We define and develop an approach for risk budgeting allocation - a risk diversification portfolio strategy - where risk is measured using a dynamic time-consistent risk measure. Specifically, a dynamic risk budgeting strategy is a portfolio allocation where at every trading point each asset contributes a predefined percentage to the future risk of the portfolio. For this, we introduce a notion of dynamic risk contributions that generalise the classical Euler contributions and which allow us to obtain dynamic risk contributions in a recursive manner. We prove that, for the class of dynamic coherent distortion risk measures, the risk allocation problem may be recast as a sequence of strictly convex optimisation problems. Moreover, we show that any self-financing dynamic risk budgeting strategy with initial wealth of 1 is a scaled version of the unique solution of the sequence of convex optimisation problems. Furthermore, we develop an actor-critic approach, leveraging the notion of conditional elicitability of dynamic risk measures, to solve for risk budgeting strategies using deep learning. We illustrate the methodology on a simulated market model, with a dynamic risk measures constructed through conditional Expected Shortfalls, discuss that the dynamic risk budgeting strategy is non-Markovian and how the agent invests.

Inverse optimization is a powerful tool for understanding decision-making processes by inferring the objective functions behind observed decisions. Traditional approaches rely heavily on parametric models, which assume the objective function's structure can be precisely known. This assumption can be impractical and prone to model mis-specification in complex real-world scenarios. Addressing this challenge, we introduce a novel nonparametric framework for inverse optimization. Our approach stands out by its flexibility to model a large class of objective functions without predefined parametric structures and that it can be implemented in data-efficient fashion. We prove the computational tractability of the nonparametric models and demonstrate their advantages via numerical experiments.

In this work, we consider a multistage expected utility maximization problem where the decision maker's utility function at each stage depends on historical data and the information on the true utility function is incomplete. To mitigate {adverse impact} arising from ambiguity of the true utility, we propose a maximin robust model where the optimal policy is based on the worst-case sequence of utility functions from an ambiguity set constructed with partially available information about the decision maker's preferences. We then show that the multistage maximin problem is time consistent when the utility functions are historical-path-dependent and demonstrate with a counter example that the time consistency may not be retained when the utility functions are historical-path-independent. With the time consistency, we show the maximin problem can be solved by a recursive formula whereby a one-stage maximin problem is solved at each stage beginning from the last stage. Moreover, we propose two approaches to construct the ambiguity set: a pairwise comparison approach and a zeta-ball approach where a ball of utility functions centered at a nominal utility function under zeta-metric is considered. To overcome the difficulty arising from solving the infinite dimensional optimization problem in computation of the worst-case expected utility value, we propose piecewise linear approximation of the utility functions and derive error bound for the approximation under moderate conditions. Finally, we use the stochastic dual dynamic programming (SDDP) method and the nested Benders' decomposition method to solve the multistage historical-path-dependent preference robust problem and the scenario tree method to solve the historical-path-independent problem, and carry out comparative analysis on the efficiency of the computational schemes as well as out-of-sample performances of the historical-path-dependent and historical-path-independent models. The preliminary results show that the historical-path-dependent preference robust model solved by SDDP algorithm displays overall superiority. This is a joint work with Jia Liu and Zhiping Chen in Xi'an Jiaotong University, China.

In behavioural economics, a decision maker’s preferences are expressed by choice functions. Preference robust optimization (PRO) is concerned with problems where the true choice function which represents the decision maker’s preferences is ambiguous, and the optimal decision is based on the worst-case choice function from a set of plausible choice functions constructed with elicited preference information. In this paper, we propose a PRO model to support choice functions that are: (i) monotonic (prefer more to less), (ii) quasi-concave (prefer diversification), and (iii) multi-attribute (have multiple objectives/criteria). As a main result, we show that the robust choice function can be constructed efficiently solved by solving a sequence of linear programming problems. In decision making with worst-case choice function, we demonstrate how the maximin problem can be efficiently solved by a sequence of convex optimization problems. To examine the the behavior and scalability of the proposed model and computational schemes, we apply them to a portfolio optimization problem and a capital allocation problem and report the numerical results.