Open research questions in Risk and Portfolio Optimization

What the literature leaves open

• While the proposed E-MOPSO algorithm demonstrates significant improvements in reinsurance contract optimization, some limitations should be acknowledged, which future work can address. Firstly, the scalability is limited since the experiment was conducted in a 7-layered real-world contract and a synthetically extended 15-layered contract. While the results are promising, the scalability of E-MOPSO to contracts with significantly more layers, for instance, 50 or 100 layers, remains untested. Secondly, the 15-layered dataset was synthetically generated from the original 7-layered dataset. While this approach allows for testing scalability, it may not fully capture the complexities and nuances of real-world 15-layered contracts. Thirdly, the optimization process relies on specific risk metrics, such as Value at Risk (VaR), which may not fully capture all aspects of risk in real-world scenarios. For instance, VaR does not account for tail risk or extreme events beyond the chosen confidence level. Fourthly, the generalization to other domains is not totally clear. While E-MOPSO is designed for reinsurance contract optimization, its applicability to other multiobjective optimization problems, such as mean-variance portfolio optimization and supply chain management, has not been explored. Finally, due to the scope of this work, the hyperparameters for E-MOPSO were set to the canonical, literaturebacked values of ω = 0.9 and c1 = c2 = 2.0 to ensure stability and a balanced search behavior. However, the performance of the algorithm could potentially be enhanced by conducting a comprehensive parameter sensitivity analysis.

  A Novel Enhanced Multiobjective PSO Algorithm for Reinsurance Analytics · 2026 · DOI

• In this paper we presented a new generalized version of the generalized by two parameters, (η, p, h)-convex stochastic process, which generalizes and brings together that of a number of existing concepts of convexity in the stochastic context. Basic properties of this class were stud- ied and new Ostrowski-, Jensen-, and Hermite-type inequalities were obtained. The theoretical conclusions were demonstrated by non-trivial cases and graphs, which demonstrated the role of stochastic elements, especially, the Wiener processes, as compared to deterministic elements. Int. J. Anal. Appl. (2026), 24:164 21 We establish that in addition to a number of classical stochastic convexity theorems being recovered as special cases of the proposed framework, the proposed framework permits a degree of modeling and analysis flexibility in the analysis and description of stochastic systems under uncertainty. To conduct future research, a number of directions are prospective: • Generalizations to multiple dimensions Multidimensional stochastic processes and gener- alized to multidimensional Wiener processes. • Developing uses in stochastic optimization, stochastic differential equations and financial mathematics, where the convexity of (η,p,h) can provide a better bound and error estimate. • The further validation of the theoretical constraints in high-dimensional stochastic models through development of numerical methods and simulation-based research. • The investigation of relationships with other generalized convexity notions and stochastic integral inequalities, which may generate new classes of stochastic inequalities. On the whole, the suggested framework not only opens up the possibilities of the theoretical aspects but also has numerous practical applications to the stochastic analysis and other areas. Conflicts of Interest: The authors declare that there are no conflicts of interest regarding the publication of this paper.

  Novel Unified Variants, Properties, and Applications of Ostrowski, Jensen, and Hermite–Hadamard Inequalities for Generalized (η, p, h)-Convex Stochastic Processes · 2026 · DOI

• Multi-Agent Equilibria: Future work will explore how the presence of ”loss-averse” agents affects market volatility and equilibrium prices compared to fully rational agents. This is particularly relevant for mechanistic interpretability and safety analysis, where identifying ”risk-aware” components in complex architectures is an open challenge [3, 8].

  Perturbed Utility Functionals: A Functional-Analytic Framework for Adaptive Decision-Making · 2026 · DOI

• LIMITATIONS OF THE STUDY The overriding limitation of this study is that the fractional refinement is based on the convexity of the exponential integral function on the support of the underlying random variable, which restricts the possible distributions to those with support on an interval where the exponential integral function is convex.

  Fractional form of Jensens Inequality for the Exponential Integral Function with Applications to Modeling Utility · 2026 · DOI

• The proofs rely on the assumption that γ^{-1/2} and related processes have almost-surely continuous paths, but discontinuities arising from market dislocations, liquidity crises, or regime breaks could violate this regularity condition. Robustness of the optimal execution strategy to jumps or discontinuities in market impact and correlation matrices remains unexplored.

