Impulse Control in Finance: Numerical Methods and Viscosity Solutions

Abstract

The goal of this thesis is to provide efficient and provably convergent numerical methods for solving partial differential equations (PDEs) coming from impulse control problems motivated by finance. Impulses, which are controlled jumps in a stochastic process, are used to model realistic features in financial problems which cannot be captured by ordinary stochastic controls. In this thesis, we consider two distinct cases of impulse control: one in which impulses can occur at any time and one in which they occur only at “fixed” (i.e., nonrandom and noncontrollable) times. The first case is used to model features in finance such as fixed transaction costs, liquidity risk, execution delay, etc. In this case, the corresponding PDEs are HamiltonJacobi-Bellman quasi-variational inequalities (HJBQVIs). Other than in certain special cases, the numerical schemes that come from the discretization of HJBQVIs take the form of complicated nonlinear matrix equations also known as Bellman problems. We prove that a policy iteration algorithm can be used to compute their solutions. In order to do so, we employ the theory of weakly chained diagonally dominant (w.c.d.d.) matrices. As a byproduct of our analysis, we obtain some new results regarding a particular family of Markov decision processes which can be thought of as impulse control problems on a discrete state space and the relationship between w.c.d.d. matrices and M-matrices. Since HJBQVIs are nonlocal PDEs, we are unable to directly use the seminal result of Barles and Souganidis (concerning the convergence of monotone, stable, and consistent numerical schemes to the viscosity solution) to prove the convergence of our schemes. We address this issue by extending the work of Barles and Souganidis to nonlocal PDEs in a manner general enough to apply to HJBQVIs. We apply our schemes to compute the solutions of various classical problems from finance concerning optimal control of the exchange rate, optimal consumption with fixed and proportional transaction costs, and guaranteed minimum withdrawal benefits in variable annuities. The second case of impulse control, involving impulses occurring at fixed times, is frequently used in pricing and hedging insurance contracts. In this case, the impulses correspond to regular anniversaries (e.g., monthly, yearly, etc.) at which the holder of the contract can perform certain actions (e.g., lapse the contract). The corresponding pricing equations are a sequence of linear PDEs coupled by nonlinear constraints corresponding to the impulses. For these problems, our focus is on speeding up the computation associated with the nonlinear constraints by means of a control reduction. We apply our results to price guaranteed lifelong withdrawal benefits in variable annuities.