Quadratic Loss Minimization in a Regime Switching Model with Control and State Constraints

Abstract

In this thesis, we address a convex stochastic optimal control problem in mathematical finance, with the goal of minimizing a general quadratic loss function of the wealth at close of trade. We study this problem in the setting of an Ito process market model, in which the underlying filtration to which the market parameters are adapted is the joint filtration of the driving Brownian motion for the market model, together with the filtration of an independent finite-state Markov chain which models occasional changes in ``regime states'', that is our model allows for ``regime switching'' among a finite number of regime states. Other aspects of the problem that we address in this thesis are:

(1) The portfolio vector of holdings in the risky assets is confined to a given closed and convex constraint set;

(2) There is a ``state constraint'' in the form of a stipulated almost-sure lower bound on the wealth at close of trade.

The combination of constraints represented by (1) and (2) makes the optimization problem quite challenging. The powerful and effective method of {\em auxiliary markets}, of Cvitanic and Karatzas [Ann. Appl. Prob., v.2, 767-818, 1992] for dealing with convex portfolio constraints, does not appear to extend to problems with regime-switching, while the more recent approach of Donnelly and Heunis [SIAM Jour. Control Optimiz., v.50, 2431-2461, 2012], which deals with both regime-switching and the convex portfolio constraints (1), is nevertheless confounded when one adds state constraints of the form (2) to the problem. The reason for this is clear: state constraints of the form (2) typically involve ``singular'' Lagrange multipliers which fall well outside the scope of the ``well-behaved'' Lagrange multipliers, manifested either as random variables or stochastic processes, which suffice when one is dealing only with portfolio constraints such as (1) above. In these circumstances we resort to an ``abstract'' duality approach of Rockafellar and Moreau, which has been applied with considerable success to finite-dimensional problems of stochastic mathematical programming in which singular Lagrange multipliers also naturally arise. The main goal of this thesis is to adapt and extend the Rockafellar-Moreau approach to the stochastic optimal control problem summarized above. We find that this is indeed possible, although some considerable effort is required in view of the infinite dimensionality of the problem. We construct an appropriate space of Lagrange multipliers, synthesize a dual optimization problem, establish optimality relations which give necessary and sufficient conditions for the given optimization problem and its dual to each have a solution with zero duality gap, and use the optimality relations to synthesize an optimal portfolio in terms of the Lagrange multipliers.