  Multi-asset optimal trade execution with stochastic cross-effects: An Obizhaeva–Wang-type framework · 2026 · DOI

• The framework incorporates stochastic cross-effects through the operator O and matrix Q, but empirical validation against real market data demonstrating the relevance of these cross-effect terms versus simpler single-asset or independent multi-asset models has not been provided. Calibration of the cross-effect parameters to actual equity, FX, or commodity market microstructure is absent.

  Multi-asset optimal trade execution with stochastic cross-effects: An Obizhaeva–Wang-type framework · 2026 · DOI

• Lemma 3.8 establishes a convergence result using truncated processes (H^N) that approximate the full operator H, but the rate of convergence and approximation error bounds are not explicitly quantified. A concrete analysis of how the truncation parameter N affects execution cost and optimal trading paths for realistic portfolio sizes is needed.

  Multi-asset optimal trade execution with stochastic cross-effects: An Obizhaeva–Wang-type framework · 2026 · DOI

• The paper proves well-definedness of the cost functional J_pm(X) and optimal execution paths under L2 admissibility constraints, but does not address numerical implementation or computational algorithms for solving the optimization problem with cross-asset execution effects. Practical solution schemes for the multi-asset system with general correlation structures remain unspecified.

  Multi-asset optimal trade execution with stochastic cross-effects: An Obizhaeva–Wang-type framework · 2026 · DOI

• The framework assumes bounded coefficients (μ, σ, and γ) across all proofs, but the behavior of the multi-asset optimal trade execution model under time-varying or unbounded market impact coefficients has not been rigorously analyzed. Extension to regimes where Obizhaeva–Wang-type parameters exhibit stochastic volatility or regime-switching dynamics requires new theoretical development.

  Multi-asset optimal trade execution with stochastic cross-effects: An Obizhaeva–Wang-type framework · 2026 · DOI

• The analysis of SMD performance for risk budgeting with ES focused on median absolute deviations computed over 100 repetitions from 100 distinct fitted models; the sensitivity of results to different model specification choices and the robustness of SMD across various market regimes and correlation structures remains unexplored.

  Mirror Descent Algorithms for Risk Budgeting Portfolios · 2026 · DOI

• While Table 6 and Figure 10 demonstrate SMD's superiority in computing portfolio weights via distance error (DE) metric, the practical impact of these improvements on out-of-sample portfolio performance, Sharpe ratio, and tail risk metrics for ES-based risk budgeting portfolios has not been evaluated.

  Mirror Descent Algorithms for Risk Budgeting Portfolios · 2026 · DOI

• The numerical experiments for risk budgeting portfolio computation via SMD were restricted to samples of size 10^6 and specific iteration points (k=3×10^5, 6×10^5, 9×10^5); the performance of SMD across varying sample sizes and a continuous range of iteration steps needs investigation to determine optimal stopping criteria and sample efficiency.

  Mirror Descent Algorithms for Risk Budgeting Portfolios · 2026 · DOI

• The comparison of SMD and t-SGD algorithms using alternative risk measures (MAD, volatility, and variantile as deviation measures) was initiated only under the assumption of centered normal distribution with three assets; this needs to be extended to Student t mixture models and larger asset universes to validate whether SMD's computational advantage generalizes across different risk measure classes.

  Mirror Descent Algorithms for Risk Budgeting Portfolios · 2026 · DOI

• The SMD algorithm's convergence advantage over t-SGD was demonstrated only on a three-asset model in Figure 8; this analysis must be extended to general cases and larger portfolio dimensions (beyond d=250) to establish broader conclusions about SMD's practical advantages in computing risk budgeting portfolios.

  Mirror Descent Algorithms for Risk Budgeting Portfolios · 2026 · DOI

• The paper assumes log-normal stock price dynamics and CRRA utility functions; applicability to other utility functions and alternative stochastic models for asset prices remains unexplored.

  Comparative statics of trading boundary in finite-horizon portfolio selection problem with proportional transaction costs · 2026 · DOI

• Sensitivity analysis, combined with parametric optimization, is often presented as a way of checking if the solution of a deterministic linear program is reliable—even if some of the parameters are not fully known but are instead replaced by a best guess, often a sample mean.

  Decision Making Under Uncertainty: Is Sensitivity Analysis of Any Use? · 2000 · DOI

• Any policy limited to actions {0, 1} necessarily loses value on the second interval; any policy limited to actions {0, 2} necessarily loses value on the first.

  A Risk–Utility Optimization Framework for Governing Large Language Model Responses · 2026 · DOI

• In summary, the isomorphism between belief and probability in Keynes’ framework cannot be demanded, as it remains undetermined when the notion of weak (relative) complementation is not predefined.

