Stochastic optimal control in mathematical finance

在过去的几十年中，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分配经济中的资本。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）的中心方面的金融市场的资本配置发生。然后，参考这个数学模型，人们可以用精确的术语来表述财务决策所关心的问题。特别重要的是分配资本的问题，以最小化风险或最大化投资资本的回报（这被称为投资组合优化）。 我们的研究计划的重点将是投资组合优化受到资本投资的约束。限制的形式可以是对投资的直接限制（通常称为“投资组合限制”），这种限制往往由监管机构施加（例如禁止卖空某些指定证券），但也可以是对投资的间接限制，特别是规定一个“下限”，投资者的财富无论市场如何演变都不得低于这个下限。后一种约束对金融交易过程中可能出现的损失实施了硬性限制（这被称为“投资组合保险”）。资本投资的约束是金融决策的一个自然组成部分，因此具有明显的重要性，实际上，投资组合优化只受到投资的直接限制（即仅投资组合约束）在现有文献中受到了极大的关注。当投资者同时受到这两种直接限制时，（即投资组合限制）以及以规定财富下限的形式对投资的间接限制（即投资组合保险），这可能是因为这种约束条件的组合确实表现出一些明确和确定的挑战，而这些挑战在只处理直接投资限制（即投资组合限制）。在这个建议中的研究的目标是解决投资组合优化与这样的组合约束，建立在一个方法和一些部分的结果，我们已经建立了这个问题。我们希望这将有助于金融工程知识的进步。 特别是，界定财富下限的限制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.