This thesis presents the decoupled copula model, a novel theoretical framework for aggregating expert judgments in structured expert judgment (SEJ) studies. The model's key innovation lies in transforming expert assessments into a "decoupled space" where systematic biases can be identified and corrected while capturing potential inter-expert dependencies. Unlike existing Bayesian SEJ methods, which are limited to linear error metrics, our framework accommodates flexible dependency measures with rigorous theoretical criteria and practical tests for their evaluation. While previous Bayesian approaches acknowledged the possibility of bias correction, they lacked practical procedures for implementing these corrections using historical test data. Our framework addresses these limitations by enabling flexible metric choices for measuring biases and dependencies. Captured biases include under- and overconfidence as well as consistent over- and underestimation. Additionally, we introduce novel calibration criteria that have been proven necessary for perfect aggregation methods, along with interpretable calibration metrics that measure discrepancies from these criteria.
Empirical evaluation on 47 real-world SEJ studies demonstrates superior calibration properties while maintaining competitive predictive performance compared to established methods like the Classical Model. The empirical analysis reveals that inter-expert dependency modeling provides limited benefits, suggesting that systematic bias correction, rather than dependency modeling, drives improvements in aggregation performance in practical applications of our model.

The aim of this thesis is the study of inference problems in Pair Copula Bayesian Networks (PCBN). To this end, certain sub-structures called arteries are identified in the PCBNs and Arterial Sample Propagation, a sample-based extension of Pearl's Belief Propagation Algorithm, is developed for single arteries. This proposed inference methodology incorporates properties unique in PCBNs as well as information on the graph structure, thus avoiding unnecessary computations and boosting the algorithm's performance. Furthermore, an extension of Arterial Sample Propagation is proposed for PCBNs with multiple arteries under some additional assumptions on the graph structure.


This thesis also explores the structural properties of PCBNs, with this examination moving in two separate directions. On the one hand we analyze inference problem reduction through pruning, building up to a pruning algorithm for PCBNs that removes a larger number of variables than existing BN pruning methods. Additionally, we study the implications that the existence of arcs in arteries have on the PCBN's structure. We prove a Theorem used as a background for Arterial Sample Propagation algorithms developed in this thesis, which has potential applications in the development stage of PCBNs.

In this thesis, we aim to improve the application of deep reinforcement learning in portfo- lio optimization. Reinforcement learning has in recent years been applied to a wide range of problems, from games to control systems in the physical world and also to finance. While reinforcement learning has shown success in simulated environments (e.g. matching or exceeding human performance in games), its adoption in practical applications (non- simulated environments) has lagged. Dulac-Arnold et al. [2019] suggest this is caused by a discrepancy in the experimental set-up in research and the conditions in practice. Specifically, they present a list of challenges that make the application of reinforcement learning in real-world settings more difficult. One of these challenges is non-stationary environments, which is common in financial environments. It is a challenge since, given an observed state, the optimal action may not always be the same as it may change over time due to non-stationarity. Therefore, more specifically, the goal of this thesis is to overcome the challenge of non-stationarity in the application of reinforcement learning to portfolio optimization. In this thesis, we use reinforced deep Markov models (RDMM) introduced by Ferreira [2020] (applied to an optimal execution problem and later used by Cartea et al. [2021] for statistical arbitrage on simulated price movements of an FX triplet) for its data efficiency and ability to handle complex environments. RDMM involve a partially observ- able Markov decision process (POMDP) which is also the setting used by Xie et al. [2021] to model non-stationarity in reinforcement learning. We extend RDMM to incorporate non-stationarity, using the framework suggested by Xie et al. [2021], and apply it to port- folio optimization. Our implementation is sample efficient which allows for quick learning, by doing this we attempt to improve on another challenge of reinforcement learning — i.e. sample-inefficiency [Dulac-Arnold et al., 2019]. Moreover, our implementation can handle continuous state and action spaces.
We compare the performance of our algorithms to classical portfolio optimization tech- niques such as Mean-Variance (MV) and Equal Risk Contribution (ERC), and to popular reinforcement learning techniques such as Deep Deterministic Policy Gradient (DDPG) and Soft Actor-Critic (SAC). We observe our implementation has higher sample-efficiency compared DDPG and SAC, and higher cumulative returns on the test set compared to MV, ERC, DDPG, and SAC.

