Quasi-maximum likelihood estimation for scalar SDEs driven by stable Lévy noise

Abstract

We develop a computational quasi-maximum likelihood procedure for drift-parameter estimation in discretely observed stochastic differential equations driven by α-stable Lévy noise. The transition density is evaluated numerically through a Malliavin representation and Monte Carlo simulation of the corresponding Poisson-space weights, avoiding Fourier inversion of the stable characteristic function. The quasi-log-likelihood and estimator are defined explicitly. Conditional Malliavin score expectations are approximated by adaptive local bridge regression and are used to construct a Rao-type one-step correction. For scalar parameters, a second-order Malliavin representation and an eight-equation variation system also provide the curvature required for a Newton-type one-step correction. Finite-sample behavior is studied for several nonlinear drift models under symmetric and asymmetric stable noise. The procedure is additionally applied to experimental optical phase-locked-loop measurements. For the experimental data, the stable-noise nuisance parameters are estimated from reconstructed innovations before the Malliavin likelihood is evaluated. The fitted stability index is 1.9366. For the effective linear phase-locked-loop model, the quasi-maximum likelihood estimates of the damping and restoring coefficients are 4.5535 × 105 s −1 and 8.4331 × 1010 s −2 , while the corresponding Rao-type estimates are 4.6022 × 105 s −1 and 8.4779 × 1010 s −2 . Robustness is confirmed with respect to the bridge neighborhood and leave-one-trace-out perturbations. The emphasis is on explicit construction, numerical implementation, and empirical validation; no new general consistency or asymptotic-normality theorem is claimed.

Keywords

• Malliavin calculus
• parameter estimation
• stable Lévy noise
• stochastic differential equations
• quasi-maximum likelihood estimation.

References

1. [1] S. Bodnarchuk and D. Ivanenko, A method for checking efficiency of estimators in statistical models driven by Lévy noise, Theory of Probability and Mathematical Statistics 92 (2016), 1–15.