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

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## Date

2015-04-09

## Authors

Ramchandani, Pradeep

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## Publisher

University of Waterloo

## 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.

## Description

## Keywords

Quadratic loss minimization, variance minimzation, Rockafellar-Moreau Approach, perturbation, Lagrangian, dual problem, Kuhn-Tucker optimality relations, portfolio process, Yodida-Hewitt decomposition, normal integrands, canonical martingales, martingale representation theorem.