  A new approach for imprecise probabilities · 2026 · DOI

• There remain many open problems. For example, if unobservable drift follows a general prior distribution, not necessarily a Bernoulli distribution, the current change of measure approach no longer works, how can we solve such a problem? We leave this and other open problems for future research.

  S-shaped Utility Maximization with VaR Constraint and Partial Information · 2026 · DOI

• The condition β ≤ 1 is assumed in Lemma 2 for certain results; the case when β > 1 (higher discount rates) and its implications for trading boundaries remain unresolved.

  Comparative statics of trading boundary in finite-horizon portfolio selection problem with proportional transaction costs · 2026 · DOI

• Monotonicity results for other market parameters (beyond risk premium, risk aversion, and volatility) such as interest rates, time horizon effects, or asymmetric transaction costs are not systematically analyzed.

  Comparative statics of trading boundary in finite-horizon portfolio selection problem with proportional transaction costs · 2026 · DOI

• The theoretical results are demonstrated primarily through numerical examples with specific parameter values; broader empirical validation across diverse market conditions and parameter ranges is not provided.

  Comparative statics of trading boundary in finite-horizon portfolio selection problem with proportional transaction costs · 2026 · DOI

• Moreover, we find that simply using LASSO is insufficient to lower turnover when the model’s tuning parameter can change over time.

  Sparsity and stability for minimum-variance portfolios · 2022 · DOI

• The aim of this paper is to review various concepts of extremal positive and negative dependence, including several recently established results, reconstruct their history, link them to probabilistic optimization problems, and provide a list of open questions in this area.

  Extremal Dependence Concepts · 2015 · DOI

• In this paper, we propose tractable methods of addressing a general class of multistage stochastic optimization problems, which assume only limited information of the distributions of the underlying uncertainties, such as known mean, support, and covariance.

  A Linear Decision-Based Approximation Approach to Stochastic Programming · 2007 · DOI

Most-cited papers in Risk and Portfolio Optimization

• Distributionally Robust Convex Optimization · Operations Research · 2014 · 803 citations
• A Linear Decision-Based Approximation Approach to Stochastic Programming · Operations Research · 2007 · 195 citations
• Decision Making Under Uncertainty: Is Sensitivity Analysis of Any Use? · Operations Research · 2000 · 143 citations
• A survey of contextual optimization methods for decision-making under uncertainty · European Journal of Operational Research · 2024 · 111 citations
• Robust Satisficing · Operations Research · 2022 · 110 citations
• Extremal Dependence Concepts · Statistical Science · 2015 · 102 citations
• Technical Note—Data-Driven Chance Constrained Programs over Wasserstein Balls · Operations Research · 2022 · 95 citations
• Efficient Portfolio Selection with Quadratic and Cubic Utility · The Journal of Business · 1970 · 81 citations
• High dimensional minimum variance portfolio estimation under statistical factor models · Journal of Econometrics · 2020 · 71 citations
• Finite-Sample Guarantees for Wasserstein Distributionally Robust Optimization: Breaking the Curse of Dimensionality · Operations Research · 2022 · 55 citations

Most recent work

• A new approach for imprecise probabilities · Mathematical Social Sciences · 2026
• Counter-monotonic risk sharing with heterogeneous distortion risk measures · Insurance: Mathematics and Economics · 2026
• Nash Social Welfare with Submodular Valuations: Approximation Algorithms and Integrality Gaps · 2026
• Comparative statics of trading boundary in finite-horizon portfolio selection problem with proportional transaction costs · Mathematics and Financial Economics · 2026
• Dual Representations for Quasiconvex Compositions with Applications to Systemic Risk Measures · SIAM Journal on Financial Mathematics · 2026
• Mirror Descent Algorithms for Risk Budgeting Portfolios · Mathematics of Operations Research · 2026
• Covariance Matrices Under Sublinear Expectation and Application to Robust Portfolio Selection · Methodology and Computing in Applied Probability · 2026
• Multi-asset optimal trade execution with stochastic cross-effects: An Obizhaeva–Wang-type framework · Mathematics and Financial Economics · 2026
• Dual adaptive stochastic block projection algorithm for solving convex feasibility problem in support vector machines · Computational Mathematics and Modeling · 2026
• Stochastic Maximum Principle with Default · Applied Mathematics & Optimization · 2026

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