Stochastic optimal control in mathematical finance

数学金融中的随机最优控制

• 批准号: RGPIN-2018-03978
• 负责人: Heunis, Andrew
• 金额: $ 1.31万
• 依托单位: University of Waterloo
• 依托单位国家: 加拿大
• 项目类别: Discovery Grants Program - Individual
• 财政年份: 2022
• 资助国家: 加拿大
• 起止时间: 2022-01-01 至 2023-12-31
• 项目状态: 已结题
• 来源: https://www.nserc-crsng.gc.ca/ase-oro/Details-Detailles_eng.asp?id=751595
• 关键词: Stochastic; optimal; control; mathematical; finance

Over the last several decades the role of mathematical methods in financial decision making has steadily increased, to the point that a distinct discipline of research and practice has developed under the rubric of "mathematical finance" or "financial engineering", the general goal of which is to establish a scientific approach for the efficient allocation of capital in an economy. In this approach one first builds an idealized mathematical model (or description) of the central aspects of the financial market within which the allocation of capital takes place. Then, with reference to this mathematical model, one can formulate in precise terms the problems with which financial decision making is concerned. Of particular importance is the problem of allocating capital in order to minimize risk or maximize returns on the capital invested (this is known as portfolio optimization).The focus of our research program will be on portfolio optimization subject to constraints on capital investment. Constraints can take the form of direct restrictions on investment (usually known as "portfolio constraints"), which are often imposed by regulatory agencies (for example a prohibition on short selling certain designated securities), but may also take the form of indirect restrictions on investment, in particular the stipulation of a "floor-level" below which the wealth of an investor must never fall regardless of how the market evolves. This latter constraint enforces a hard limit on possible losses in the course of financial trading (this is known as "portfolio insurance"). Constraints on capital investment are a natural part of financial decision making and are therefore of clear importance, and indeed portfolio optimization subject to only direct restrictions on investment (i.e. portfolio constraints alone) has received significant attention in the established literature. Considerably less attention has been devoted to portfolio optimization when one has the combination of both direct restrictions on investment (i.e. portfolio constraints) together with indirect restrictions on investment in the form of a stipulated lower bound on wealth (i.e. portfolio insurance), possibly because this combination of constraints does exhibit some clear and definite challenges which are not encountered when one deals with only direct restrictions in investment (i.e. portfolio constraints). The goal of the research in this proposal is to address portfolio optimization with such combined constraints, building on an approach and some partial results which we have already established for this problem. We expect that this will contribute to the advancement of knowledge in financial engineering. In particular, constraints which define a lower bound on wealth serve to limit losses in the course of portfolio optimization, thus helping to stabilize financial trades, with clear benefits to the Canadian economy.

在过去的几十年里，数学方法在金融决策中的作用稳步增强，以至于在“数学金融”或“金融工程”的标题下发展出了一门独特的研究和实践学科，其总体目标是是建立一个科学的方法来有效配置经济中的资本。在这种方法中，人们首先建立资本配置发生的金融市场核心方面的理想化数学模型（或描述）。然后，参考这个数学模型，人们就可以精确地表述财务决策所涉及的问题。特别重要的是为了最小化风险或最大化投资资本回报而分配资本的问题（这称为投资组合优化）。我们研究计划的重点将是在资本投资约束下的投资组合优化。限制可以采取直接限制投资的形式（通常称为“投资组合限制”），通常由监管机构施加（例如禁止卖空某些指定证券），但也可能采取间接限制的形式投资，特别是“最低水平”的规定，无论市场如何发展，投资者的财富都不得低于该水平。后一个约束对金融交易过程中可能的损失施加了严格限制（这称为“投资组合保险”）。对资本投资的约束是财务决策的自然组成部分，因此具有明显的重要性，事实上，仅受投资直接限制（即仅投资组合约束）的投资组合优化在现有文献中受到了极大的关注。当一个人同时具有对投资的直接限制（即投资组合约束）和以规定的财富下限形式对投资进行间接限制（即投资组合保险）时，对投资组合优化的关注就会少得多，这可能是因为限制的组合确实表现出一些清晰明确的挑战，而当仅处理投资的直接限制（即投资组合限制）时，不会遇到这些挑战。本提案中的研究目标是在我们已经针对此问题建立的方法和部分结果的基础上，解决具有此类组合约束的投资组合优化问题。我们期望这将有助于金融工程知识的进步。 特别是，定义财富下限的约束有助于限制投资组合优化过程中的损失，从而有助于稳定金融交易，对加拿大经济有明显的好处。