Noether Theorem in Stochastic Optimal Control Problems via Contact Symmetries

We establish a generalization of the Noether theorem for stochastic optimal control problems. Exploiting the tools of jet bundles and contact geometry, we prove that from any (contact) symmetry of the Hamilton–Jacobi–Bellman equation associated with an optimal control problem it is possible to build a related local martingale. Moreover, we provide an application of the theoretical results to Merton’s optimal portfolio problem, showing that this model admits infinitely many conserved quantities in the form of local martingales.

Effect of the Profile of the Decision Maker in the Search for Solutions in the Decision-Making Process

Many real-world optimization problems involving several conflicting objective functions frequently appear in current scenarios and it is expected they will remain present in the future. However, approaches combining multi-objective optimization with the incorporation of the decision maker’s (DM’s) preferences through multi-criteria ordinal classification are still scarce. In addition, preferences are rarely associated with a DM’s characteristics; the preference selection is arbitrary. This paper proposes a new hybrid multi-objective optimization algorithm called P-HMCSGA (preference hybrid multi-criteria sorting genetic algorithm) that allows the DM’s preferences to be incorporated in the optimization process’ early phases and updated into the search process. P-HMCSGA incorporates preferences using a multi-criteria ordinal classification to distinguish solutions as good and bad; its parameters are determined with a preference disaggregation method. The main feature of P-HMCSGA is the new method proposed to associate preferences with the characterization profile of a DM and its integration with ordinal classification. This increases the selective pressure towards the desired region of interest more in agreement with the DM’s preferences specified in realistic profiles. The method is illustrated by solving real-size multi-objective PPPs (project portfolio problem). The experimentation aims to answer three questions: (i) To what extent does allowing the DM to express their preferences through a characterization profile impact the quality of the solution obtained in the optimization? (ii) How sensible is the proposal to different profiles? (iii) How much does the level of robustness of a profile impact the quality of final solutions (this question is related with the knowledge level that a DM has about his/her preferences)? Concluding, the proposal fulfills several desirable characteristics of a preferences incorporation method concerning these questions.

Machine learning-based allocation of renewable power production

<p>The integration of renewable energy sources into the power grid is of the utmost importance for achieving the goal of zero carbon emission. Although there are feasibility studies showing that renewable energy might be able to cover 2050 global energy demand using less than 1 % of the world's land for footprint and spacing, see Jacobson and Delucchi (2011), nowadays renewable energy production is known to be highly intermittent due to substantial uncertainties in the weather conditions. One possibility to reduce such uncertainty (besides storage and employing hydrogen technologies) is spatiotemporally diversified allocation of renewable power capacities which (alongside with the transmission infrastructure) should guarantee that the power demand is met at any given time with a certain (high) probability. We treat the question of spatiotemporal diversification of renewable capacities as a Markowitz portfolio problem with the difference that instead of n = 1, …, N stocks we have geographical locations each with a certain expected level of renewable power production (instead of expected returns for stocks) and the corresponding variance. Another difference to a classical Markowitz portfolio problem is that we require additionally that at each given time point t = 1, …, T, we can reach a predetermined level of renewable power production with a certain probability, i. e. we solve so called chance-constrained problem. Finally, instead of solving one-step problem as it is the case with a Markowitz portfolio we reformulate our problem in the optimal control framework in continuous time and solve it with a reinforcement learning algorithm as suggested in Lillicrap et al. (2019). The advantage of this approach is that the optimal capacities (control) are updated continuously as a response to changing weather conditions (state). We exemplify our approach with the data from ERA5 data, see Hersbach et al. (2020), and suggest possible allocation of renewable energy sources across the European Union.</p>

The Probabilistic Risk Measure VaR as Constraint in Portfolio Optimization Problem

The paper realizes inclusion of probabilistic measure for risk, VaR (Value at Risk), into a portfolio optimization problem. The formal analysis of the portfolio problem illustrates the evolution of the portfolio theory in sequentially inclusion of different market characteristics into the problem. They make modifications and complications of the portfolio problem by adding various constraints to consider requirements for taxes, boundaries for assets, cardinality constraints, and allocation of the investment resources. All these characteristics and parameters of the investment participate in the portfolio problem by analytical algebraic relations. The VaR definition of the portfolio risk is formalized in a probabilistic manner. The paper applies approximation of such probabilistic constraint in algebraic form. Geometrical interpretation is given for explaining the influence of the VaR constraint to the portfolio solution. Numerical simulation with data of the Bulgarian Stock Exchange illustrates the influence of the VaR constraint into the portfolio optimization problem.

Explicit Value at Risk Goal Function in Bi-Level Portfolio Problem for Financial Sustainability

The mean-variance (MV) portfolio optimization targets higher return for investment period despite the unknown stochastic behavior of the future asset returns. That is why a risk is explicitly considering, quantified by algebraic characteristics of volatilities and co-variances. A new probabilistic definition of portfolio risk is the Value at Risk (VaR). The paper makes explicit inclusion and minimization of VaR as a quantitative measure of financial sustainability of a portfolio problem. Thus, the portfolio weights as problem solutions will respect not only the MV requirements for risk and return, but also the additional minimization of risk defined by VaR level. The portfolio problem is defined in a new, bi-level form. The upper level minimizes and evaluates the VaR value. The lower level evaluates the optimal assets weights by minimizing portfolio risk and maximizing the return in MV form. The bi-level model allows to have extended set of portfolio solutions with the portfolio weights and the value of VaR. Graphical interpretation of this bi-level definition of the portfolio problem explains the differences with the MV portfolio definition. Thus, the bi-level portfolio problem evaluates the optimal weights, which makes maximization of portfolio return and minimization of the risk in its algebraic and probabilistic form of definition.