Mathematics and Control of Systemic and High-Frequency Trading Risks

了解金融机构违约之间的依赖结构，对于制定旨在增强金融稳定的政策具有至关重要的意义。公司之间的合同、法律和商业关系可以作为传递和放大风险的渠道。在发生金融危机的情况下，溢出效应对企业对手方乃至更广泛的实体经济部门的影响可能是巨大的。因此，持有大量固定收益证券投资组合的金融机构，在做出对冲和投资决策之前，能够量化这些风险及其系统性影响，这一点至关重要。对于金融稳定来说，同样重要的是解释资产价格的大幅波动飙升。在高频交易员主导的市场中，闪电崩盘期间会定期观察到资产价格的大幅波动。如果这些事件发生在一家太大而不能倒的机构身上，市场可能会崩溃，政府救助将是必要的，这将给纳税人带来巨大的后果。该项目将开发随机控制技术，以解决存在传染风险的固定收益投资组合选择问题，并分析高频交易市场的日内流动性和价格动态。在项目的第一部分，调查者将研究在固定收益投资组合的最优选择和对冲中产生的非线性抛物型偏微分方程组的递归系统。在项目的第二部分，他将开发新的数值技术来分析带跳的二阶正倒向随机微分方程，这是一类在控制库存风险时高频交易中自然产生的方程。这些问题的独特结构将在随机分析中提出新的数学挑战，包括研究完全非线性偏积分-微分方程，带跳的二阶向前-向后随机微分方程，以及与这些方程相关的随机对策。

Understanding the dependence structure among defaults of financial institutions is of fundamental importance for the design of policies aiming to enhance financial stability. Contractual, legal, and business relationships among firms may act as a conduit for the transmission and amplification of risks. Under scenarios of financial distress, spillover effects on business counterparties and more broadly on the real economic sectors can be significant. It is thus fundamentally important that financial institutions, holding large portfolios of fixed income securities, can quantify these risks and their systemic implications before making hedging and investment decisions. Equally important for financial stability is to explain the huge volatility spikes of asset prices, periodically observed during flash crash events in markets dominated by high-frequency traders. If these events were happening to a too-big-to-fail institution, the market could be wrecked and government bailout would be necessary, with enormous consequences on taxpayers.This project will develop stochastic control techniques for solving fixed income portfolio selection problems in the presence of contagion risk, and analyzing intraday liquidity and price dynamics in high-frequency trading markets. In the first part of the project, the investigator will study recursive systems of nonlinear parabolic partial differential equations arising in the optimal selection and hedging of fixed income portfolios. In the second part of the project, he will develop new numerical techniques for analyzing second order forward-backward stochastic differential equations with jumps, a class of equations arising naturally in high frequency trading when controlling for the inventory risk. The unique structure of these problems will present new mathematical challenges in stochastic analysis, including the study of fully nonlinear partial integro-differential equations, second order forward-backward stochastic differential equations with jumps, and stochastic games associated with these equations.