Mean Field Stochastic Linear Quadratic Differential Games With Markov Regime-Switching

As we known,game theory is the study for multiple agents making their strategies.Roughly speaking,we are able to regard it as a high dimensional optimal control problem from the control theory.There are many kinds of game mathematical models,for in-stance,the relationship among agents can be cooperation or conflict.It has wide-world applications in different fields such as finance market,management science,computer science,physics,chemistry,etc.The earliest study is about zero-sum games,where all agents' profit always equals to their loss.This is a special case for non-cooperative game,and now we term the Nash equilibrium strategy as a kind of "optimal" strategy with-in such non-cooperative game.With the development of game theory,there are more and more scientists applying the game theory to deal with the problems in their own fields.The agents involved always have conflicting goals in many mathematical models.Therefore,Nash equilibrium analysis becomes important in such settings.Combined with the stochastic analysis,the game theory gradually developed a new branch called stochastic differential games(SDG,for short).This is a big progress from determinacy to random.The mathematical models of SDG are useful in modeling dy-namic systems involving noise.In the literature,the studies of SDG may be traced to the 1960s(refer to[6,7,9,50,80,92]).Recently,the controlled mean-field SDG(MFG,for short)has been studied in decision analysis,engineering,portfolio selection,financial market,etc.One application of MFG is dealing with the large population(LP,for short)system.Many studies have been conducted on MFGs.Since the independent research of Huang-Caines-Malhame[43,44]and Lasry-Lions[58,59,60],MFG theory and its applications have experienced rapid growth.Related developments in MFG theory may include those in the research of Bardi[8],Bensoussan-Frehse-Yam[13],Carmona-Delarue[23],Garnier-Papanicolaou-Yang[38],Gueant-Lasry-Lions[37],and the references there-in.Note that mean-field game differs from the mean-field type control such as[2,30].Among SDGs,the Stackelberg game(also called leader-follower game)is firstly pro-posed by H.Von Stackelberg in 1934[81].Stackelberg game describes the case where the positions of agents are asymmetric.It divides the agents into leaders and follow-ers.A lot of studies have been conducted on Stackelberg SDGs.Basar[9]considered Stackelberg game under the linear quadratic system.Bensoussan-Chau-Yam[10]stud-ied the Stackelberg MFGs.The maximum principle for Stackelberg SDGs was given by Bensoussan-Chen-Sethi[12].Demiguel-Xu[29]studied multiple leaders case in Stackel-berg SDGs.Du-Huang-Qin[30]studied the maximum principle for delayed Stackelberg MFGs.A very similar subject to the Stackelberg MFG is major-minor MFG.This is a concept in LP system,where the minors' individual influence can be negligible.minors can influence the LP system by changing the state average but the major will direct-ly influence the LP system by changing its own strategy.There are lots of literature studying the major-minor MFGs.To the best of my knowledge,Huang[46]proposed the scheme first.Since then,Nourian-Caines[70]verify the Nash certainty equivalence theory.Huang-Wang-Wu[41]studied major-minor MFGs in backward-forward stochas-tic differential equation system(BFSDE,for short).One significant feature of MFG is that there is a kind of weak coupled structure in the state dynamics and cost functionals via the mean-filed term.When,we solve the MFG problems,the first idea we would put forward is decoupling the coupled structure and therefore we may introduce some kinds of the Riccati equations.An interesting result is that if we focus on the Stackelberg MFG and set the dynamics be forward stochastic differential equation(SDE,for short),then the dynamics of the leader agents will still end up with a forward-backward stochastic differential equation(FBSDE,for short).This thesis mainly discuss the linear-quadratic(LQ,for short)case where the state dynamic is driven by a linear equation and the cost functional is quadratic form.It is a classical and fundamental problem in the field of game theory and control theory.In the past few decades,both deterministic and stochastic linear quadratic control problems are widely studied.Stochastic linear quadratic(SLQ,for short)optimal control problem was firstly studied by Kushner[50]using dynamic programming principle.Since then,Wonham[92]studied the extended version of the matrix Riccati equation arose in the problems of SLQ and filtering.Taking advantage of functional analysis theory,Bismut[6]proved the existence of solution to the Riccati equation and derived the existence of the optimal control in a random feedback form for SLQ optimal control with ran-dom coefficients.Based on the good structure of LQ system,there are many works on MFG modeled by LQ scheme.Li-Sun-Yong[54]studied the open-loop(OL,for short)solvability for LQ-MFG;Sun[85]studied the closed-loop(CL,for short)solvability for LQ-MFG.Besides,the LQ games in a large-population system are similar to LQ-MFG,there are also lots of existing paper on LQ game in LP system.The LQ games in LP systems where the participants' dynamics are nonuniform were studied and a kind of e-Nash equilibrium property was proved in[44].In[45],Huang-Caines-Malhame studied a kind of LQ games with N agents,where the common objective is to minimize the cost functional as the sum of N agents' cost functionals called social optimal problem.This is a kind of cooperate MFG and has its own applications in reality.For more literature about LQ-MFG,please refer to[41,42,31]etc.Another extension to SLQ problems is to considering the cases where the coefficients in dynamics and cost functionals contain random jumps,such as Poisson jumps or the regime switching jumps.Recently,more and more people have studied the applications of regime-switching models in finance and SLQ problems and have published lots of literature.For instance,Wu-Wang[93]was the first to consider the SLQ problems with Poisson jumps and obtain the existence and uniqueness of the deterministic Riccati equation.Besides,the existence and uniqueness of the stochastic Riccati equation with jumps and connections between the stochastic Riccati equation with jumps and the associated Hamilton systems of SLQ optimal control problem were also presented.Yu[103]investigated a kind of infinite horizon backward SLQ optimal control problems and SDGs under the jump-diffusion model state system.Li et al.