Stochastic Optimal Control Theory Under Nonlinear Expectation And Its Application In Finance

Motivated by the research on the pricing and risk control of financial derivatives under volatility uncertainty of stock,Professor Shige Peng established Gexpectation theory,which includes the central limit theorem under the sublinear expectation,the limit distribution is called G-normal distribution,the Gexpectation space and the related G-Brownian motion were constructed based on this distribution,the corresponding stochastic analysis theories were established for G-Brownian motion.Here G is a monotone sublinear function,which describes the strength of volatility uncertainty.At present,more and more experts at home and abroad have begun to focus on the study of this field.No matter in the theoretical research or in the practical application,this field has a very good prospect.It is inevitable to consider the optimization problem in finance,for this purpose,Professor Mingshang Hu and Shaolin Ji established the maximum principle and dynamic programming principle for stochastic recursive optimal control problem under non-degenerate G-expectation.In this paper,we first extend the above stochastic recursive optimal control problem to nonlinear expectation dominated by G-expectation and degenerate G-expectation respectively,and then study the relationship between maximum principle and dynamic programming principle,finally study the maximum principle for discrete-time stochastic optimal control problem under distribution uncertainty corresponding to the above continuoustime control problem.The main research content consists of the following four parts.In the first part,we study the dynamic programming principle for stochastic recursive optimal control problem under nonlinear expectation.We first obtain the comparison theorem for backward stochastic differential equation under nonlinear expectation,and prove that the sequence of simple random variables in the G-expectation space can approximate any random variable in the sense of mean square convergence.Based on this,we give a novel method to prove dynamic programming principle,which is simpler than the implied partition method.Based on dynamic programming principle,we further obtain the corresponding Hamilton-Jacobi-Bellman(HJB)equation and stochastic verification theorem.In the second part,we study the dynamic programming principle for stochastic recursive optimal control problem under degenerate G-expectation.Firstly,we study the backward stochastic differential equation driven by degenerate GBrownian motion(G-BSDE).Compared with the non-degenerate case,the difficulty lies in that the solution of the corresponding partial differential equation is not smooth.To overcome this difficulty,we obtain the solution for degenerate G-BSDE in the extended space,and propose a new probabilistic method based on the representation theorem of G-expectation and weak convergence to obtain the uniform lower bound estimate for second-order partial derivative,which can not be obtained by Krylov’s estimate.We further establish the existence and uniqueness theorem for degenerate G-BSDE,and obtain the application of this theory to the regularity of fully nonlinear partial differential equation and the pricing of financial derivatives under volatility uncertainty.Secondly,we obtain the viscosity solution for a class of degenerate G-heat equation by using stochastic control method.We further obtain the corresponding G-capacity,and prove that the indicate function of degenerate G-Brownian motion is not in G-expectation space,which is completely different from the non-degenerate case.Finally,in order to overcome this difficulty that the indicate function can not be used directly under the degenerate case,we give the approximation of admissible control by simple stochastic processes and random variable by simple random variables for two kinds of multidimensional degenerate G-expectation,and then obtain the dynamic programming principle and HJB equation for stochastic optimal control problem based on degenerate G-BSDE.In the third part,we study the relationship between maximum principle and dynamic programming principle for stochastic recursive optimal control problem under non-degenerate G-expectation,and give its application in finance.Under the smooth assumption for the value function,we obtain the relationship between maximum principle and dynamic programming principle under a reference probability,and show that this relationship may not hold for other probability in the representation theorem of G-expectation by an example.In the sense of viscosity solution,we establish the relation between the first-order super-jet,sub-jet of the value function and the solution to the adjoint equation in maximum principle,and show that the super-jet may be empty by an example,which is completely different from the relationship between maximum principle and dynamic programming principle under one probability case.In the fourth part,we study discrete stochastic optimal control problem under distribution uncertainty of noise.By using the weak convergence method,we obtain the variational equation for cost functional of this control problem.Furthermore,we obtain the variational inequality on a reference probability by Sion’s minimax theorem.Under the square integrability condition for noise and control,we obtain the integrability of the solution to the adjoint equation.Based on this,we establish the maximum principle under a reference probability,and prove that this maximum principle is also a sufficient condition under usual convex assumptions.It should be pointed out that the integrability requirements of noise and control for the existing results under one probability depend on the number N of controls.In particular,we introduce a backward algorithm to calculate the reference probability in maximum principle and the optimal control,and further give its application in finance.It is worth emphasizing that this reference probability is part of maximum principle and the key to applying maximum principle.