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，对于设计旨在增强金融稳定的政策具有重要意义。公司之间的合同、法律和商业关系可能成为风险传递和放大的渠道。在金融危机的情况下，对商业对手方以及更广泛的实体经济部门的溢出效应可能是显著的。因此，持有大量固定收益证券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。对金融稳定同样重要的是，解释在高频交易员主导的市场中，在闪电崩盘事件中周期性观察到的资产价格大幅波动。如果这些事件发生在一家“大到不能倒”的机构身上，市场可能会崩溃，政府将有必要出手纾困，这将给纳税人带来巨大后果。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，研究者将研究非线性抛物型偏微分方程的递归系统，该系统出现在固定收益投资组合的最优选择和对冲中。在项目的第二部分，他将开发新的数值技术来分析二阶正反向随机微分方程，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给随机分析带来新的数学挑战，包括研究完全非线性偏积分微分方程、二阶正反向带跳跃的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.