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

Abstract

We consider a Markov chain approximation scheme for utility maximization problems in continuous time, which uses, in turn, a piecewise constant policy approximation, Euler-Maruyama time stepping, and a Gauß-Hermite approximation of the Gaußian increments. The error estimates previously derived in A. Picarelli and C. Reisinger, Probabilistic error analysis for some approximation schemes to optimal control problems, arXiv:1810.04691 are asymmetric between lower and upper bounds due to the control approximation and improve on known results in the literature in the lower case only. In the present paper, we use duality results to obtain a posteriori upper error bounds which are empirically of the same order as the lower bounds. The theoretical results are confirmed by our numerical tests.

Probabilistic error analysis for some approximation schemes to optimal control problems

“…The error bound obtained in this way allows us to improve one side of previous results from the literature. In ongoing work [22], we are investigating the use of duality to obtain symmetric bounds.…”

An algorithm to construct subsolutions of convex optimal control problems

“…The authors discuss a wide class of estimators for the conditional expectation of the value function that exploit this assumption. Picarelli and Reisinger (2020) consider a maximization problem over a finite time horizon where only the terminal costs are non-zero. They approximate the continuous-time problem by discretizing space in steps of size ∆x, time in steps of h, and by approximating the expectations with respect to the Gaussian noise via the Gauss-Hermite quadrature formula.…”