This thesis investigates the Ground Motion Model (GMM) for the Groningen Seismic Hazard & Risk Analysis. We look at various aspects of the model to see where improvements can be made. We start by looking at model calibration and validation, where we check to what extent the proposed model along with its parameter fits the data. Here we come to the conclusion that both the parameters and the model itself have room for further optimization to reflect the data set more accurately. In addition to this, a new model for correlations of site-response amplifications is presented. The topic of how the dependence of these quantities should be modeled is still unclear. The lack of coherent solution to this modeling problem makes the proposed model valuable, as it is simple in nature and reflects the data we have the best.
Lastly, the main improvements that come out of this research are in on the computational front, by finding a novel method for calculating average spectral accelerations, which is the main quantity used in risk assessment. This method generates the distribution of this quantity in one step using numerical integration rather than the previously used Monte Carlo method. The method speeds up computations by a factor of 550 times.

Counterparty Credit Risk (CCR) refers to the risk that a counterpary involved in a financial contract will default before the final settlement of the contract, resulting in unrealized financial gains. One risk measure for managing counterparty credit risk is the Potential Future Exposure (PFE). The PFE is defined as the 97.5%-quantile of the exposure distribution. The traditional numerical method for computing the PFE is the Monte Carlo simulation method. Recent research has produced a semi-analytical method based on Fourier-cosine series expansion which produce PFE estimates with at least five times the accuracy of the Monte Carlo simulation in one-tenth of the CPU time. In this thesis this COS-PFE framework is combined with Monte Carlo simulation with the goal of reducing the variance of the PFE estimates.
In this thesis three methods are developed. Firstly, the Control Variate method that uses the COS-PFE framework to retrieve the CDF of the portfolio's exposure from which control variates are sampled. The second method developed uses Adaptive Importance Sampling. This method uses the COS-PFE framework to estimate the PFE and expected exposure (EE) of the portfolio. Using the risk factor that has the highest correlation with the exposure of the portfolio a shift is found such that the new joint probability distribution, from which the risk factors are sampled, has an EE that coincides with the portfolio's PFE. The third method developed in this thesis uses Adaptive Importance Sampling based on the Cross-Entropy method. This method aims to find an auxiliary probability distribution which minimizes the Kullbeck-Leibler divergence between itself and the theoretical zero-variance estimator.
The methods were tested using portfolios containing 100, 1000 and 10000 derivatives, both with and without collaterals. Testing showed that the control variate method was not able to produce a variance reduction compared to the straight forward Monte Carlo simulation. However, it did demonstrate the ability to produce a variance reduction for the EE.
The method that finds an optimal shift was successful in producing a variance reduction. The variance of the PFE estimates produced by this method was, on average, 3.5 times lower than the variance found using the straight forward Monte Carlo simulation. However, the algorithm required between 96 and 1773 seconds to produce a PFE estimate.
The adaptive importance sampling method using the cross-entropy approach was tested on all portfolio, both with and without collateral. This method was shown to produce PFE estimates with a significantly lower variance than the straight forward Monte Carlo simulation. For the uncollateralized portfolios containing 100, 1000 and 10000 derivatives the variance of the PFE estimates were on average 35.4, 38.6 and 37.2 times smaller compared to straight forward Monte Carlo simulation. For the portfolios with collateral this performance remains. We conclude that this method produces more accurate PFE estimations with significant lower variance than straight forward Monte Carlo simulation within the same CPU time.

During this bachelorproject, more research is done to investigate the properties of the Hanea’s and Nane’s calibration score and to look at slight adjustment of Hanea’s and Nane’s calibration score. Both simulated data and real data will be used to investigate the behaviour of these calibration scores.