Unbiased and Multilevel Monte Carlo Methods for Parameter Inference in Latent Stochastic Systems

Overview

Many systems in science, engineering and finance are modelled as stochastic processes that are only partially observed, and whose unknown parameters must be inferred from noisy data. This thesis develops efficient Monte Carlo methods for score-based parameter inference in such systems, mainly hidden Markov models driven by diffusion processes, and for the related problem of conditional stochastic optimization. Because the score function is generally intractable and requires time discretization, the thesis combines sequential Monte Carlo (particle filter) methods with multilevel and unbiased Monte Carlo techniques to control both the statistical error and the discretization bias at a reduced computational cost.

The thesis makes three main contributions. First, it develops multilevel and unbiased particle filters for diffusions observed at random times, where the observation times follow a Cox process that depends on the diffusion itself. To reach a mean squared error of ε², the multilevel estimator costs O(ε^-2.5), compared with O(ε^-3) for a standard particle filter, while the unbiased variant removes the discretization bias entirely.

Second, it addresses parameter estimation when the diffusion coefficient itself depends on the unknown parameter, a setting that existing score-based methods cannot handle. Combining a continuous-time, bridge-based backward-sampling particle filter with a double-randomization stochastic approximation scheme yields a parameter estimator that is unbiased, has finite variance, and achieves the optimal Monte Carlo complexity.

Finally, it considers conditional stochastic optimization when the relevant distributions can only be accessed through Markov chain Monte Carlo (MCMC) rather than exact simulation. Combining Markovian stochastic approximation with level randomization, it constructs an unbiased gradient estimator with finite second moment, and applies it to model-averaged parameter estimation for state-space models and to high-dimensional portfolio selection on real equity data.