Optimal dynamic portfolio with risk limits

The paper studies connections between arbitrage and utility maximization in a discrete-time financial market. The market is incomplete. Thus one has several choices of equivalent martingale measures to price contingent claims. Davis determines a unique price for a contingent claim which is based on an optimal dynamic portfolio by use of a `marginal rate of substitution' argument. Here conditions will be given such that this price is determined by a martingale measure and thus by a consistent price system. The underlying utility function U is defined on the positive half-line. Then dynamic portfolios are admissible if the terminal wealth is positive. In case of the logarithmic utility function, the optimal dynamic portfolio is the numeraire portfolio.

This thesis consists of three essays in financial economics, more precisely in the field of asset pricing and optimal portfolio choice. The first essay studies the asset pricing implications of heterogeneity in preferences coupled with long run risk in a dynamic general equilibrium pure exchange economy. It is concerned with the understanding of how risk sharing among different types of agents affects the assets returns moments in equilibrium and moreover, how these agents reacts dynamically to the long run risk that they perceive. The second and third essay approach the question of optimal portfolio with stochastic opportunity sets. The second essay proposes an approximation methodology for deriving approximate closed form solutions of dynamic optimal portfolio allocation problems. The third essay examines the optimal consumption, portfolio rule and life insurance purchase problem of a wage earner having uncertain lifetime when the market price of risk follows a mean-reverting stochastic process.

The paper studies connections between risk aversion and martingale measures in a discrete‐time incomplete financial market. An investor is considered whose attitude toward risk is specified in terms of the index b of constant proportional risk aversion. Then dynamic portfolios are admissible if the terminal wealth is positive. It is assumed that the return (risk) processes are bounded. Sufficient (and nearly necessary) conditions are given for the existence of an optimal dynamic portfolio which chooses portfolios from the interior of the set of admissible portfolios. This property leads to an equivalent martingale measure defined through the optimal dynamic portfolio and the index 0 < b≤ 1. Moreover, the option pricing formula of Davis is given by this martingale measure. In the case of b= 1; that is, in the case of the log‐utility, the optimal dynamic portfolio defines the numéraire portfolio.

In this paper, we consider the problem of an optimal pension fund portfolio given the heterogeneous risk preferences of pension fund participants. The relative risk aversion of a pension fund tends to be a decreasing function of the level of aggregate wealth. We find that the dynamic optimal portfolio is simply characterized as the weighted sum of the optimal portfolio for each participant. Our model helps successfully establish the microfoundation of asset liability management models. A numerical example using recent Japanese data indicates the significant total welfare losses of adopting a suboptimal portfolio strategy and an inefficient risk‐sharing rule.

Realized Covariance Estimation in Dynamic Portfolio Optimization

Dynamic portfolio optimization with transaction costs and state-dependent drift

Deep reinforcement learning (DRL) has been extensively used to address portfolio optimization problems. DRL agents acquire knowledge and make decisions through unsupervised interactions with their environment without requiring explicit knowledge of the joint dynamics of portfolio assets. Among these DRL algorithms, the combination of actor-critic algorithms and deep function approximators is the most widely used DRL algorithm. Here, we find that training the DRL agent using the actor-critic algorithm and deep function approximators may lead to scenarios where the improvement in the DRL agent's risk-adjusted profitability is insignificant. We argue that such situations primarily arise from the following two problems: sparsity in positive reward and the curse of dimensionality. These limitations prevent DRL agents from comprehensively learning asset price change patterns in the training environment. As a result, the DRL agents cannot effectively explore the dynamic portfolio optimization policy to improve the risk-adjusted profitability in the training process. To address these problems, we propose a novel multi-agent learning system based on the hierarchical deep reinforcement learning (HDRL) algorithmic framework in this research. Under this framework, the agents work together as a learning system for portfolio optimization. Specifically, by designing an auxiliary agent that works together with the executive agent for optimal policy exploration, the learning system can focus on exploring the policy with higher risk-adjusted return in the action space with positive return and low variance. The performance of the proposed learning system is evaluated using a portfolio of 29 stocks from the Dow Jones index in four different experiments. In the training process, the objective functions of the actor and critic both ultimately achieve stable convergence in the training process. The risk-adjusted profitability of our learning system in the training environment is significantly improved. Hence, we prove that the policies executed by our learning system in out-sample experiments originate from the DRL agents' comprehensive learning of asset price change patterns in the training environment. Furthermore, we find that adopting the auxiliary agent and HDRL training algorithm can efficiently overcome the issue of the curse of dimensionality and improve the training efficiency in the positive reward sparse environment. In each back-test experiment, the proposed learning system is compared to sixteen traditional strategies and ten strategies based on machine learning algorithms in the performance of profitability and risk control ability. The empirical results in the four evaluation experiments demonstrate the efficacy of our learning system, which outperforms all other strategies by at least 8.2% in terms of Sharpe ratio, Sorino ratio, and Calmar ratio. This indicates that the policies learned in the training environment can exhibit excellent generalization ability in the back-testing experiments.

We study the optimal dynamic portfolio exposure to predictable default risk, taking inspiration from the search for yield by means of defaultable assets observed before the 2007–2008 crisis and in its aftermath. Under no arbitrage, default risk is compensated by an ‘yield pickup’ that can strongly attract aggressive investors via an investment-horizon effect in their optimal non-myopic portfolios. We show it by stating the optimal dynamic portfolio problem of Kim and Omberg (Rev Financ Stud 9:141–161, 1996) for a defaultable risky asset and by rigorously proving the existence of nirvana-type solutions. We achieve such a contribution to the portfolio optimization literature by means of a careful, closed-form-yielding adaptation to our defaultable asset setting of the general convex duality approach of Kramkov and Schachermayer (Ann Appl Probab 9(3):904–950, 1999; Ann Appl Probab 13(4):1504–1516, 2003).

