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Assume $X_t$ is an Ornstein-Uhlenbeck process in the form of $$ d X_t = -\alpha X_t dt + \sigma dB_t $$ Is there an exponential bound (large-deviation bound) for $$ P\left( \max_{t\le T} |X_t| \ge z \right) \le ? $$

P.S. when $\alpha = 0$ above $X_t$ is a Brownian motion, and hence $$ P\left( \sup_{t\le T} |\sqrt{\varepsilon} B_t | \ge c \right) \le 4d \exp\left( - \frac{c^2 }{2dT \varepsilon} \right) $$ for Brownian motion (Nonasymptotics result wanted). See e.g. the last display in

https://en.wikipedia.org/wiki/Schilder%27s_theorem

Is there an analogous version for the OU process?

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    $\begingroup$ What is the relation between $T$ and $z$ in the asymptotic you are after? $\endgroup$
    – ofer zeitouni
    Commented Jul 12, 2017 at 4:26
  • $\begingroup$ There is no relations, we want nonasymptotic result analogous to the Brownian motion version in the (new) P.S. part in the question $\endgroup$
    – Nikolayevich
    Commented Jul 13, 2017 at 17:56
  • $\begingroup$ The asymptotics you wrote is meaningful when $T/z^2\to 0$. To me this looks like a relation. Anyway, you already got two answers in different asymptotic regimes, unless you clarify what you are after let me guess that you won't get a third. $\endgroup$
    – ofer zeitouni
    Commented Jul 13, 2017 at 22:25
  • $\begingroup$ I actually wondered if we can resolve this problem using Girsanov transformation type of argument ... $\endgroup$
    – Nikolayevich
    Commented Jul 18, 2017 at 1:24
  • $\begingroup$ @Koltchinskii My answer is already along these lines; one derives the LDP for the scaled Wiener process and then uses a Girsanov type argument to reduce white noise perturbations of a dynamical system to the Wiener process case. $\endgroup$
    – Ian
    Commented Jul 18, 2017 at 3:13

2 Answers 2

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Fernique's theorem, valid for all continuous Gaussian processes, implies that there are constants $C,c$ (depending on $T$) such that $P(\max_{0 \le t \le T} |X_t| \ge z) \le C e^{-c z^2}$.

See also Deviation bound for the maximum of the norm of Wiener process.

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Yes: for white noise perturbations of the 1D dynamical system $\dot{x}=b(x)$, the action functional is $S(\phi)=\frac{1}{2} \int_0^T |b(\phi(s))-\dot{\phi}(s)|^2 ds$ for $\phi \in H^1$ and otherwise infinity. The LDP says that $\sigma^2 \log(P(X_\cdot \in A))$ is asymptotically between $-\inf_{f \in \mathrm{Int}(A)} S(f)$ and $-\inf_{f \in \mathrm{Cl}(A)} S(f)$ as $\sigma \to 0$.

In your case $A$ is the exterior of a ball in $C([0,T])$ centered at zero, so these two are the same, and so the problem is (presumably) to compute $I=\inf_{\phi \in C([0,T]) : \| \phi \|_\infty \geq z,\phi(0)=0} \frac{1}{2} \int_0^T |\dot{\phi}(s)+\alpha \phi(s)|^2 ds$. Then the logarithmic asymptotic for your quantity is $e^{-\sigma^{-2} I}$. I suspect the minimizer is attained by putting $\phi(T)=z$ (giving yourself the maximum possible amount of time to get to a magnitude of $z$) so that by the Euler-Lagrange equation $\phi(t)=z \frac{\sinh(\alpha t)}{\sinh(\alpha T)}$ is a minimizer; you should check this guess though. If I'm right then that means the minimum of $I$ is $\int_0^T \left ( z \alpha \left ( \frac{\cosh(\alpha s) + \sinh(\alpha s)}{\sinh(\alpha T)} \right ) \right )^2 ds = \alpha z^2(\coth(\alpha T)+1)$. This is large compared to $z^2$ for $T \ll \alpha^{-1}$ but quickly settles toward $2\alpha z^2$; unsurprisingly this means that in the long run you should expect to see maximum deviations from zero on the order of $\frac{\sigma}{\sqrt{\alpha}}$.

The situation is not much different in higher dimensions provided the noise covariance matrix is constant and nonsingular (even if it isn't isotropic).

Cf. Random Perturbations of Dynamical Systems by Freidlin and Wentzell, chapter 3 in the 3rd edition for more details.

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