Unbiased estimation of the Hessian for partially observed diffusions

Area of Science:

• Statistics
• Computational Mathematics
• Stochastic Processes

Background:

• Partially observed diffusion processes are crucial in various scientific fields.
• Accurate derivative information (Jacobian, Hessian) is vital for parameter estimation.
• Time-discretization of diffusions introduces bias in derivative calculations.

Purpose of the Study:

• To develop an unbiased estimator for the Hessian of the log-likelihood function for diffusion processes.
• To address the bias introduced by time-discretization methods.
• To provide a statistically sound method for derivative estimation in complex models.

Main Methods:

• Utilizing Girsanov's Theorem for unbiased estimation.
• Applying randomization schemes based on Mcleish (2011) and Rhee & Glynn (2016).
• Developing a novel Hessian estimator for partially observed diffusions.

Main Results:

• The proposed Hessian estimator is demonstrated to be unbiased.
• The estimator is shown to have finite variance.
• Numerical comparisons validate the methodology against particle filtering techniques.

Conclusions:

• The developed unbiased Hessian estimator effectively addresses bias in diffusion process analysis.
• The method offers a reliable tool for derivative estimation in applications like neuroscience.
• This work advances the statistical methodology for analyzing complex stochastic systems.