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First exit times of SDEs driven by stable Lévy processes. (English) Zbl 1104.60030
The authors consider the exit problem of solutions of the stochastic differential equation \(\mathrm{d}X_t^\varepsilon = U'(X_t^\varepsilon)\mathrm{d}t +\varepsilon \mathrm{d}L_t\), \(\varepsilon > 0\), on \(I=[-b,a]\) or \(J=(-\infty,a]\). Here \(U\) is a potential function and \(L\) is a Lévy process with intensity \(\varepsilon\). The bounded or unbounded interval \(I\) and \(J\), respectively, contain the unique asymptotically stable critical point of the deterministic system \(\dot{Y}_t= -U'(Y_t)\). The authors perform a large deviation analysis of the exit times and base it on a noise intensity dependent decomposition of \(L\) into a sum of two independent processes: first, a compound Poisson process with large jumps and second another process which is a sum of the Brownian motion and a Lévy motion with small jumps.
The authors prove that asymptotically exits from the intervals above are due to large jumps of the first process. The second process will essentially not affect the solution of the system without noise. In the Gaussian noise case, estimates of exit times of particles depend on the height of the potential well, in the case considered here, these depend on the distance to the boundary. Also, the mean values of exit times are different to the Gaussian noise case, here they depend polynomially on \(\varepsilon\) in the small noise limit.

MSC:
60H10 Stochastic ordinary differential equations (aspects of stochastic analysis)
60J75 Jump processes (MSC2010)
60F10 Large deviations
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