...

More precisely, if $a\geq 0$ and $\tau_a$ is the hitting time defined by $$ \tau_a\triangleq inf \left\{ t>0: \, |Y_t|\leq a \right\}, $$ then is there a result quantifying $f$ where $f$ satisfies a large-deviation principle of the form $$ \ln \mathbb{P}\left( \tau_a >x \right) \leq f(x) , $$ and $f(x)$ is a $C^1$-function of $x$.

Improve this question
$\endgroup$
6
  • $\begingroup$ Can you state more precisely the inequality you want to prove? $\endgroup$ Commented Sep 2, 2019 at 9:17
  • $\begingroup$ Sure, I've made it more precise. $\endgroup$ Commented Sep 2, 2019 at 9:23
  • $\begingroup$ Please proofread your definition of $\tau_a$ $\endgroup$ Commented Sep 2, 2019 at 21:25
  • $\begingroup$ Are you sure you want $|Y_t| \le a$ rather than $|Y_t| \ge a$? If so then $W_0$ must be nonzero. $\endgroup$ Commented Sep 8, 2019 at 5:55
  • $\begingroup$ That's fine, I adjusted for that to make things clear $\endgroup$ Commented Sep 8, 2019 at 17:24

1 Answer 1

Reset to default
1
$\begingroup$

Since they are independent, lets start with treating $W_{t}$ as a deterministic Holder-continuous function (and so lets interpret it as the t-dependent mean of the first term).

Indeed, in "Hitting times for Gaussian processes" they develop an LT estimate for hitting time. But you have a lot more eg. Markov structure. In "On the first-passage time of an integrated Gauss-Markov process" they study integrated processes and rewrite them as a BM

enter image description here

From here they can extract some formulas.

Generally though there are quite a few hits even under "integrated Ornstein Uhlenbeck" and hitting times/exists eg. "The one-sided barrier problem for an integrated ornstein-uhlenbeck process.".

Improve this answer
$\endgroup$

You must log in to answer this question.

Start asking to get answers

Find the answer to your question by asking.

Ask question

Explore related questions

See similar questions with these tags.