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

随机控制理论可以看作是控制随机过程的数学理论，它模拟了物理现象的动力学，以优化某一准则。它在金融、经济学、物理学、工程学和生物学中都有应用，这使得该理论的任何新发展都非常重要。本提案将集中于两种新型的问题，分别来自最优停止理论的子类，其中控制采用一次性停止的形式，以及随机控制博弈理论。这项研究将从不同的市场角度，即个人、企业、金融机构和政府机构，在不利和压力的条件下，在以前没有数学处理的情况下，解决决策的时机。因此，它还将扩大这一成熟理论在金融和经济领域的应用范围，并试图将其与社会科学，如行为经济学、政府政策和宏观经济学联系起来。主要目标是，这项工作的结果将对不同市场参与者的反应和他们的决策对其战略成功的影响以及对一般公众的影响进行审查。特别是，当决策者有时间限制时的最佳决策时机，由于他们对不利的市场运动的不容忍或当他们的资产在很长一段时间内表现不佳时的不耐烦，将被数学表述并作为两个创新的时间约束最佳停止问题来解决。不同的优化标准将被考虑处理不同的金融环境，例如对信贷事件的不容忍，交易账户的关闭或资产经理在表现不佳时的冗余，需要早期清算，强制退出不可持续的项目或自愿放弃表现不佳的项目。此外，将使用不同的随机过程来模拟资产价值的演变，例如（连续）扩散模型或具有跳跃的Levy模型。最后，本文将研究政府本身与债券持有人之间控制政府债务与gdp比率的博弈，其行为分别影响水平（单一控制）和动态（经典控制）。鉴于衍生成本最小化，政府的目标是控制其债务与gdp之比，同时它还需要考虑政府债券持有人的不利行为，他们交易政府债券以优化他们的个人标准。从数学上讲，这转化为经典奇异随机控制的非零和博弈，这是现有文献中的一个新颖设置。政府的这项任务对政府本身很重要，因为政府希望防止高债务与gdp之比带来的直接、多重不愉快的后果，对该国公民也很重要，因为他们的生活在经济上受到了间接的负面影响。在现代金融中，这项工作也可以应用于控制公司的股价，投资组合的价值，或公司的债务与权益比率，仅举几例。

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.