Backward (Doubly) Stochastic Differential Equations With Markov Chains And Associated (Stochastic) PDEs

In 1992,Peng and Pardoux[70]firstly gave the existence and uniqueness result of adapted solutions of nonlinear backward stochastic differential equations(BSDEs).Since then,due to the good structure of BSDEs and forward-backward stochastic differ-ential equations(FBSDEs),they have not only been widely studied in basic fields such as stochastic analysis([18,74])and partial differential equations([68,12]),but also have provided solid theoretical support for financial mathematics([27,20]),stochastic control([72,75])and other application fields.However,a forward or backward system driven only by Brownian motion can only effectively describe the continuous parameters in the model.However,in the real world,there are many events that occur with low frequency(incidental)but have a long-term and profound impact on the system.Taking the stock market as an example,market trend changes(bull market and bear market transition),policy changes,etc.are all discontinuous and sporadic,but they have a profound im-pact on the stock market.The classic diffusion model driven by Brownian motion alone cannot describe the above events well.As a random jumping process with discrete s-tates and continuous time,Markov chains have unique properties that can describe the above events.This idea was first proposed by Hamilton[31]and has received extensive attention and research([105,87]).Therefore,we introduced a type of continuous-time finite state Markov chain and couple it with the BSDE system,and then studied a type of hybrid system driven by Brownian motion and Maxkov chain.In this type of system,the coefficients include Markov chains,and the state of Markov chains is used to describe incidents,etc.,so that the state of the system depends on incidents.For example,in the stock market,we add a Markov chain with two states to the coefficients of the stock price model.In the Markov chain,the two states represent the bull market and the bear market respectively.At this time,the system can describe the impact of the conversion from the bull market to the bear market on the stock price.In addition,the filtering method with Markov chain plays a key role when studying the problem of partially ob-servable information disturbed by noise([2,88]).The mathematical theoretical support for the above problems is essentially a forward or backward system with Markov chains.Therefore,this article will focus on studying backward stochastic systems with Markov chains,including backward differential equations with Markov chains(BSDEMs),back-ward double stochastic differential equations with Markov chains(BDSDEMs),and the associated partial differential equations(PDEs),stochastic partial differential equations(SPDEs).This article mainly consists of the following six chapters;The first chapter of the thesis describes the research background and significance of the issues involved in this article,and details the main academic contributions of each chapter thereafter.In chapter 2,we mainly studied the random flow properties of the solutions of the stochastic differential equations with Markov Chains(SDEMs)and the Malliavin analysis on it,preparing for the next study of backward and backward double stochastic differential equations.First,we obtained that the solutions of SDEMs can form a random flow.Then,using the classical estimation method,we obtained a high-order estimate of the solutions of SDEMs,and using the homotopy theory,we obtain that the solution can form a differential homeomorphism.In the end,we got a generalized equivalence norm theorem,which plays a key role in studying the solution of the coupled BSDEM and BDSDEM in Sobolev space.In addition,in order to study the regularity of the solution of SDEM in Winner space and the representation of "Z" in the following chapters,we studied the Malliavin differentiability of a class of random variables,that not only contain information about the Wiener process,but also contain information independent of the Wiener process.Using independence,we obtained the Winner-Ito chaos expansion of such random variables,and finally extended the famous Clark-Ocone formula.Using the approximation method,the regularity of the solutions of SDEMs in the Wiener space is finally obtained.In Chapter 3,we mainly studied the smooth solutions and Sobolev weak solutions of PDEs associated with BSDEMs.First of all,using classic estimation techniques,we obtained the high-order estimation of the solutions of BSDEMs and the continuous dependence and smoothness of the solution on the parameters.Using the approximation technique,we get the representation of "Z" in BSDEM,from which we get the existence and uniqueness of the classical solution of PDEs.Under the classical Lipschitz condition,there have been many studies on the existence and uniqueness of the solution of BSDEMs.However,when studying the Sobolev weak solution of the corresponding PDEs,we found that if the classical Lipschitz condition is weakened into a functional Lipschitz condition with a weight function,the solutions of BSDEMs can describe the solution of the PDEs more naturally.Therefore,using the equivalent norm theorem obtained in Chapter 2,Riesz representation theorem,method of mollifier and some classic estimation methods,we firstly obtained the existence and uniqueness of the solutions of BSDEMs under the functional Lipschitz condition.Finally,using the approximation technique,we obtained the existence and uniqueness of the Sobolev weak solution of PDE.In chapter 4,we mainly studied the existence and uniqueness of the solutions of BDSDEM with Markov chain and a comparison theorem.First,we gave a generalized Ito formula,then,under the classical Lipschitz condition,using martingale representation theorem and approximation technique,we got the existence and uniqueness result of the solutions of BDSDEMs.Then,using the Yosida approximation,we studied the BDSDEMs under the monotonic conditions.Moreover,we gave comparison theorems under the above two conditions respectively.Finally,we studied the BDSDMs under local monotonic conditions.By constructing a sequence of globally monotonous BDSDEM,we proved that its limit is the unique solution of BDSDEMs under local monotonic conditions.In Chapter 5,we mainly studied the smooth solution and Sobolev weak solution of SPDEs associated with BDSDEMs.First,using classic estimation techniques,we ob-tained high-order estimates of solutions of the BDSDEMs and the continuous dependence and smoothness of its solutions on parameters.Using the Malliavin analysis in Chapter 2,we got the representation of "Z" in BSDEMs,from which we got the existence and uniqueness of the smooth solution of SPDE.Similarly,under the Lipschitz condition of functional form,the solution of BDS-DEMs can describe the solution of SPDEs more naturally.Therefore,using the equiva-lent norm theorem obtained in Chapter 2,Riesz representation theorem and some classic estimation methods,we firstly obtained the existence and uniqueness of solutions the BDSDEMs under the condition of functional Lipschitz.Then using the approximation method,we got the existence and uniqueness of the Sobolev weak solution of correspond-ing SPDEs.Finally,we gave a numerical result of this type of SPDEs.Finally,we summarized the research results of this article and gave some research prospects in Chapter 6.