Probabilistic error analysis for some approximation schemes to optimal control problems

Abstract

We introduce a class of numerical schemes for optimal stochastic control problems based on a novel Markov chain approximation, which uses, in turn, a piecewise constant policy approximation, Euler-Maruyama time stepping, and a Gauß-Hermite approximation of the Gaußian increments. We provide lower error bounds of order arbitrarily close to 1/2 in time and 1/3 in space for Lipschitz viscosity solutions, coupling probabilistic arguments with regularization techniques as introduced by Krylov. The corresponding order of the upper bounds is 1/4 in time and 1/5 in space. For sufficiently regular solutions, the order is 1 in both time and space for both bounds. Finally, we propose techniques for further improving the accuracy of the individual components of the approximation.

Duality-based a posteriori error estimates for some approximation schemes for optimal investment problems

“…While standard finite difference schemes are in general non-monotone, semi-Lagrangian (SL) schemes (see [22,6,10]) are monotone by construction. The basic scheme considered in this paper belongs to this family and has been previously analyzed in [25].…”

“…We focus here on computable error bounds for the solution. Many of the published error bounds for this kind of maximisation problem, including those in [25], are asymmetrical in the sense that a more accurate lower bound can be given than the upper bound. In this work, we construct an upper bound which consists of two additive contributions: a term which can be computed a priori from the model parameters and is of the same order in the mesh parameters as the known lower bounds; and a term which can be computed a posteriori from the solution of the dual problem.…”

“…These results are mainly derived by PDE techniques and strongly rely on the comparison principle between viscosity sub-and super-solutions of the HJB equation and the consistency properties of the scheme. For the scheme considered in the present paper, the probabilistic proof in [25] exploits the fact that the numerical scheme is based on a discrete approximation of the optimal control problem, specifically by a piecewise constant policy approximation, Euler-Maruyama time stepping, and a Gauß-Hermite approximation of the Gaußian increments. This yields the desired error bounds by a direct comparison between two value functions and leads to an improvement of the error contribution of the second and third of these approximations by avoiding the use of the truncation error.…”

“…This yields the desired error bounds by a direct comparison between two value functions and leads to an improvement of the error contribution of the second and third of these approximations by avoiding the use of the truncation error. The piecewise constant policy approximation, however, introduces an asymmetry between the upper and the lower bound of the error and, as a result, the bounds in [25] give only a partial improvement of the classical PDE-based results.…”

“…More specifically, a direct application of the results in [25] to this problem gives one-sided (lower) error bounds for the considered Markov chain approximation of order h (M −1)/2M + ∆x (M −1)/(3M −1) (1.1) for timestep h, spatial mesh size ∆x and number of Gaußian points M , for Lipschitz viscosity solutions. They coincide with the two-sided bounds in [10] for the standard linear-interpolation SL scheme, i.e.…”