General Quadratic Risk Minimization: a Variational Approach
Mean-variance portfolio selection and mean-variance hedging are mainstream research topics in mathematical nance, which can be subsumed within the framework of a general problem of quadratic risk minimization. We study this quadratic risk minimization problem in the setting of an It^o process market model with random market parameters. Our particular contribution is to introduce a combination of constraints on both the trading strategy (i.e. portfolio) and the wealth process, which includes in particular portfolio insurance in the form of a stipulated lower-bound on the wealth process over the entire trading interval (this is also called an American wealth constraint). The result is a stochastic control problem which includes the combination of a portfolio constraint (i.e. a \control constraint") and a wealth constraint over the trading interval (i.e. a \state constraint"). The goal of the present thesis is to address this stochastic control problem. Even in the setting of deterministic (or non-random) optimal control it is well known that a combination of control constraints and state constraints over the control interval presents some particular challenges, and of course these challenges increase considerably for stochastic control problems with the same combination of constraints. In this thesis we shall take advantage of the convexity of the problem and apply a powerful variational method of Rockafellar which has proved to be very e ective in the deterministic optimal control of partial di erential equations, convex optimization in continuum mechanics, and stochastic convex programming over nite dimensional spaces. The variational approach of Rockafellar enables one to systematically construct an appropriate vector space of dual variables, together with a dual problem on this space of dual variables, and gives conditions which ensure that there is zero duality gap (i.e. the values of the primal and dual problems are equal) as well as existence of a solution of the dual problem (i.e. existence of Lagrange multipliers for the constraints in the problem). The key to applying the Rockafellar variational approach to the stochastic control problem outlined above turns out to be a mild feasibility condition on the wealth process which is very reminiscent of \Slater-type" conditions familiar from convex optimization. With this condition in place we are able to construct an associated dual problem, and establish existence of a solution of the dual problem, together with Kuhn-Tucker optimality conditions which relate putative solutions of the primal and dual problems. We then use these optimality conditions to construct an optimal portfolio in terms of the solution of the dual problem.