This paper concerns the problem of portfolio optimization in the context of ultra-high frequency environment with dynamic and frequent changes in statistics of financial assets. It aims at providing Pareto fronts of optimal portfolios and updating them when estimated return rates or risks of financial assets change. The problem is defined in terms of dynamic optimization and solved online with a proposed evolutionary algorithm. Experiments concern ultra-high frequency time series coming from the London Stock Exchange Rebuilt Order Book database and the FTSE100 index.

This paper introduces a novel hybrid framework, ICEEMDAN-PatchTST+FDPO, which integrates advanced signal decomposition, Transformer-based time series forecasting, and forecast-driven portfolio optimization (FDPO). Improved Complete Ensemble Empirical Mode Decomposition with Adaptive Noise (ICEEMDAN) enhances signal stationarity by decomposing noisy financial series into intrinsic mode functions, while PatchTST employs patch-wise self-attention to capture complex temporal dependencies. These components enable precise multi-step return forecasting, which is subsequently operationalized through FDPO to generate dynamically optimized allocations aligned with predicted market movements. Extensive out-of-sample evaluations from January 2022 to December 2024 on a rule-based selection of ten representative NIFTY 50 constituents demonstrate that the proposed framework delivers substantial gains over conventional baselines (EqualWeight, Mean–Variance Optimization, Risk Parity) and alternative deep learning models (LSTM, TimesNet, NBEATSx, Informer) in both forecasting accuracy and portfolio performance. The model achieves the lowest MAE (38.25), RMSE (55.92), and MAPE (2.06%), with an <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">R</i><sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">2</sup> of 0.9776. It also attains a CAGR of 1.8453, a Sharpe ratio of 3.615, a Sortino ratio of 3.985, and the lowest maximum drawdown of –0.0971. Alignment metrics such as Cosine Similarity and Jensen–Shannon Divergence further indicate that its weight vectors closely approximate ideal allocations based on actual future returns. A supplementary validation on a rule-based S&P 500 subset corroborates these trends, suggesting that the framework maintains its performance advantages across distinct market environments. Overall, the findings highlight the practical viability of the ICEEMDAN-PatchTST+FDPO framework for robust and adaptive asset management under real-world market dynamics.

This chapter presents a dynamic optimization model under regime-switching market uncertainty to manage stock portfolios. The novelty of the approach is that stocks for long and short investments are selected based on the notion of fundamental business strength and consistent with specific regime scenarios. Firm-strength is determined via the quarterly financial statements of a firm, relative to other firms in the same market sector/industry. Referred to as Relative Firm Strength (RFS), it is determined using Data Envelopment Analysis (DEA) in which various financial metrics from accounting statements are specified as inputs and outputs. It is shown that such an RFS measure has predictive power of stock returns in the historical data period 1971–2010. The regime-switching market model is combined with the RFS metric of stock selections to test the performance of a two-quarter stochastic portfolio optimization model during Jan–Jun, 2011. The out-of-sample results demonstrate that the proposed methodology is superior to using sector-based ETF portfolios or the market index itself.

Optimal dynamic portfolio with risk limits

By this study, we want to provide tools for an economic agent who wants to maintain her maximized intertemporal consumption rate over time period regularly by trading activity in capital market. She would use the diversification principle to compose portfolio that is consisted of risky and risk free assets, with initial wealth by using principle of dynamic programming in stochastic control problem. While our target function is to maximize the discounted utility function over consumption rate, the agent may use this portion of wealth for hedonistic lifestyle without thinking the bequest aim. Motivated by observation that seminal work of Samuelson (1969)’s optimal investment on risky asset was independent of time and wealth process, we were replacing the source of uncertainty in the risky asset by binomial process in order to improve Samuelson's solution. The risky asset price evolution is determined to move in two possible directions of up and down in the next time instantly with certain probability, following binomial trees scheme. Working within this setting, we manage to obtain trading strategies as explicit solution for the optimal investment and consumption decisions which were being expressed as time-dependent functions as well as wealth process feedback form. This solution goes along with inter-temporal concept that our consumption and investment strategies tomorrow will be affected with our wealth today. We exemplify the results by numerical examples and perform Monte Carlo simulation.

This study addresses the challenges of market non-stationarity, risk uncertainty, and dynamic inter-asset relationships in dynamic portfolio optimization by proposing an adaptive investment decision model based on multi-agent reinforcement learning. The research constructs a multi-agent architecture that combines centralized training with decentralized execution, allowing each agent to perform local strategy optimization while sharing global information to achieve a dynamic balance between return and risk. The model embeds multi-dimensional features such as market states, price fluctuations, trading volume, and risk indicators into the reinforcement learning framework, where a reward function drives policy iteration and enhances learning efficiency and decision stability in complex environments. A dynamic risk penalty mechanism and return adjustment term are introduced to effectively suppress excessive risk-taking behavior under high volatility and improve system robustness. Experimental results show that the proposed model outperforms traditional reinforcement learning methods in annualized return, maximum drawdown, Sharpe ratio, and Hit Ratio, maintaining strong return stability and risk control in multi-asset and multi-timescale trading environments. The findings confirm the effectiveness of multi-agent collaborative learning in dynamic asset allocation and risk-constrained optimization, providing methodological support for building intelligent and data-driven investment decision systems.

New Models and Methods in Dynamic Portfolio Optimization