Stochastic optimal control in mathematical finance

在过去的几十年里，数学方法在金融决策中的作用稳步增加，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，其总体目标是建立一种科学的方法来有效地配置经济中的资本。在这种方法中，人们首先建立一个理想化的数学模型（或描述），描述资本配置发生的金融市场的中心方面。然后，参考这个数学模型，人们可以用精确的术语来表述与金融决策有关的问题。特别重要的是分配资本的问题，以使风险最小化或资本投资回报最大化（这被称为投资组合优化）。******我们研究项目的重点将放在资本投资约束下的投资组合优化上。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”），这通常是由监管机构施加的（例如禁止卖空某些指定的证券），但也可以采取间接限制投资的形式，特别是规定一个“最低限度”，无论市场如何发展，投资者的财富都不得低于这一限度。后一种约束对金融交易过程中可能出现的损失施加了严格的限制（这被称为“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（即仅受投资组合约束）的投资组合优化在现有文献中受到了极大的关注。当对投资的直接限制（即投资组合约束）与以规定的财富下限（即投资组合保险）形式对投资的间接限制相结合时，对投资组合优化的关注就少得多。可能是因为这种约束的组合确实表现出一些清晰而明确的挑战，而这些挑战在只处理投资中的直接限制（即投资组合约束）时是不会遇到的。本提案的研究目标是在我们已经为这个问题建立的方法和一些部分结果的基础上，解决具有这种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。***********************

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.***********************