Differential Games Of Linear Quadratic Forward Backward Stochastic System And Their Applications In Finance

Game theory is the study of mathematical models of strategic interaction between rational decision-makers,and it has applications in fields of economic and finance,as well as in logic and computer science.Modern game theory originated from John von Neumann's research about the two-person zero-sum games.He and Morgenstern firstly discussed the cooperative game problem with multiple players in detail in their book Game Theory and Economic Behavior.After that,in virtue of its excellent properties and wide application prospects,game theory got the sustained attention and in-depth research and achieved constant development and progress,such as:Nash equilibrium under the non-cooperative game,the leader-follower game theory,as well as the mean-field game of large population problems.When participants are disturbed by the noise,people try to describe the state system by using the stochastic differential equations(SDEs),and then the corresponding game model which called stochastic differential game was built.As we know,since introduced by Pardoux-Peng,the nonlinear backward stochastic differential equation(BSDE)becomes a powerful tool to price contingent,and it is widely used to deal with some optimal problems.Therefore,the study of backward stochastic differential games has attracted consistent and extensive research attentions because of its significant theoretical values and broad practical applications.Inspired by previous studies,this paper concentrates on studying the differential games of several types of linear quadratic forward and backward stochastic systems,and it also attempts to apply the relevant theoretical results to solve some practical problems faced by us.The structure of this paper is as follows:In Chapter 1,we introduce the research background and elaborates on the main contributions of each chapter.The second chapter of this thesis is concerned with a dynamic game of N weakly-coupled linear BSDE systems involving mean-field interactions.Here,in our formulation,we consider the influence of the common noise,and it can be regard as that all agents should be affected by the same economic big environment.Furthermore,as a benchmark,all agents' cost functional conclude the weakly-coupled terms of state and control.These are different,from existing conclusions.In this chapter,the backward mean-field game is introduced to establish the backward decentralized strategies.To this end,we introduce the notations of Hamiltonian-type consistency condition and Riccati-type consistency condition in BSDE setup.Then,the backward mean-field game strategies are derived based on two consistency condition respectively.Under mild conditions,these two consistency condition solutions are shown to be equivalent.Next,the approximate Nash equilibrium of derived mean-field game strategies are also proved.In addition,the scalar-valued case of backward mean-field game is solved explicitly.As an illustration,one example from quadratic hedging with relative performance is further studied.In Chapter 3.Herein,motivated by problems facced by insurance firms,we consider the dynamic games of N weakly-coupled linear forward stochastic systems with terminal costraints involving mean-field interactions.which can relate to the above backward mean-field game.By penalization method,the associated mean-field game is formulated and its consistency condition is given by a fully coupled forward-backward stochastic differential equation(FBSDE).Moreover,the decentralized strategies are obtained,and the ?-Nash equilibrium is verified.In addition,we study the connection of backward linear quadratic mean-field game and forward linear quadratic mean-field game with terminal constraint.Furthermore,the decoupled optimal strategies of this linear quadratic are solved explicitly by introducing some Riccati equations.As an illustration.some simulations of the optimal asset allocation for firm and pension funds are furher are further studied.The fourth chapter studies the linear quadratic social optimal problems with control input constraint,where the basic objective is to minimize a social cost as the sum of individual costs containing mean field coupling.In Chapter 2 and Chapter 3,we study the mean-field game of the large population system,and its essence is a kind of non-cooperate game.However,in this chapter,within the mean field modeling we study different situation where all agents are cooperative and seek the same socially optimal decisions to minimize their common social cost,and its essence is a type of cooperate game.Therefore,in this case,to achieve the social optimal strategies,every agent should maintain a delicate balance between the reducing its own cost.and the effect of the sum of all other agents' costs.Firstly,in virtue of the variational method,we can obtain the corresponding auxiliary stochastic control problem by introducing the limiting state-average process.In addition,by using the maximal principle,we get the auxiliary problem's optimal control,which is given by a fully coupled non-liner forward-backward stochastic differential equation.Furthermore,we prove that the optimal control is a approximate social optimal strategy of the primal problem.And on this basis,we further deal with the social optimal problems with linear constraints and get the corresponding feedback form social optimal strategy.Finally,we use the above conclusions in dealing with the optimal portfolio with linear portfolio constraints and obtain the optimal investment strategy.In addition,the results are compared with those of the mean-field game of large population problems.Chapter 5 is concerned with a new kind of Stackelberg differential game of mean-field backward stochastic differential equations.There are two agents of different positions in the Stackelberg differential game.By means of four Riccati equations,the follower first solves a backward mean-field stochastic linear quadratic optimal control problem and gets the corresponding open-loop optimal control with the feedback representation.Then,the leader turns to solve an optimization problem for a 1 × 2 mean-field forward-backward stochastic differential system.In virtue of some high dimensional and complicated Riccati equations,we obtain the open-loop Stackelberg equilibrium,and it admits a state feedback representation.Finally,as applications,a class of stochastic pension fund optimization problems which can be viewed as a special case of our formulation is studied and the open-loop Stackelberg strategy is obtained.The sixth chapter studies the pricing problem for callable-puttable convertible bonds with call-put protections and call notice period as a Dynkin game via the backward stochastic differential equation with two reflecting barriers.By virtue of reflected backward stochastic differential equations,we first reduce such a Dynkin game to an optimal stopping time problem and establish the formulae for the fair price of puttable convertible bond.Based on that,we further obtain the valuation model of callable-puttable convertible bonds and investigate how the call and put clauses affect the value of convertible bonds.In virtue of the comparison theorem of doubly reflected backward stochastic differential equations,the sensitivity analysis of callable-puttable convertible bonds' price about some key parameters is considered and verified by numerical simulations through an obstacle problem for a parabolic partial differential equation.In the last chapter,we conclude this paper and give some future research directions.