This thesis contributes to the field of statistical and mathematical finance by introducing novel Bayesian and game-theoretic methods in discrete-time portfolio theory. These methodologies enhance the precision and adaptability of investment strategies and risk management, particularly in complex market environments. The work is structured around five papers.

Paper I introduces a Bayesian framework for optimizing portfolio allocation, utilizing value at risk (VaR) and conditional value at risk (CVaR) as risk measures. This approach leverages the posterior predictive distribution to derive portfolio weights directly from observed data, contrasting with traditional methods that rely on estimates of unobserved variables. The benefit of the Bayesian method is demonstrated through simulations and empirical comparisons, particularly in predicting out-of-sample VaR.

Paper II presents a dynamic Bayesian approach to incorporate volatility clustering into VaR and CVaR estimation, utilizing hyperparameters based on different rolling windows to adapt quickly to changing market conditions. This method shows distinct advantages over existing models by adjusting the certainty and expected values of prior distributions in response to volatility changes, offering improved risk estimates during market turbulence.

Paper III develops a Bayesian inference procedure for tangency portfolios by establishing a new conjugate prior directly for the optimal portfolio weights, integrating high-frequency returns and a market condition metric, such as the CBOE Volatility Index (VIX) or Economic Policy Uncertainty Index (EPU). This approach enables direct inference on portfolio weights, and backtesting suggests potential advantages over traditional strategies in real-world scenarios.

Paper IV addresses the construction of tangency portfolios under short-selling constraints, using the same reparameterized asset return model within a Bayesian context as in Paper III. An innovative prior enforces positive weight constraints. The effectiveness of this method is empirically validated with selected stocks, highlighting its potential to enhance risk-adjusted returns.

Paper V innovates within a game-theoretic framework by introducing a recursive scheme for asset allocation using a mean-semivariance reward functional to better reflect investors' aversion to downside risk. This approach resolves the time-inconsistency problem in multi-period investments through an extended Bellman equation, effectively reaching a Nash equilibrium as demonstrated by an extensive numerical study.

Collectively, these studies provide a cohesive advancement in statistical and mathematical finance, demonstrating the effectiveness of Bayesian methods and game-theoretic approaches in improving the theoretical and practical aspects of portfolio optimization and risk management.