Reflected Backward Stochastic Differential Equations And Recursive Optimal Mixed Control Problems

Stochastic differential recursive utilities which not only depend on some control processes corresponding to consumption or investment,but also rely on the future utility generalize the classic additive utilities.In 1992,Duffie and Epstein[27]firstly introduced this kind of utilities.After that,recursive utilities have been widely used in economic,financial and other fields.El Karoui,Peng and Quenz[38]formulated it from the perspective of backward stochastic differential equation(BSDE)and applied it to stochastic control as the cost functional of an optimal control problem.Since then,the stochastic recursive optimal control problem has gradually become a new research hot spot.Reflected backward stochastic differential equation,also known as backward stochastic differential equation with obstacle constraints,is generated on the basis of BSDE by introducing a continuous increasing process to keep the solution above the given obstacle process.It can also be given by a special BSDE,that is,penalty backward stochastic differential equation.When the filtration contains the information of some jump process or the obstacle process has jumps,the solution of the reflected BSDE or its increasing process is no longer a continuous process,but is correspondingly transformed into a process with jumps.Obstacle constraints make the reflected BSDE have a good correspondence with the optimal stopping problem.The solution of the reflected BSDE can be used to describe the optimal stopping time and solve the optimal mixed control problem involving both control strategy and stopping time,which makes the reflected BSDE get continuing concern and profound study.On the basis of previous studies,combined with zero-sum differential games,diffusion control and Poisson stopping constraints,this thesis makes further research on several optimization problems related to reflected BSDEs and recursive utilities,obtains the sufficient condition for saddle point strategy,the sufficient condition for optimal control and the differential equation representation for the value function of the optimal stopping problem,respectively.The structure of the thesis is as follows:Chapter 1 introduces the research background and our main contributions.Chapter 2 concerns a class of stochastic recursive zero-sum differential game problem with recursive utility related to a backward stochastic differential equation with double obstacles.This BSDE is coupled with a forward backward differential equation via its terminal condition,generator and double obstacles.A sufficient condition is provided to obtain the saddle-point strategy under some assumptions.Then,using the above result,we get a sufficient condition for the saddle-point mixed strategy of a kind of stochastic recursive mixed differential game problem and obtain the explicit saddle-point strategy as well as the saddle-point stopping time for the linear recursive mixed differential game problem under certain conditions by virtue of the corresponding relationship of doubly reflected BSDE and Dynkin game problem.Compared with some existing literature on stochastic recursive mixed differential game problem,our model admits random coefficients and allows the diffusion coefficients of system equations depending on the game strategies explicitly.In addition,the stopping framework of the problem studied in this chapter requires that the costs for agents to stop the system is dependent on the state process which means these costs also depend on control strategies.This discussion can be used to solve the pricing problem of some callable-putable convertible bonds.At last,a numerical example is given to demonstrate our theoretical results visually by virtue of the finite difference method,because of the corresponding relationship between BSDEs with double obstacles and double obstacles problem for nonlinear partial differential equations.Chapter 3 concerns about a kind of stochastic recursive optimal control problems with obstacle constraint involving diffusion type control,where the cost functionals were described by some reflected BSDEs with regular control and diffusion type control.The diffusion type control is a right continuous with left limitation process with locally bounded variation paths.By the Lebesgue decomposition,this kind of control processes can be divided into the singular part which could be regard as a impulse control process and the pure jump part corresponding to a singular control process.Its pure jump part is likely to bring some jumps to the reflected BSDE.Considering the right continuous with left limitation obstacle process and obstacle constraint condition,we replaces the continuous increasing process in reflected BSDE with a right continuous with left limitation increasing process to offset some negative jumps of the diffusion type control term,and derive the existence and uniqueness of solution for the reflected BSDE involving diffusion type control.Hence,we introduce the related stochastic recursive optimal control problem with obstacle constraint involving diffusion type control and provide a sufficient condition to obtain its optimal regular control and diffusion type control.Then,a corresponding relation between recursive optimal mixed control problems involving diffusion type control and recursive optimal control problems with obstacle constraint involving diffusion type control is proposed.A sufficient condition for the optimal mixed control of this recursive optimal mixed control problem is obtained.Finally,to illustrate the results,we consider a class of linear recursive optimal mixed control problem involving diffusion type control,and obtain the explicit optimal stopping time,optimal regular control and optimal diffusion type control with optimal impulse moments.This result can be used to solve a class of utility maximization problems involving continuous andinstantaneous consumption processes.Chapter 4 focuses on the recursive optimal stopping problem with Poisson stopping constraints,in the sense that the stopping is only allowed at Poisson random intervention times.To begin with,we formulate this constrained recursive optimal stopping problem and introduce some relevant penalized BSDEs with jumps.We only consider a sequence of special extraneous stopping times,Poisson arrival times,because of the nice properties of Poisson arrival times and the intuition of the multiplier in penalization term.The Poisson stopping constraints make the equation from which we define recursive utility involving jumps and driven not only by a Brownian motion but also by the Poisson process.Therefore,the recursive utility now is described by a BSDE in a random time duration with the random jump measure generated by the Poisson process.Hence,the Poisson random interventions may be able to spill over into the coefficients of the dynamic model and some decomposition results with respect to the progressive enlargement of filtration are introduced to separately describe the dependence of the recursive utility on these interventions.Next,we initially focus on the case where the random generator of recursive utility is independent of the two hedge processes which are the last two components of the solution of BSDE in random time duration with jumps,and derive the connection between the solution of penalized BSDE with jumps and the value function of the above constrained optimal stopping problem in the whole time interval under the Brownian-Poisson filtration.The comparison theorem of BSDEs with jumps is critical to represent the value function of the above constrained optimal stopping problem when the initial time moves between two adjacent Poisson arrival times.And then,a similar penalized BSDE interpretation result for the case where the random generator is convex or concave with respect to the three components of the solution of BSDE in random time duration with jumps is given as a supplement.Finally,we consider four different kinds of optimal control problems,namely,recursive optimal control problem with intensity control constraints,recursive optimal mixed control problem involving distance penalization with Poisson stopping constraints,recursive optimal switching problem with Poisson switching constraints and risk-sensitive optimal stopping problem with Poisson stopping constraint,as application examples to illustrate our results and give the corresponding penalized equation representations.This result can be used to solve the pricing problem of American options with exercise time constraints in the market with credit default risk and liquidity risk.Chapter 5 gives a brief summary and prospect of this research.