Limit order markets and Bayesian learning

Abstract

This thesis explores two distinct research directions in mathematical finance and statistical learning. The first part develops microstructure models of trading in limit order markets with asymmetric information. The second part introduces a Bayesian deep learning framework designed to improve predictive performance and uncertainty quantification. The first part investigates equilibrium formation, belief updating, and price impact in markets where liquidity suppliers face uncertainty about the presence and number of informed traders. We begin with a one-period model in which equilibrium arises from a fixed-point mapping that accommodates general distributions for asset value and insider count. We show that the resulting market impact function exhibits a power law under fat-tailed fundamentals and a logarithmic form under light-tailed ones, with exponents and coefficients characterised by numerically solvable fixed-point equations. Extending to a multiperiod setting, we study sequential trading with myopic insiders and heavy-tailed t-distributed noise. The dynamic equilibrium is characterised by a recursive system of fixed-point equations, and we prove that liquidity suppliers’ beliefs about the asset value converge to the truth over time. The resulting price impact again follows a power-law decay, shaped by the noise tail index, insider competition, and the learning dynamics. The second part proposes a flexible framework for Bayesian neural networks by introducing probabilistic activation functions and mixture-of-experts architectures. Leveraging a semiparametric variational inference scheme, the model improves prediction accuracy, uncertainty quantification, and interpretability. Applications to real-world regression and classification tasks demonstrate performance gains over standard deep learning models.