Numerical methods for decision: dynamic preferences and multivariate risks – DREAMES

In presence of abrupt (financial crisis or epidemics) or long-term (environmental or demographic) changes, one needs to use dynamic tools, to detect such changes from observable data, and to re-estimate models and risk quantification parameters, based on a dynamic and long-term view. Classical decision theory relies on a backward approach with given deterministic utility criteria. The drawbacks are twofold: first it does not incorporate any changes in the agents’ preferences, or any uncertain evolution of the environment variables. Furthermore, it leads to time-inconsistency and to optimal choices that depend on a fixed time-horizon related to the optimization problem. The framework of dynamic utilities is adapted to solve the issues raised above, by taking into account various risks and by proposing long-term, time-coherent policies. Dynamic utilities allow us to define adaptive strategies adjusted to the information flow, in non-stationary and uncertain environment. Therefore, the dynamic preferences framework provides a general and flexible framework in order to evaluate the impacts of short and long-term changes and to combine various risk parameters. Members of the team have worked since several years on this notion of dynamic utilities and they are now recognized as experts on this field.
In a complex and random environment, decision rules cannot be based on too simple criteria, and some economic approximations lead to optimal choices that are based on linear, or at best quadratic, cost-benefit analysis over time, and which can result in an underestimation of extreme risks. A general stochastic formulation and numerical estimation is useful to question the robustness of the theory.
The aim of this research project consists in proposing efficient numerical methods based on this theoretical framework. The main objectives are optimal detection of tendency changes in the environment, and optimization of economic actors’ decisions using dynamic preference criteria.
1) First, we aim at simulating dynamic utilities, which leads to various numerical challenges. They are related to non-linear forward second order HJB-Stochastic Partial Differential Equations, for which the standard numerical schemes are complex and unstable. We propose different methods for simulating these SPDEs, based on the stochastic characteristic method and neural networks.
2) We detect the transition point and study extreme scenarios in a context of multivariate risks. One possibility to overcome the short-term view of insurance and financial regulations is to consider hitting probabilities over a long-term or infinite horizon. In a multivariate setting, one quickly faces problems of estimation of the dependence structure between risks, as well as heavy computation times. It is non-trivial to detect changes in the risk processes as quickly as possible in presence of multiple sensors. We develop computing algorithms for hitting probabilities and other risk measures in a multivariate setting and in presence of changes in the parameters. We also obtain optimal risk mitigation techniques, using numerical methods.
3) We aim at calibrating dynamic utilities, that should be adapted to the evolving environment characterized by a multivariate source of risks. It consists in learning the decision maker’s preferences, and predict her behavior, based on an observed sequence of decisions. In the meantime, one also need to implement advanced statistical tools to calibrate the multivariate stochastic processes governing the environment.
4) We develop robust decision-making tools, for better handling model uncertainty in the worst case, including uncertainties on volatilities and correlations as well as jumps and moral hazard. We aim to study theoretical and numerical aspects for dynamic utilities under model uncertainty. It addresses the issues of moral hazard and ambiguity in model specification as well as in preferences and investment horizon specification.