Integrals of Bessel processes and multi-dimensional Ornstein--Uhlenbeck processes:

V. R. Fatalov

Izvestiya. Mathematics , Tome 82 (2018) no. 2, pp. 377-406

Résumé

We prove results on exact asymptotics of the expectations $\mathbf{E}_a \exp \bigl(-\int_0^T \xi_q^p(t) \,dt \bigr)$, $\mathbf{E}_a \bigl[ \exp \bigl(-\int_0^T \xi_q^p(t) \,dt \bigr) \bigm| \xi_q(T)=b \bigr]$ as $T\to\infty$ for $p>0$, $a\geqslant 0$, $b\geqslant 0$, where $\xi_q(t)$, $t\geqslant 0$, is a Bessel process of order $q\geqslant-1/2$. We also find exact asymptotics of the probabilities $\mathbf{P} \bigl\{ \int_0^1 \sum_{k=1}^n |Y_k(t)|^p \,dt \leqslant \varepsilon^p \bigr\}$, $\mathbf{P} \bigl\{ \int_0^1 \bigl[ \sum_{k=1}^n Y_k^2(t) \bigr]^{p/2} \,dt \leqslant \varepsilon^p \bigr\}$ as $\varepsilon\to 0$, where $\mathbf{Y}(t)=(Y_1(t),\dots, Y_n(t))$, $t\geqslant 0$, is the $n$-dimensional non-stationary Ornstein–Uhlenbeck process with a parameter $\gamma=(\gamma_1, \dots, \gamma_n)$ starting at the origin. We also obtain a number of other results. Numerical values of the asymptotics are given for $p=1$, $p=2$.

Détail

Keywords: Bessel processes, multi-dimensional Wiener process, Girsanov's theorem, small deviations, Schrödinger operator, Airy function, Bessel function.
Mots-clés : Feynman–Kac formula