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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 $x$.

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  • $\begingroup$ Can you state more precisely the inequality you want to prove? $\endgroup$
    – Yuval Peres
    Commented Sep 2, 2019 at 9:17
  • $\begingroup$ Sure, I've made it more precise. $\endgroup$
    – ABIM
    Commented Sep 2, 2019 at 9:23
  • $\begingroup$ Please proofread your definition of $\tau_a$ $\endgroup$
    – Yuval Peres
    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. $\endgroup$
    – Yuval Peres
    Commented Sep 8, 2019 at 5:55
  • $\begingroup$ That's fine, I adjusted for that to make things clear $\endgroup$
    – ABIM
    Commented Sep 8, 2019 at 17:24

1 Answer 1

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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.".

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