The Stochastic Maximum Principle For Relaxed Control Problem With Regime-Switching And Two Kinds Of Mean-Field Stochastic Differential Equations

In recent years,the models with regime-switching and mean field are deeply investigated and become research hotspots.These kinds of extended models,which have wide applications in financial mathematics,can be closer to the problems in real world.Inspired of that,in this paper we study the stochastic relaxed maximum principle with regime-switching and the theory of mean-field forward-backward stochastic differential equations and doubly reflected backward stochastic differential equations.In the first chapter,we give the background of the problem,including relaxed control,maximum principle,mean-field,then we illustrate the research background and our contribution of each chapters respectively.In the second chapter,we study the relaxed maximum principle with regime-switching.Our paper does not have the boundedness of the derivative of the coefficients of the cost functional,so we give the estimates of the solution of variational equation and the state of relaxed control problem with application of integrability of admissible relaxed controls to get the variational inequality.Since Clarke’s derivatives cannot be used in the verification theorem of relaxed control problems,we introduce the L-derivative to obtain the verification theorem of relaxed control problems when the strict control domain is convex.For the case of non-convex strict control domain,we obtain another kind of verification theorem after making more strict assumptions on the coefficients.Using the relaxed maximum principle and the corresponding verification theorem,we solved two linear quadratic problems with regime switching,where the optimal relaxed control in the problem is not unique but can be expressed explicitly.In the third chapter,we give the existence and uniqueness of mean-field forwardbackward stochastic differential equations,in which the mean-field term is given in the form of distribution.Here we need to assume that the Lipschitz constant with respect to distribution is small enough relative to the constant of the monotonicity assumption,so that we can use the continuity method to obtain the existence and uniqueness of the equation.If the diffusion term of the positive process does not depend on the distribution term,we can relax the monotonicity conditions.On the other hand,we can prove that the existence and uniqueness of this equation can be related to a class of nonzero-sum stochastic differential games.We can obtain the existence of Nash equilibrium point of the corresponding nonzero sum stochastic differential game problem by the existence and uniqueness of the equation.Finally,we give an example to show that when the assumptions of constants in the equation is not satisfied,the Nash equilibrium point of the corresponding nonzero sum stochastic differential game problem may not exist.In chapter 4,we consider a class of doubly reflected stochastic differential equations,where reflection depends on Y and the expectations of Y.In the first part,the generator of the equation does not depend on Z.At this point,we can construct a contraction to obtain the existence and uniqueness of the solution of the equation,where the Lipschitz constants in the assumptions of the equation needs to meet some conditions.In the second part,the generator depends on Z.At this point,we need to make the equation satisfy the improved Mokobodski’s condition,and then through the penalty method,we obtain the estimates of a series of penalization equations.Finally,we obtain the continuity of the limit process and the convergence of the solution of the penalization equations through the properties of the predictable process,and then we obtain the existence of the solution of the equation.In chapter 5,we summarize the innovation of our research,and give some prospects for further researches.