深度学习、机器学习、人工智能炙手可热，这些研究对数据是强依赖的，对数据深入了解，做到知己知彼，可以说是很有必要的。数据中，有一种数据叫时间序列数据，是很重要的一种数据。这种数据在各行各业都占有很大的比重。目前在金融界，时间序列的研究是极热的。
随机模型在金融数据中的应用就包括这项研究，可以将所得模型用于场景生成，风险管理和预测的基本工具。
我们提出以下模型来支撑理论：起初的模型是一个有交互作用的随机偏微分方程。我们证明了方程的弱解可由一对给定生灭率和转移率的马尔科夫链逼近。基于此逼近结果,我们可求得方程的解。第二个模型是一个以马氏调节的布朗运动作为输入流的存储过程。我们重点分析了它们负荷的极限性质。第三,作为随机模型在金融中的应用,我们考建立了一类依赖于波动类型的随机偏微分方程的远期利率模型,并将其用于信用违约互换等衍生品的定价。本研究分为三个部分：起初一章研究了一类具有交互作用分支扰动的随机偏微分方程,此方程也称之为竞争的随机Lotka-Volterra方程。通过用一对时空尺度变换的粒子系统对方程进行逼近,我们证明了方程弱解的存在性。具体地,我们由给定的生灭率构造了一对马氏链,进而通过合适的时空尺度变换和Dynkin公式,我们得到了一对取值于离散函数空间的随机微分方程。而此方程的鞅部分根据跳的构造可以分解为反应、扩散和分支跳三项和。我们首先证明了关于它们上界和收敛的一些结果。利用这些结果我们可以证明方程各个构成项在合适的Sobolev空间的胎紧性。基于这些胎紧性结果,应用Prohorov定理和Skorohod表示定理等,我们可以得到一个一对收敛到方程的子序列,证明了方程弱解的存在性。第二章考虑了一个由马氏调节的布朗运动作为输入流的存储过程。近年来,马氏调节的布朗运动在金融模型和排队模型中有着广泛的应用。本章研究了关于此存储过程的两个重要性质。首先我们证明了在某些技术性条件下,过程平稳分布的存在性。进而假设过程依此平稳分布作为初始分布,我们分析了其运行极大过程的渐进增长率。证明了其以对数速度增长。第三章提出了一个对远期利率期限结构进行建模的新方法。避开即期利率模型,Heath et al.(1992)转而对远期利率的期限结构直接进行建模。进而金融学者们对此模型在许多方面进行了推广,这其中包含无穷维随机模型和带跳的模型。我们发现以上模型在处理两个时间指标的方式上是不同的:一个作为变量而另一个作为参数。在本章中,我们用一波动类型的随机偏微分方程来对远期利率的期限结构进行建模。在此模型中我们可以相对一致的处理这两个指标。使用波动类型方程的另一方面考虑是,方程本身能很自然地描述随机扰动性质。进而,我们推导出了在此模型下,市场是无套利的充要条件。与HJM条件类似,此条件可由模型的漂移项表示。进而,应用此远期利率对应的债券,我们考虑了几类可违约衍生品的定价问题,例如信用违约掉期。
Deep learning, machine learning and artificial intelligence are hot, and these researches are strongly dependent on data. It is necessary to have a deep understanding of data to know one's enemy and oneself. Among the data, there is a kind of data called time series data, which is a very important kind of data. This kind of data occupies a large proportion in all walks of life. At present, the study of time series is the hottest in the financial world.
The application of stochastic models to financial data includes this research, and the resulting models can be used as basic tools for scenario generation, risk management and prediction.
We propose the following model to underpin the theory: The first model is a stochastic partial differential equation with interactions. We show that the weak solution of the equation can be approximated by a pair of Markov chains with given birth and death rates and transfer rates. Based on this approximation result, we can obtain the solution of the equation. The second model is a stored procedure with Mahalanobis regulated Brownian motion as input stream. We focus on the analysis of the limiting properties of their loads. Third, as an application of stochastic models in finance, we establish a class of forward rate models with stochastic partial differential equations that depend on volatility types and apply them to the pricing of derivatives such as credit default swaps. This study is divided into three parts: In Chapter 1, we study a class of stochastic partial differential equations with branch perturbations of interaction, which is also called competitive stochastic Lotka-Volterra equations. We prove the existence of weak solutions to the equation by approximating it by a pair of particle systems with spatio-temporal scaling. Specifically, we construct a pair of Markov chains with given birth and death rates, and then obtain a pair of stochastic differential equations with values in the space of discrete functions by using appropriate spatio-temporal scaling and Dynkin's formula. The martingale part of this equation can be decomposed into the sum of reaction, diffusion and branch jump according to the structure of jump. We first prove some results about their upper bounds and convergence. Using these results, we can prove the tire compactness of each component of the equation in a suitable Sobolev space. Based on these results of tire compactness, we can obtain a pair of subsequences converging to the equation by using Prohorov theorem and Skorohod representation theorem, and prove the existence of weak solutions to the equation. In Chapter 2, we consider a stored procedure with Markov regulated Brownian motion as input stream. In recent years, Markovian adjusted Brownian motion has been widely used in financial models and queuing models. This chapter examines two important properties of this stored procedure. First, we prove the existence of stationary distribution of process under some technical conditions. Then, assuming the stationary distribution as the initial distribution of the process, we analyze the gradual growth rate of the process with maximum operation. And we show that it grows logarithmically. Chapter 3 presents a new method to model the term structure of forward rates. Instead of the spot rate model,Heath et al.(1992) directly modeled the term structure of forward rates. Furthermore, financial scholars have extended this model in many aspects, including infinite-dimensional stochastic model and model with jump. We find that the above models differ in the way they treat the two time indicators: one as a variable and the other as a parameter. In this chapter, we model the term structure of forward rates using a stochastic partial differential equation of the wave type. In this model, we can deal with these
two indicators relatively consistently. Another consideration of using wave type equations is that the equations themselves naturally describe the properties of stochastic disturbances. Furthermore, we derive the necessary and sufficient conditions for the market to be arbitrage-free under this model. Similar to the HJM condition, this condition can be represented by the drift term of the model. Furthermore, USING THE BONDS corresponding TO this FORWARD RATE, WE CONSIDER the pricing problem of several types of defaultable derivatives, such as credit default SWAPS.