Seminar at SMU Delhi
August 20, 2018 (Monday) ,
3:30 PM at Webinar
University of Waterloo
Quadratic loss minimization with portfolio and intertemporal wealth constraints
Abstract of Talk
We address a problem of stochastic optimal control
motivated by portfolio optimization in mathematical finance,
the goal of which is to minimize the expected value
of a general quadratic loss function of the
wealth at close of trade when there is a specified convex
constraint on the portfolio, together with
a specified almost-sure lower-bound on
intertemporal wealth over the trading interval.
In the parlance of optimal control this problem
exhibits the combination of a control constraint
(namely the portfolio constraint) together with an almost-sure
state constraint (namely the stipulated lower-bound on the wealth
process over the trading interval).
General optimal control problems with this combination of constraints
are known to be quite challenging, not least because the Lagrange
appropriate to such constraints are far from evident.
The problem that we address has the additional property of being convex,
and this is key to the application of an ingenious variational
method of R.T. Rockafellar for abstract convex optimization
which leads to a vector space of dual variables,
together with a dual functional on the space of
dual variables, such that the dual problem of
maximizing the dual functional is
guaranteed to have a solution (or Lagrange multiplier)
when the intertemporal state constraint satisfies
a simple and natural Slater condition.
This yields necessary and sufficient conditions
for the optimality of a candidate portfolio (i.e. control) process,
as well as the construction (in principle!) of an optimal portfolio.