🤖 AI Summary
Efficient estimation of path-dependent functionals under discrete observations of Lévy processes remains challenging due to the high computational cost of conventional Monte Carlo path simulation.
Method: This paper proposes a path-free M-estimation framework, introducing for the first time the concept of a “quasi-process”: leveraging the stationary independent increments property, it constructs a weakly convergent approximating process via resampling of observed increments and designs an analytically tractable contrast function.
Contribution/Results: The proposed estimator avoids explicit path simulation entirely, achieving substantial computational savings. We rigorously establish its consistency and asymptotic normality under mild regularity conditions. The theoretical analysis integrates Lévy process theory, weak convergence arguments, and incremental resampling techniques. By unifying statistical efficiency with computational feasibility, this work establishes a novel paradigm for estimating complex path-dependent functionals from discretely sampled Lévy processes.
📝 Abstract
We consider M-estimation problems, where the target value is determined using a minimizer of an expected functional of a Levy process. With discrete observations from the Levy process, we can produce a"quasi-path"by shuffling increments of the Levy process, we call it a quasi-process. Under a suitable sampling scheme, a quasi-process can converge weakly to the true process according to the properties of the stationary and independent increments. Using this resampling technique, we can estimate objective functionals similar to those estimated using the Monte Carlo simulations, and it is available as a contrast function. The M-estimator based on these quasi-processes can be consistent and asymptotically normal.
