Stochastic modelling in mathematical and computational finance

数学和计算金融中的随机建模是提出的研究计划的重点。数学和计算金融考虑金融或经济变量和系统的不确定未来行为；提出了衍生证券的估值与风险管理理论；并为检查各种投资、管理和监管决策提供了一个定量框架。应考虑由金融应用驱动的各种概率模型、统计估计方法和计算算法。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领域的新理论进步。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数量的随机模型开始，如利率和资产波动。我们将开发新的定价和风险管理理论方法，使市场参与者能够评估和对冲暴露于信用风险的金融衍生品。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，在可能的情况下推导显式解决方案，但重点放在现实建模上，这需要创建新的有效的计算算法来解决这些方程。***提出的研究计划的第一个目标是研究正反向随机微分方程（FBSDEs）及其在数学金融中的应用。FBSDE是一个随机方程的耦合系统，随机方程的分量从一个特定的初始条件开始随时间向前发展，而分量从一个随机的最终条件开始随时间向后发展。我们将使用FBSDEs来描述信用风险衍生品（如违约债券）的新定价方法，并扩展该方法。第二个目标是发展求解FBSDEs的数值方法，因为具有显式解的FBSDEs类是有限的。我们将进一步发展一种新的数值方法，基于快速傅里叶变换，到更高的维度。第三个目标涉及数学金融问题的研究，这些问题可以被描述为产生或依赖于大量高维数据。我们的目标是将高维数据的某些建模和统计技术扩展到金融环境，这些技术可以有效地降低维度，从而实现精确的低维模型。远期利率过程的无限维模型将是第一个考虑的例子，这样，通过降维，我们可以创建新的简洁的金融模型，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.**