Online drift estimation for jump-diffusion processes

Journal article

We show the convergence of an online stochastic gradient descent estimator to obtain the drift parameter of a continuous-time jump-diffusion process. The stochastic gradient descent follows a stochastic path in the gradient direction of a function to find a minimum, which in our case determines the estimate of the unknown drift parameter. We decompose the deviation of the stochastic descent direction from the deterministic descent direction into four terms: the weak solution of the non-local Poisson equation, a Riemann integral, a stochastic integral, and a covariation term. This decomposition is employed to prove the convergence of the online estimator and we use simulations to illustrate the performance of the online estimator.

Authors: Bhudisaksang, T; Cartea, A

• Publication status: Published
• Peer review status: Peer reviewed
• Publisher: Bernoulli Society for Mathematical Statistics and Probability
• Journal: Bernoulli - Journal of the Bernoulli Society
• Volume: 27
• Issue: 4
• Pages: 2494-2518
• Publication date: 2021-11-01
• Acceptance date: 2020-12-29
• DOI: 10.3150/20-BEJ1319
• ISSN: 1350-7265
• Language: English
• Keywords: Levy process; SGDCT; non-local Poisson equation; online estimation; extended Ito lemma; jump-diffusion