Stochastic maximum principle and dynamic convex duality in continuous-time constrained portfolio optimization

This thesis seeks to gain further insight into the connection between stochastic optimal control and forward and backward stochastic differential equations and its applications in solving continuous-time constrained portfolio optimization problems. Three topics are studied in this thesis. In the first part of the thesis, we focus on stochastic maximum principle, which seeks to establish the connection between stochastic optimal control and backward stochastic differential differential equations coupled with static optimality condition on the Hamiltonian. We prove a weak neccessary and sufficient maximum principle for Markovian regime switching stochastic optimal control problems. Instead of insisting on the maxi- mum condition of the Hamiltonian, we show that 0 belongs to the sum of Clarkes generalized gradient of the Hamiltonian and Clarkes normal cone of the control constraint set at the optimal control. Under a joint concavity condition on the Hamiltonian and a convexity condition on the terminal objective function, the necessary condition becomes sufficient. We give four examples to demonstrate the weak stochastic maximum principle. In the second part of the thesis, we study a continuous-time stochastic linear quadratic control problem arising from mathematical finance. We model the asset dynamics with random market coefficients and portfolio strategies with convex constraints. Following the convex duality approach,we show that the necessary and sufficient optimality conditions for both the primal and dual problems can be written in terms of processes satisfying a system of FBSDEs together with other conditions. We characterise explicitly the optimal wealth and portfolio processes as functions of adjoint processes from the dual FBSDEs in a dynamic fashion and vice versa. We apply the results to solve quadratic risk minimization problems with cone-constraints and derive the explicit representations of solutions to the extended stochastic Riccati equations for such problems. In the final section of the thesis, we extend the previous result to utility maximization problems. After formulating the primal and dual problems, we construct the necessary and sufficient conditions for both the primal and dual problems in terms of FBSDEs plus additional conditions. Such formulation then allows us to explicitly characterize the primal optimal control as a function of the adjoint processes coming from the dual FBSDEs in a dynamic fashion and vice versa. Moreover, we also find that the optimal primal wealth process coincides with the optimal adjoint process of the dual problem and vice versa. Finally we solve three constrained utility maximization problems and contrasts the simplicity of the duality approach we propose with the technical complexity in solving the primal problem directly.Open Acces

Canonical Dual Algorithms for Global Optimization with Applications

Canonical duality theory provides a unified framework which can transform a nonconvex primal minimization problem to a canonical dual maximization problem over a convex domain without duality gap. But the global optimality is guaranteed by a certain positive definite condition and such condition is not always satisfied. The goal of this thesis aims to explore possible techniques that can be used to solve global optimization problems based on the canonical duality theory. Firstly, an algorithmic framework for canonical duality theory is established, which shows that the canonical dual algorithms can be developed in four aspects under the positive definite condition explicitly or implicitly, namely, (i) minimizing the primal problem, (ii) maximizing the canonical dual problem, (iii) solving a nonlinear equation caused by total complementary function, and (iv) solving a nonlinear equation caused by canonical dual function. Secondly, we show that if there exists a critical point of the canonical dual problem in the positive definite domain, by solving an equivalent semidefinite programming (SDP) problem, the corresponding global solution to the primal problem can be obtained easily via off-the-shelf software packages. A specific canonical dual algorithm is given for each problem, including sum of fourth-order polynomials minimization, nonconvex quadratically constrained quadratic program (QCQP), and boolean quadratic program (BQP). Thirdly, we propose a canonical primal-dual algorithm framework based on the total complementary function. Convergence analysis is discussed from the perspective of variational inequalities (VIs) and contraction methods. Specific canonical primal-dual algorithms for sum of fourth-order polynomials minimization is given as well. And a real-world application to the sensor network localization problem is illustrated. Next, a canonical sequential reduction approach is proposed to recover the approximate or global solution for the BQP problem. By fixing some previously known components, the original problem can be reduced sequentially to a lower dimension one. This approach is successfully applied to the well-known maxcut problem. Finally, we discuss the canonical dual approach applied to continuous time constrained optimal control. And it shows that the optimal control law for the n-dimensional constrained linear quadratic regulator can be achieved precisely via one-dimensional canonical dual variable, and for the optimal control problem with concave cost functional, an approximate solution can be obtained by introducing a linear perturbation term.Ph

Constrained Quadratic Risk Minimization via Forward and Backward Stochastic Differential Equations

In this paper we study a continuous-time stochastic linear quadratic control problem arising from mathematical finance. We model the asset dynamics with random market coefficients and portfolio strategies with convex constraints. Following the convex duality approach, we show that the necessary and sufficient optimality conditions for both the primal and dual problems can be written in terms of processes satisfying a system of FBSDEs together with other conditions. We characterise explicitly the optimal wealth and portfolio processes as functions of adjoint processes from the dual FBSDEs in a dynamic fashion and vice versa. We apply the results to solve quadratic risk minimization problems with cone-constraints and derive the explicit representations of solutions to the extended stochastic Riccati equations for such problems.Comment: 22 page