Stochastic modelling in mathematical and computational finance

数学和计算金融中的随机建模是拟议研究计划的重点。 数学和计算金融学考虑金融或经济变量和系统的不确定未来行为;为衍生证券的估值和风险管理提供理论;并为检查各种投资，管理和监管决策提供定量框架。各种概率模型，统计估计方法和计算算法，这是金融应用的动机应予以考虑。 在现实的框架中解决应用问题也推动了新的理论研究，并推动了概率和统计的新理论进展。 该研究整合了数学和计算金融的各个方面，从开发新的基本金融和经济量（如利率和资产波动率）的随机模型开始。我们将开发新的定价和风险管理理论和方法，使市场参与者能够对面临信用风险的金融衍生品进行估值和对冲。 我们还计划研究表征这些新的估值和风险管理方法的随机方程，在可能的情况下导出明确的解决方案，但重点是现实的建模，这需要创建新的高效计算算法来求解这些方程。 该研究计划的第一个目标是研究正倒向随机微分方程及其在数学金融中的应用。FBBACK是随机方程的耦合系统，其分量从指定的初始条件在时间上向前演化，并且分量从随机终端条件在时间上向后演化。 我们将使用FBSDES来描述一种新的信用风险衍生品（如可违约债券）的定价方法，并扩展该方法。 第二个目标是发展的数值方法的解决方案的FBSDES，因为类的FBSDES显式解决方案是有限的。 我们将进一步发展一个新的数值方法，我们创建的基础上，快速傅立叶变换，更高的维度。 第三个目标是研究数学金融中的问题，这些问题可以被描述为产生或依赖于大量的高维数据。我们的目标是扩展到金融环境中的某些建模和统计技术的高维度的数据，有效地降低了维度的一个准确的低维模型可以实现。 远期利率过程的无限维模型应该是考虑的第一个例子，这样，通过减少维度，我们可以创建新的简约金融模型，保留理论模型和观测数据的关键特征。

Stochastic modelling in mathematical and computational finance is the focus of the proposed research program.  Mathematical and computational finance considers the uncertain future behaviour of financial or economic variables and systems; presents a theory for the valuation and risk management of derivative securities; and provides a quantitative framework for examining various investment, managerial, and regulatory decisions. Various probabilistic models, statistical estimation methods, and computational algorithms which are motivated by financial applications shall be considered.  Addressing applied problems in a realistic framework also drives new theoretical research and is the impetus for novel theoretical advances in probability and statistics.  The research integrates various aspects of mathematical and computational finance starting with the development of new stochastic models for fundamental financial and economic quantities such as interest rates and asset volatility. We shall develop new pricing and risk management theories methodologies that allow market participants to value and hedge financial derivatives exposed to credit risk.  We also plan to study the stochastic equations which characterize these new valuation and risk management methods, deriving explicit solutions where possible but  focusing on realistic modelling which requires the creation of new efficient computational algorithms for solving these equations. The first objective of the proposed research program is the study of forward ­backward stochastic differential equations (FBSDEs) and applications in mathematical finance. An FBSDE is a coupled system of stochastic equations with components that evolve forward in time from a specified initial condition and components that evolve backward in time from a random terminal condition.  We shall use FBSDEs to characterize a new pricing methodology for credit risk derivatives such as defaultable bonds and extend this method.  The second objective is the development of numerical methods for the solution of FBSDEs since the class of FBSDEs with explicit solutions is limited.  We shall further develop a new numerical method we created, based on the fast Fourier transform, to higher dimensions.  The third objective involves the study of problems in mathematical finance that can be characterized as producing or depending on large amounts of high­ dimensional data. Our goal is to extend to financial contexts certain modelling and statistical techniques for high­ dimensional data that effectively reduce the dimension to the point that an accurate lower dimensional model can be implemented.  Infinite dimensional models of forward interest rate processes shall be the first example considered so that, by reducing the dimension,  we can create new parsimonious financial models that preserve the key features of the theoretical model and the observed data.