# 105年第1學期-1762 隨機過程 課程資訊

## 評分方式

Mid-term Exam 30
Project Report 40

## 教育目標

The objective of this course is to introduce basic concepts for stochastic processes. The main focus will be on studying analytical models for systems which change states stochastically with time. The topics include: Poisson processes, discrete-time Markov chains, continuous-time Markov chains, renewal processes, Brownian motion, etc.(1) Poisson process (basic definitions and properties), (2) nonhomogeneous Poisson processes (properties and applications), (3) discrete-time Markov chains (classification of states, periodicity, ergodic, finite-state Markov chain), (4) hidden Markov models, (5) continuous-time Markov chains (Chapman-Kolmogorov equations, infinitesimal generator, birth and death Processes), (6) Brownian motion (properties, Black-Scholes formula)

## 課程概述

The objective of this course is to introduce basic concepts for stochastic processes. The main focus will be on studying analytical models for systems which change states stochastically with time. This is a course for studying probabilistic models rather than statistical models. Thus, background on probability and mathematical statistics are necessary. We will begin right after �onditional probability�and �onditional expectation� Students will learn concepts and techniques for characterizing models stochastically. This will help students for further study. The topics of this course include basic processes, stochastic models, and diffusion processes. Contents of this course might be adjusted according to time limitation and students�interests. They are: 1.Preliminaries: lack of memory property, transformations, inequalities, limit theorems, notations of stochastic processes 2.Markov chains: Chapman-Kolmogorov equation, classification of chains, long run behavior of Markov chains, branch processes, random walk 3.Poisson processes: Inter-arrival time distributions, conditional waiting time distributions, non-homogeneous Poisson processes 4.Continuous-time Markov chains: birth-death processes, compound Poisson processes, finite-state Markov chains 5.Renewal processes: renewal functions, limit theorems, delayed and stationary renewal processes, queueing 6.Stochastic models: Markov renewal processes, marked processes 7.Martingales: conditional expectations, filtrations, stopping time, martingale CLT 8.Diffusion Processes: Brownian motions, It�s formula, Black-Scholes Model, Girsanov Theorem

## 參考書目

1. Sheldon M. Ross (2014) Introduction to Probability Models, 11th ed, Academic Press
2. Sheldon M. Ross (1996) Stochastic Processes, 2nd ed, John Wiley , New York.