Peter A. Forsyth

Parsiad Azimzadeh, Peter A. Forsyth

This work is motivated by numerical solutions to Hamilton-Jacobi-Bellman quasi-variational inequalities (HJBQVIs) associated with combined stochastic and impulse control problems. In particular, we consider (i) direct control, (ii) penalized, and (iii) semi-Lagrangian discretization schemes applied to the HJBQVI problem. Scheme (i) takes the form of a Bellman problem involving an operator which is not necessarily contractive. We consider the well-posedness of the Bellman problem and give sufficient conditions for convergence of the corresponding policy iteration. To do so, we use weakly chained diagonally dominant matrices, which give a graph-theoretic characterization of weakly diagonally dominant M-matrices. We compare schemes (i)--(iii) under the following examples: (a) optimal control of the exchange rate, (b) optimal consumption with fixed and proportional transaction costs, and (c) pricing guaranteed minimum withdrawal benefits in variable annuities. We find that one should abstain from using scheme (i).

Peter A. Forsyth, George Labahn

The decumulation of a defined contribution (DC) pension plan is well known to be one of the hardest problems in finance. We model this decumulation challenge as an optimal stochastic control problem. The control problem is solved, at each rebalancing date, by alternatively solving a linear partial-integro differential equation (PIDE) followed by an optimization step. We solve the PIDE by using a $δ$-monotone Fourier method, which ensures that monotonicity holds to $O(δ)$. We allow for the use of leverage (i.e. borrowing to invest in stocks), as well as minimum constraints on bond holdings. We pay particular attention to minimizing wrap-around error, an issue which is endemic for Fourier methods and central to the effective use of these methods for optimal control problems. Rather unexpectedly, we find that restricting the portfolio equity fraction to a maximum of 50\% does not reduce portfolio efficiency noticeably. This may be a useful strategy for risk-averse retirees.