General Quadratic Risk Minimization: a Variational Approach
Loading...
Date
2016-07-05
Authors
Zhu, Dian
Journal Title
Journal ISSN
Volume Title
Publisher
University of Waterloo
Abstract
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.
Description
Keywords
quadratic loss minimization, portfolio selection, American wealth constraint, European wealth constraint, Portfolio constraint, Stochastic Control, Convex optimization, Conjugate duality, Slater condition, Singular linear functionals, Lagrange multipliers, variational approach