Recently, machine learning (ML) models have gained prominence in predicting asset returns, selecting portfolios, and estimating stochastic discount factors, with significant success in these areas. ML techniques, by capturing complex and nonlinear relationships in financial data, are particularly well-suited for enhancing portfolio management decisions. For example, within the mean-variance portfolio framework, ML methods are increasingly used to estimate expected returns and (co)variances, often leading to more effective portfolio allocations. The literature consistently demonstrates the effectiveness of machine learning in these and other applications (e.g., Gu, Kelly, and Xiu (2020); Bianchi, B uchner, and Tamoni (2021); Cong, Tang, Wang, and Zhang (2021); Kelly, Malamud, and Zhou (2021); Patton and Weller (2022); Didisheim, Ke, Kelly, and Malamud (2023); Filipovic and Schneider (2024)). Despite the success of machine learning in asset pricing, existing literature typically treats ML predictions as point estimates and conducts asset pricing analyses as if they were true values, overlooking the associated uncertainty. This is surprising, given that uncertainty about input parameters is widely acknowledged as critical in portfolio selection (e.g., DeMiguel, Garlappi, and Uppal (2009)), and Garlappi, Uppal, and Wang (2007) show that incorporating forecast uncertainty in mean-variance portfolio allocation leads to distinct economic insights. However, quantifying prediction uncertainty in ML forecasts, particularly with neural networks, remains a complex challenge, limiting their broader application in asset pricing.

Understanding the structure of real data is paramount in advancing modern deep-learning methodologies. Natural data such as images are believed to be composed of features organised in a hierarchical and combinatorial manner, which neural networks capture during learning. Recent advancements show that diffusion models can generate high-quality images, hinting at their ability to capture this underlying structure. We study this phenomenon in a hierarchical generative model of data. We find that the backward diffusion process acting after a time $t$ is governed by a phase transition at some threshold time, where the probability of reconstructing high-level features, like the class of an image, suddenly drops. Instead, the reconstruction of low-level features, such as specific details of an image, evolves smoothly across the whole diffusion process. This result implies that at times beyond the transition, the class has changed but the generated sample may still be composed of low-level elements of the initial image. We validate these theoretical insights through numerical experiments on class-unconditional ImageNet diffusion models. Our analysis characterises the relationship between time and scale in diffusion models and puts forward generative models as powerful tools to model combinatorial data properties.

Socially responsible investors build investment portfolios intending to incite social and environmental advancement alongside a financial return. Although Mean-Variance (MV) models successfully generate the highest possible return based on an investor's risk tolerance, MV models do not make provisions for additional constraints relevant to socially responsible (SR) investors. In response to this problem, the MV model must consider Environmental, Social, and Governance (ESG) scores in optimisation. Based on the prominent MV model, this study implements portfolio optimisation for socially responsible investors. The amended MV model allows SR investors to enter markets with competitive SR portfolios despite facing a trade-off between their investment Sharpe Ratio and the average ESG score of the portfolio.

The multi-armed bandit (MAB) problem is a classical learning task that exemplifies the exploration-exploitation tradeoff. However, standard formulations do not take into account risk. In online decision making systems, risk is a primary concern. In this regard, the mean-variance risk measure is one of the most common objective functions. Existing algorithms for mean-variance optimization in the context of MAB problems have unrealistic assumptions on the reward distributions. We develop Thompson Sampling-style algorithms for mean-variance MAB and provide comprehensive regret analyses for Gaussian and Bernoulli bandits with fewer assumptions. Our algorithms achieve the best known regret bounds for mean-variance MABs and also attain the information-theoretic bounds in some parameter regimes. Empirical simulations show that our algorithms significantly outperform existing LCB-based algorithms for all risk tolerances.

Classical planning has been notably successful in synthesizing finite plans to achieve states where propositional goals hold. In the last few years, classical planning has also been extended to incorporate temporally extended goals, expressed in temporal logics such as LTL, to impose restrictions on the state sequences generated by finite plans. In this work, we take the next step and consider the computation of infinite plans for achieving arbitrary LTL goals. We show that infinite plans can also be obtained efficiently by calling a classical planner once over a classical planning encoding that represents and extends the composition of the planning domain and the Buchi automaton representing the goal. This compilation scheme has been implemented and a number of experiments are reported.