Statistical Inference And Stock Price Simulation Based On Birth-death Process With Immigration

The birth-death process, as a kind of widely used Markov process, can model a variety of phenomena in numerous fields. In biology, the migration, the reproduction and the interaction between species can be modeled by the birth-death process. In queuing theory, in order to arrange the service staff and service facilities efficiently, the birth-death process can be used to study traffic in stores. In economics, the birth-death process can be applied to deal with stock price volatility problems and claim risk problems.Besides its strong application background, the birth-death process also plays an important role in theory. It is often the research entry point of general Markov processes. So far, scholars have obtained many profound theoretical results about it. Based on the previous work, this thesis mainly studies a special linear birth-death process-the birth-death process with immigration (BDI process) and gives its transition possibility distribution in different time. Then exploring its applicability in simulation of stock price. Finally, using the BDI process to simulate parameters and predict the changes of HS300index.By exploring the distribution and applicability of BDI process, we can understand the birth-death process more clearly. First, take the population of a region as an example to make numerical simulation and statistical infer-ence. Second, introduce how to simulate stock price with birth-death process. It verifies that the market does not have validity. By introducing the rela-tive deviation, it indicates that birth-death processes are more appropriate than geometric Brownian motion (GBM) theoretically to describe stock price volatility. Third, use the BDI process to simulate the HS300index volatility.This thesis mainly consists of four chapters.In chapter1, we introduce the background and some preparations of BDI process and stock price volatility.In chapter2, we present the concept of BDI process, the equilibrium dis-tribution of BDI process and three important transition probability formulae, which establish the relationship between BDI process and three common types of distribution in probability.In chapter3, numerical simulations are carried out for BDI process. Statis-tical inference under complete observation is given. The maximum likelihood estimation is used to estimate parameters.In chapter4, the birth-death process is used to analyze the number changes of rational traders and irrational traders, thus to study the evolution of the market. It indicates that the market does not meet the validity. By comparing the relative deviation of GBM and bounded birth-death process, we prove that birth-death processes are more appropriate to describe stock price volatility. As an example, HS300index is simulated by BDI process.