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 full 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 multipliers 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.
Andrew Heunis was born in Johannesburg, South Africa, and educated at the University of the Witwatersrand, Johannesburg, and Imperial College, London, receiving the PhD degree in Electrical Engineering from the latter institution in 1984. Since 1985 he has been at the Department of Electrical & Computer Engineering, University of Waterloo, Ontario, Canada. During 1992 he was Chercheur Invite at IRISA/INRIA, Rennes, France. His main interests are in applied probability, with particular focus on limit theorems (for system identification, averaging principles and recursive stochastic algorithms), nonlinear filtering, and more recently, problems of stochastic optimal control motivated particularly by portfolio optimization in mathematical finance.