Stochastic Analysis with Applications to Finance

这是一个由三部分组成的建议。在第一部分中，电子交易所的限价订单将被建模为一个排队系统，演变为一个度量值过程，其中所讨论的度量是位于价格轴上的订单大小集。像许多排队系统一样，在离散订单级别上订单簿的演变是非常复杂的。因此，排队系统的大流量限制理论将被用于离散系统的连续逼近。在提案的第二部分，将开发资产价格波动的模型。目前使用的局部波动率模型将波动率描述为时间和标的资产价格的函数。我们建议将这一概念扩展到波动性是时间、标的资产价格和一个或多个因素的函数的模型中，这些因素决定了标的资产上的衍生证券的支付。提案的第三部分继续研究存在交易成本的最优投资和消费。这些问题很好理解，并且在交易成本为零的情况下存在封闭形式的解决方案，但在交易成本为正的更现实的情况下没有封闭形式的解决方案。提出了未知解的小交易成本参数在零交易成本已知解周围的渐近展开式。综上所述，本文提出建立旨在降低金融市场风险和提高金融市场效率的数学模型。金融市场的创新已经改变了它们的结构，分析需要跟上步伐。特别是，订单驱动的电子交易所的出现催生了算法交易的增长，公司通过算法交易快速交易大量资产。这会使市场具有流动性吗？它会抑制还是放大波动性？这些问题很难回答，因为我们没有很好的数学模型来解释这些交流。我们也无法准确预测这些交易所的各种订单匹配协议将如何影响波动性、买卖价差和一般市场效率。这个建议的一个主要部分是开发电子交换的数学模型。该提议的第二个方面是改进资产价格波动的数学模型。著名的布莱克-斯科尔斯（Black-Scholes）模型假设波动性是恒定的，但金融市场对这一假设的偏离太过显著，不容忽视。本文的第三部分是更好地理解交易成本对最优投资和最优消费问题的影响。这个问题是金融市场均衡分析的核心，但迄今为止，大多数研究都不切实际地假设交易成本为零。

This is a three-part proposal. In the first part, the limit order book of an electronic exchange will be modeled as a queueing system, evolving as a measure-valued process, where the measure in question is the set of order sizes located on the price axis. Like many queueing systems, the evolution of the order book at the level of discrete orders is prohibitively complex. Therefore, the theory of heavy traffic limits for queueing systems will be used to obtain continuous approximations to the discrete systems. In the second part of the proposal, models for asset price volatility will be developed. Local volatility models currently in use describe volatility as a function of time and the underlying asset price. We propose to extend this concept to models in which volatility is a function of time, the underlying asset price, and one or more factors that determine the payoff of derivative securities written on the underlying asset. The third part of the proposal continues work on optimal investment and consumption in the presence of transaction costs. These problems are well understood and there are closed-form solutions in cases where transaction costs are zero, but there are no closed-form solutions in the more realistic case when transaction costs are positive.It is proposed to develop asymptotic expansions in the small transaction cost parameter of the unknown solution around the zero-transaction-cost known solution.In overview, it is proposed to build mathematical models designed to reduce risk and increase efficiency in financial markets. Innovations in the financial markets have changed their structure, and analysis needs to keep pace. In particular, the advent of order-driven electronic exchanges has spawned the growth of algorithmic trading, whereby firms trade large quantities of assets quickly. Does this make markets liquid? Does it dampen or amplify volatility? It is difficult to answer these questions because we do not have good mathematical models for these exchanges. Neither can we predict with any accuracy how various order-matching protocols on these exchanges will affect volatility, bid-ask spreads, and general market efficiency. A major part of this proposal is the development of mathematical models for electronic exchanges. A second aspect of this proposal is to improve mathematical models for asset price volatility. The celebrated Black-Scholes model assumes that volatility is constant, but departures from this assumption in financial markets are too significant to ignore. The third part of this proposal is to better understand the effect of transaction costs on problems of optimal investment and consumption. This problem is at the heart of equilibrium analysis of financial markets, but most studies to date make the unrealistic assumption that transaction costs are zero.