Theoretical Analysis And Application Of Stochastic Differential Equations Under G-Brownian Motion In Option Pricin

With the rapid development of modern mathematics,the relationship between mathematics and finance is getting closer and closer.In particular,the development of stochastic differential equation has made it the theoretical basis for solving financial problems.The Black Scholes equation is used to solve option pricing problems.However,this model is idealized,and in real life,using G-Brownian motion to describe volatility is more accurate.Therefore,studying option pricing under G-Brownian motion has strong practical significance.Financial security is an important manifestation of national security,and the Asian financial crisis had a significant impact on the Chinese financial market.However,there is still a certain gap between the research on option pricing in China and that in developed countries.Barrier option is a very important option.Therefore,it is of great significance to study barrier option pricing by using the stochastic differential equation under the G-Brownian motion to provide a theoretical basis for domestic option pricing and thus protect national financial security.Many stochastic differential equations have no explicit solutions,or the solutions are complex,so numerical simulation is an important means to solve them.Therefore,the purpose of the study is to use multi-level Monte Carlo method to simulate option pricing and compare it with other methods to obtain a more accurate numerical simulation method.Based on this,this paper uses the stochastic differential equation under the G-Brownian motion to make a theoretical analysis of option pricing and apply it to an example.This paper gives the necessary preparatory knowledge,and gives several mathematical models of stochastic differential equation under G-Brownian motion to solve barrier options.In order to provide theoretical basis for numerical solutions,the existence and uniqueness theorem of solutions for stochastic differential equation under G-Brownian motion is given and proved.By combining specific examples and improving the Monte Carlo method,a more accurate method is obtained by numerically simulating and comparing the multi-level Monte Carlo method with other methods.