In the chapter, two new priors, designed directly for the tangency portfolio weights, are developed. While the first approach is based on the extension of the Laplace prior, the second procedure generalizes the spike and slab prior. The posterior distributions of the tangency portfolio weights under both priors are characterized in terms of stochastic representations. These findings are used to establish exact sampling schemes for drawing samples of tangency portfolio weights from the corresponding posterior distributions, from which both the Bayesian point and interval estimators of the tangency portfolio weights are constructed.

This paper introduces Bayesian inference procedures for tangency portfolios, with a primary focus on deriving a new conjugate prior for portfolioweights. This approach not only enables direct inference about the weightsbut also seamlessly integrates additional information into the prior specification. Specifically, it automatically incorporates high-frequency returns and amarket condition metric (MCM), exemplified by the CBOE Volatility Index(VIX) and Economic Policy Uncertainty Index (EPU), significantly enhancing the decision-making process for optimal portfolio construction. While theJeffreys prior is also acknowledged, emphasis is placed on the advantages andpractical applications of the conjugate prior. An extensive empirical studyreveals that our method, leveraging this conjugate prior, consistently outperforms existing trading strategies in the majority of examined cases.

This paper introduces a novel recursive scheme for optimal asset allocation based on a mean–semivariance reward functional and a game-theoretic approach in a discrete-time setting. Unlike established frameworks that can handle variance as a risk measure, this study shifts focus to semivariance, which cannot be handled by existing theory due to aspects of its definition, including the use of an indicator function. To address this problem and the corresponding challenges of time inconsistency in multi-period investment decisions, we propose an extended Bellman equation to find a Nash equilibrium. The main contribution of this paper is a computational framework and a numerical investigation of a semivariance-based allocation strategy, based on an extended Bellman equation. Our analysis is restricted to the two-asset case — one risky and one risk-free asset — as a proof of concept, leaving multi-asset extensions for future work. The results of the numerical study indicate that our proposed method shows potential in achieving favorable investment outcomes.

This article is devoted to the problem of detection of forest and nonforest areas on Earth images. We propose two statistical methods to tackle this problem: one based on multiple hypothesis testing with parametric distribution families, another one—on nonparametric tests. The parametric approach is novel in the literature and relevant to a larger class of problems—detection of natural objects, as well as anomaly detection. We develop mathematical background for each of the two methods, build self-sufficient detection algorithms using them and discuss practical aspects of their implementation. We also compare our algorithms with each other and with those from standard machine learning using satellite data.

We address the challenge of constructing tangency portfolios in the context of short-selling restrictions. Utilizing Bayesian techniques, we reparameterize the asset return model, enabling direct determination of priors for the tangency portfolio weights. This facilitates the integration of non-negative weight constraints into an investor’s prior beliefs, resulting in a posterior distribution focused exclusively on non-negative values. Portfolio weight estimators are subsequently derived via the Markov Chain Monte Carlo (MCMC) methodology. Our novel Bayesian approach is empirically illustrated using the most significant stocks in the S&P 500 index. The method showcases promising results in terms of risk-adjusted returns and interpretability.

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.

We suggest a new method for integrating volatility information for estimating the value-at-risk and conditional value-at-risk of a portfolio. This new method is developed from the perspective of Bayesian statistics and is based on the idea of volatility clustering. By specifying the hyperparameters in a conjugate prior based on two different rolling window sizes, it is possible to quickly adapt to changes in volatility and automatically specify the degree of certainty in the prior. This gives our method an advantage over existing Bayesian methods, which are less sensitive to such changes in volatilities and usually lack standardized ways of expressing the degree of belief. We illustrate our new approach using both simulated and empirical data. The new method provides a good alternative to other well-known homoscedastic and heteroscedastic models for risk estimation, especially during turbulent periods, when it can quickly adapt to changing market conditions.

We study the optimal portfolio allocation problem from a Bayesian perspective using value at risk (VaR) and conditional value at risk (CVaR) as risk measures. By applying the posterior predictive distribution for the future portfolio return, we derive relevant quantities needed in the computations of VaR and CVaR, and express the optimal portfolio weights in terms of observed data only. This is in contrast to the conventional method where the optimal solution is based on unobserved quantities which are estimated. We also obtain the expressions for the weights of the global minimum VaR (GMVaR) and global minimum CVaR (GMCVaR) portfolios, and specify conditions for their existence. It is shown that these portfolios may not exist if the level used for the VaR or CVaR computation are too low. By using simulation and real market data, we compare the new Bayesian approach to the conventional plug-in method by studying the accuracy of the GMVaR portfolio and by analysing the estimated efficient frontiers. It is concluded that the Bayesian approach outperforms the conventional one, in particular at predicting the out-of-sample VaR.