Optimal timing for financial and economic decisions under adverse and stressful conditions

Stochastic control theory can be viewed as the mathematical theory of controlling a stochastic process, which models the dynamics of a physical phenomenon, in view of optimising a certain criterion. It has found applications in finance, economics, physics, engineering and biology, which makes any new development in the theory quite important. This proposal will focus on two novel types of problems from the subclasses of optimal stopping theory, where control takes the form of an one-off stopping, and of the theory of stochastic control games. This research will address the timing of decision making by different market perspectives, namely by individuals, businesses, financial institutions and governmental bodies, in the setting of adverse and stressful conditions that have not been mathematically treated before. It will therefore also extend the application span of this well-established theory in the world of finance and economics, as well as attempt to bridge it with social sciences, such as behavioural economics, government policy and macroeconomics. The main objective is that the results of this work will give a review of different market participants' reactions and the impact of their decisions' on the success of their strategies, but also on the general public.

In particular, the optimal decision timing when the decision makers have time-restrictions, due to their intolerance of adverse market movements or their impatience when their assets do not perform well for a significant amount of time, will be mathematically formulated and solved as two innovative time-constrained optimal stopping problems. Different optimisation criteria will be considered dealing with a diverse spectrum of financial settings, e.g. intolerance to credit events, closure of trading accounts or redundancy of an asset manager when underperforming, need for an early liquidation, compulsory exit from a non-sustainable project or voluntary abandonment of a low-performing one. In addition, different stochastic processes will be used to model the evolution of asset values, e.g. (continuous) diffusion models, or Levy models with jumps.

Finally, this proposal will study a game of controlling the government's debt-to-GDP ratio between the government itself and its bond holders, whose actions affect the level (singular control) and its dynamics (classical control), respectively. The government aims to control its debt-to-GDP ratio in view of minimising derived costs, while it also needs to consider the adverse behaviour of the holders of government bonds, who trade them to optimise their individual criterion. Mathematically, this translates to a non-zero-sum game of classical-singular stochastic control, a novel setting in the existing literature. This governmental task is important both for the government itself, which wants to prevent the direct multiple unpleasant consequences of a high debt-to-GDP ratio, and for the country's citizens, whose lives are indirectly affected in an economically negative way. In modern finance, this work may also find applications in controlling a company's share price, portfolio's value, or company's debt-to-equity ratio, only to name a few.

The UK and London in particular is one of the world's largest financial centres. As such, the academic community bears the responsibility to support the nation's strategic position with groundbreaking financial and mathematical research. This proposal aims to contribute to the vision through research on time-constrained optimal stopping and game-type stochastic control theory, which focus on the optimal timing of decision making as a response to adverse market changes. With publications in high-impact journals and actively making this work accessible though conference talks and seminars, this proposal will help advance research in this field, setting the infrastructure for cross-disciplinary collaboration between financial mathematics, behavioural economics and government policy studies.

Results will also impact practitioners (individuals, businesses and financial institutions), which are either completely intolerant to specific adverse market movements, or impatient when their assets are underperforming for a significant amount of time. The tools provided will help optimise their decision making processes, as well as support the regulatory decisions on acceptable market drops by the Commodity Futures Trading Commission and Commodity Trading Advisors. From a government perspective, results will contribute to the methodology of quantifying the effects of controlling the government's debt-to-GDP ratio, thus help shape policies regarding its optimal control against adverse trading of its bonds. The governmental, societal and economic importance of controlling this ratio is paramount.