[55]solved the indefinite SLQ problem with Poisson jumps.The SLQ optimal control problems with regime switching system are of interest and of practical importance in various fields such as option pricing,science,engineering,financial investment and economics.The regime-switching models and related topics have been extensively studied in the areas of applied probability and stochastic control theory.Recently,there has been dramatically increasing interest in studying this family of SLQ optimal control problems as well as their financial applications.For instance,Li-Zhou[53]and Li-Zhou-Ait Rami[55]introduced indefinite stochastic LQ controls with Markovian jumps,Liu-Yin-Zhou[57]considered near optimal controls of regime-switching LQ problems with indefinite control weight costs,Donnelly[32]analyzed the stochastic maximum principle for the optimal control of a regime-switching diffusion model,Tao-Wu[88]investigated the stochastic maximum principle for optimal control problems of forward-backward regime-switching systems.From the aspect of finance field,people usually might find two kinds of market regimes,one of which represents a bull market with price increases,while the other regime represents a bear market with price drops.Therefore,the regime-switching type portfolio selection problem is of great interest and importance in financial investment.Typical examples that are applicable include,but are not limited to,those presented in Yiu-Liu-Siu-Ching[102],Donnelly-Heunis[33]and etc.Motivated by above research,the main idea of the thesis is combining the LQ-MFG with regime-switching system.As we known,if we directly study the MFG with random coefficients,then we lack some necessary mathematical tools to deal with such kind of FBSDEs.With the rapid development of Markov chain theory,we are able to deal with the LQ-MFG with regime-switching system.Besides,we have interest in some other topics,such as the Stackelberg MFG driven by BFSDE system,combining the Stack-elberg game with major-minor game in the same MFG,and the financial applications in regime-switching system.This thesis consists of all of the topics mentioned above.Dealing with the random coefficients MFG problems,we can not avoid the difficulty that E[A(t,?(t))X(t)]?A(t,?(t))E[X(t)],which is always hold when deterministic co-efficients are given,i.e.E[A(t)X(t)]=A(t)E[X(t)].Although,there are some division method in discrete time,it can not be applied in continuous time.Therefore,with the limit of such difficulty,we are not able to introduce some Riccati equations to gain the feedback of optimal control.However,we can still discuss the OL solvability of MF-SLQ optimal control problems in regime-switching system.The thesis is organized as follows:Chapter 1,we introduce the abstract of each problems we are going to study,which provides readers with great convenience to get the main idea of each chapters.In chapter 2,we invest the large-population system with backward-forward setting and the related LQ-MFGs are formulated.For the leader agent and follower agents,the auxiliary limiting problem is constructed and solve out the related optimal control.Due to the feature of backward-forward setting,we are not able to decouple the consisten-cy condition(CC,for short)system which is a kind of FBSDE by introducing Riccati equations.Therefore,we give some monotonicity condition and prove its well-posedness by contraction mapping method.Moreover,the decentralized control strategies are de-rived from the CC system.In addition,the ?-Nash equilibrium property of our original problem is also verified based on some FBSDE estimates.Furthermore,we study the Stackelberg game coupled with major-minor game in Chapter 3.We divide the participants into three groups:major leader,minor leader and(minor)follower.For applications,they can represent three types of agents in financial market:major supplier,minor suppliers and minor producers.We derive the approxi-mate Stackelberg-Nash-Cournot(SNC,for short)equilibrium in such a MFG.Although we set all agents are of forward states,our SNC analysis tells us the major leader will end up with a forward-backward state naturally due to the Stackelberg scheme.This result differs from those reported in the literature on standard MFG frameworks,mainly as a result of the adoption of a Stackelberg structure.Through variational analysis,the CC system can be represented by some fully-coupled FBSDEs with a high-dimensional block structure in an OL case.To sufficiently address the related solvability,we also derive the feedback form of the SNC approximate equilibrium strategy via some coupled Riccati equations.At last,we verify the ?-SNC equilibrium property and give some applications within our model.In Chapter 4,we consider an optimal portfolio selection problem in regime-switching system.Financial models are based on the standard assumptions of frictionless markets,complete information,no transaction costs and no taxes and borrowing and short selling without restrictions.Short-selling bans around the world after the global financial crisis and in several exchanges during the COVID 19 period,become more and more important.This paper bridges the gap by providing for the first time in the literature a model that accounting explicitly and simultaneously for inflation,information costs and short sales in the portfolio performance with regime switching.Our model can be used by portfolio managers to assess the impact of these market imperfections on portfolio decisions.At last,the Chapter 5 investigates the LQ-MF-SLQ in regime switching system.The representation of the cost functional is derived using the technique of operators.It is shown that the convexity of the cost functional is necessary for the finiteness of the problem,whereas uniform convexity of the cost functional is sufficient for the OL solvability of the problem.By considering a family of uniformly convex cost functionals,a characterization of the finiteness of the problem is derived and a minimizing sequence,whose convergence is equivalent to the OL solvability of the problem,is constructed.We demonstrate with a few examples that our results can be employed for tackling some financial problems such as mean-variance portfolio selection problem.