Asymptotic evaluation of the Poisson measures for tubes around jump curves

Abstract

Abstract. We find the asymptotic behavior of P ( X − φ ≤ ε) when X is the solution of a linear stochastic differential equation driven by a Poisson process and φ the solution of a linear differential equation driven by a pure jump function.

The Onsager–Machlup function as Lagrangian for the most probable path of a jump-diffusion process

“…The Girsanov transformation, which plays a crucial role in this approach, is of little help in pure jump case, since it can not help to obtain the uniqueness in distribution. As also realized earlier in [4] that the Girsanov transformation appears not to work for their case, since it transforms the Poisson process to a general semi-martingale that cannot be handled easily.…”

“…We remark that Moret and Nualart [19] recently derived the OM function for the fractional (but still Gaussian) Brownian motion. Bardina et al [4] dealt with the asymptotic evaluation of the Poisson measure for a tube around a jump path but that work is not for a general Lévy motion.…”

Onsager-Machlup action functional for stochastic partial differential equations with Levy noise

“…It is still open to derive the Onsager-Machlup action functional for stochastic dynamical systems with pure Lévy process both in finite and infinite dimensional systems, because the Girsanov theorem fails to normalize two probability measures. Bardina et.al [3] tried to deal with jump functions directly rather than using the Girsanov theorem to obtain the asymptotic evaluation of the Poisson measure for a tube. But Lévy process is much more complex than Poisson process.…”

“…As for deriving the Onsager-Machlup action functional for systems with non-Gaussion noise, there are a few attempts. Bardina, Rovira and Tindel [3] dealt with the asymptotic evaluation of the Poisson measure for a tube. More recently, Chao and Duan [10] used the Girsanov transformation to absorb the drift term and thus derive the Onsager-Machlup action functional in one dimensional systems with Lévy process.…”

Transition pathways for a class of high dimensional stochastic dynamical systems with Lévy noise

“…So we are urgent to derive the Onsager-Machlup action functional for stochastic dynamical systems with Lévy noise. Bardina et.al [4] tried to deal with jump functions directly rather than using the Girsanov theorem to obtain the asymptotic evaluation of the Poisson measure for a tube in the path space. More recently, Chao and Duan [10] used the Girsanov transformation to absorb the vector field or drift term and thus derived the Onsager-Machlup action functional in one dimensional systems with addictive Lévy process of small jumps.…”