﻿财经数理方法（54学时）-中央财经大学管理科学与工程学院
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《财经数理方法》课程简介

Mathematical Methods in Finance and Economics

This course contends two parts. The first part introduces some concepts of ordinary differential equations and methods of solving them. Based on the pioneering course of Probability, the second part of this course redefines some basic concepts in Probability, thus deeply introduces and analyzes some classical models and processes, finally introduces an application of stochastic analysis in finance.

The content of this course mainly includes the following:

First we first introduce some basic concepts of ordinary differential equations, including the definitions and geometrical explanations of ordinary differential equations and its solutions.

Then some elementary methods of solving ordinary differential equations, including separate variable method, elementary transform method, integrating factor method, etc.

After having introduced the basics of ODE, we start with recalling and redefining some concepts in Probability. We introduce the definition and basic properties of stochastic processes, including the basic concepts, mathematical characteristics, the classification, the introduction of two basic models: normal process and Poisson process.

We give the definition of Poisson process and extensions, including the relationship between Poisson process and exponential distribution, the distributions of inter-arrival time and waiting time, compound Poisson process, the residual life and age, etc.

As an important part of stochastic process, we introduce the Markov chains, including basic concepts of discrete and continuous Markov chains, transport matrices, Chapman-Kolmogorov equations, birth and death processes, etc.

The central concept of this course, Brownian motion, will be discussed, including properties of Brownian motion paths, extensions of Brownian motion, joint distributions of Brownian motion, distributions of first hitting time and maximum value, etc.

The concept of martingales, including the rigorous definition of conditional expectations and martingales, the basic properties of martingales, will be introduced briefly.

Last but not the least, the most important part, we focus on the stochastic integrals and Ito’s formula, including the definitions and properties of stochastic integrals with respect to Brownian motion, the form of Ito’s formula and applying Ito’s formula to solve stochastic differential equations. We apply Ito’s formula to derive Black-Scholes formula.

《财经数理方法》课程教学进度计划

 课程名：财经数理方法 课时分配 大约第几周完成（教师可调整） 36学时 54学时 72学时 第一章：常微分方程的基本概念 2 1 第二章：初等积分法解常微分方程 10 4 第三章：金融中的数学模型 2 5 第四章：概率空间 2 6 第五章：随机过程 2 6.5 第六章：泊松过程 6 8 第七章：马尔科夫链 6 10 第八章：布朗运动 6 12 第九章：鞅及其应用 6 14 第十章：随机微分方程及其在金融中的应用 8 17 第十一章：答疑 4 18 第十二章： 第十三章： 合计 54