Applications of forward-backward stochastic differential equations to financial modelling

在金融建模中，假设唯一可以用于决策的信息是现在和过去的信息。否则，投资者将以正概率获得无风险利润。如果金融市场的数学模型考虑到无风险利润的正概率，通常被认为是病态的。然而，我们认识到，有关某些衍生证券价格未来走势的某些信息可供所有市场参与者获得，必须被视为一条当前信息。最简单的例子是债券的赎回价值在发行时是已知的。正-倒向随机微分方程是一种有效的数学工具，它以严格的方式结合对资产价格未来行为的现有知识或约束，同时仍保留了经济主体不能利用未来信息进行交易以获得无风险利润的限制。一组方程基于现在和过去的信息，为资产价格（或影响价格的经济因素）的未来演变建模。第二组方程可以被看作是从一个随机的最终条件在时间上向后发展的，然而，它被限制以一种基于当前和过去信息的可预测的方式这样做。本研究将正反向随机微分方程应用于各种债券、期货合约和远期合约。目标是将这些方程的存在性和唯一性结果扩展到更广泛的模型类别，从而更好地建模观察到的金融数据。我们还计划研究将这些方程应用于更广泛的衍生证券，如债券期权、期货和远期。我们计划将正反向方程的数值方法扩展到与金融应用相关的更广泛的模型中。

In financial modelling it is assumed that the only information which can be used in decision making is present and past information.  Otherwise, investors would be able to make riskless profits with positive probability.  Mathematical models of financial markets are usually considered ill-posed if they allow for riskless profits with positive probability.   Nevertheless, we recognize that some information about the future behaviour of certain derivative security prices is available to all market participants and must be regarded as a piece of current information.  The simplest example is that the redemption value of a bond is known at issue.Forward-backward stochastic differential equations are an effective mathematical tool for incorporating present knowledge of, or constraints on, the future behaviour of an asset price in a rigorous fashion while still preserving the restriction that economic agents cannot use future information to make trades which result in riskless profits.  One set of equations models the evolution of the price of an asset (or factors of the economy effecting prices) forward in time based on present and past information. The second set of equations can be viewed as evolving backward in time from a random terminal condition, however, it is constrained to do so in a way that is predictable based on current and past information.  The research in this proposal applies forward-backward stochastic differential equations to various bonds, futures contracts and forward contracts.  The goal is to extend the existence and uniqueness results for these equations to a wider class of models that better model observed financial data.  We also plan to study the application of these equations to a wider class of derivative securities such as options on bonds, futures, and forwards.  We plan to extend numerical methods for forward-backward equations to a wider class of modelsrelated to the financial applications.