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）。

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.