Bayesian inference for diffusion processes: using higher-order approximations for transition densities.

Area of Science:

• Computational Mathematics
• Stochastic Processes
• Statistical Inference

Background:

• Diffusion processes are crucial for modeling random dynamical systems in continuous time across various scientific fields.
• Parameter estimation for these processes from discrete observations often employs Markov chain Monte Carlo (MCMC) methods.
• Current MCMC methods typically use numerical approximations like the Euler-Maruyama scheme, which can be computationally expensive.

Purpose of the Study:

• To investigate the utility and performance of higher-order approximations for MCMC methods in diffusion processes.
• Specifically, to evaluate the Milstein scheme as an alternative to the standard Euler-Maruyama approximation.
• To assess the computational cost and applicability of higher-order approximations, particularly in multidimensional settings.

Main Methods:

• Implementation and analysis of MCMC methods utilizing the Milstein approximation for diffusion processes.
• Comparison of estimation results and computational efficiency against the standard Euler-Maruyama scheme.
• Examination of the impact of combining the Milstein approximation with modified bridge proposals on numerical challenges.

Main Results:

• MCMC methods employing the Milstein approximation demonstrate effective parameter estimation for diffusion processes.
• These higher-order methods incur greater computational expense compared to the Euler-Maruyama scheme.
• Application to multidimensional processes is constrained by practical limitations, and integration with modified bridge proposals presents further numerical difficulties.

Conclusions:

• The Milstein approximation offers a viable, albeit computationally demanding, alternative for MCMC parameter estimation in diffusion processes.
• Careful consideration of computational cost and dimensionality is necessary when applying higher-order approximations.
• Further research is needed to overcome the numerical challenges associated with advanced approximation schemes in complex stochastic models.