Optimization

Stochastic Dominance Constrained Optimization with S-shaped Utilities: Poor-Performance-Region Algorithm and Neural Network ArXiv ID: 2512.00299 “View on arXiv” Authors: Zeyun Hu, Yang Liu Abstract We investigate the static portfolio selection problem of S-shaped and non-concave utility maximization under first-order and second-order stochastic dominance (SD) constraints. In many S-shaped utility optimization problems, one should require a liquidation boundary to guarantee the existence of a finite concave envelope function. A first-order SD (FSD) constraint can replace this requirement and provide an alternative for risk management. We explicitly solve the optimal solution under a general S-shaped utility function with a first-order stochastic dominance constraint. However, the second-order SD (SSD) constrained problem under non-concave utilities is difficult to solve analytically due to the invalidity of Sion’s maxmin theorem. For this sake, we propose a numerical algorithm to obtain a plausible and sub-optimal solution for general non-concave utilities. The key idea is to detect the poor performance region with respect to the SSD constraints, characterize its structure and modify the distribution on that region to obtain (sub-)optimality. A key financial insight is that the decision maker should follow the SD constraint on the poor performance scenario while conducting the unconstrained optimal strategy otherwise. We provide numerical experiments to show that our algorithm effectively finds a sub-optimal solution in many cases. Finally, we develop an algorithm-guided piecewise-neural-network framework to learn the solution of the SSD problem, which demonstrates accelerated convergence compared to standard neural network approaches. ...

Optimal Benchmark Design under Costly Manipulation ArXiv ID: 2506.22142 “View on arXiv” Authors: Ángel Hernando-Veciana Abstract Price benchmarks are used to incorporate market price trends into contracts, but their use can create opportunities for manipulation by parties involved in the contract. This paper examines this issue using a realistic and tractable model inspired by smart contracts on blockchains like Ethereum. In our model, manipulation costs depend on two factors: the magnitude of adjustments to individual prices (variable costs) and the number of prices adjusted (fixed costs). We find that a weighted mean is the optimal benchmark when fixed costs are negligible, while the median is optimal when variable costs are negligible. In cases where both fixed and variable costs are significant, the optimal benchmark can be implemented as a trimmed mean, with the degree of trimming increasing as fixed costs become more important relative to variable costs. Furthermore, we show that the optimal weights for a mean-based benchmark are proportional to the marginal manipulation costs, whereas the median remains optimal without weighting, even when fixed costs differ across prices. ...

Stochastic Gradient Descent in the Optimal Control of Execution Costs ArXiv ID: 2412.12199 “View on arXiv” Authors: Unknown Abstract Bertsimas and Lo’s seminal work laid the groundwork for addressing the implementation shortfall dilemma in institutional investing, emphasizing the significance of market microstructure and price dynamics in minimizing execution costs. However, the ability to derive a theoretical Optimum market order policy is an unrealistic assumption for many investors. This study aims to bridge this gap by proposing an approach that leverages stochastic gradient descent (SGD) to derive alternative solutions for optimizing execution cost policies in dynamic markets where explicit mathematical solutions may not yet exist. The proposed methodology assumes the existence of a mathematically derived optimal solution that is a function of the underlying market dynamics. By iteratively refining strategies using SGD, economists can adapt their approaches over time based on evolving execution strategies. While these SGD-based solutions may not achieve optimality, they offer valuable insights into optimizing policies under complex market frameworks. These results serve as a bridge for economists and mathematicians, facilitating the study of the Optimum policy volatile markets while offering SGD driven implementable policies that closely approximate optimal outcomes within shorter time frames. ...

Multilevel Monte Carlo in Sample Average Approximation: Convergence, Complexity and Application ArXiv ID: 2407.18504 “View on arXiv” Authors: Unknown Abstract In this paper, we examine the Sample Average Approximation (SAA) procedure within a framework where the Monte Carlo estimator of the expectation is biased. We also introduce Multilevel Monte Carlo (MLMC) in the SAA setup to enhance the computational efficiency of solving optimization problems. In this context, we conduct a thorough analysis, exploiting Cramér’s large deviation theory, to establish uniform convergence, quantify the convergence rate, and determine the sample complexity for both standard Monte Carlo and MLMC paradigms. Additionally, we perform a root-mean-squared error analysis utilizing tools from empirical process theory to derive sample complexity without relying on the finite moment condition typically required for uniform convergence results. Finally, we validate our findings and demonstrate the advantages of the MLMC estimator through numerical examples, estimating Conditional Value-at-Risk (CVaR) in the Geometric Brownian Motion and nested expectation framework. ...

Optimal Linear Signal: An Unsupervised Machine Learning Framework to Optimize PnL with Linear Signals ArXiv ID: 2401.05337 “View on arXiv” Authors: Unknown Abstract This study presents an unsupervised machine learning approach for optimizing Profit and Loss (PnL) in quantitative finance. Our algorithm, akin to an unsupervised variant of linear regression, maximizes the Sharpe Ratio of PnL generated from signals constructed linearly from exogenous variables. The methodology employs a linear relationship between exogenous variables and the trading signal, with the objective of maximizing the Sharpe Ratio through parameter optimization. Empirical application on an ETF representing U.S. Treasury bonds demonstrates the model’s effectiveness, supported by regularization techniques to mitigate overfitting. The study concludes with potential avenues for further development, including generalized time steps and enhanced corrective terms. ...

Causal Inference on Investment Constraints and Non-stationarity in Dynamic Portfolio Optimization through Reinforcement Learning ArXiv ID: 2311.04946 “View on arXiv” Authors: Unknown Abstract In this study, we have developed a dynamic asset allocation investment strategy using reinforcement learning techniques. To begin with, we have addressed the crucial issue of incorporating non-stationarity of financial time series data into reinforcement learning algorithms, which is a significant implementation in the application of reinforcement learning in investment strategies. Our findings highlight the significance of introducing certain variables such as regime change in the environment setting to enhance the prediction accuracy. Furthermore, the application of reinforcement learning in investment strategies provides a remarkable advantage of setting the optimization problem flexibly. This enables the integration of practical constraints faced by investors into the algorithm, resulting in efficient optimization. Our study has categorized the investment strategy formulation conditions into three main categories, including performance measurement indicators, portfolio management rules, and other constraints. We have evaluated the impact of incorporating these conditions into the environment and rewards in a reinforcement learning framework and examined how they influence investment behavior. ...