Robust control and model misspecification

Abstract

A decision maker fears that data are generated by a statistical perturbation of an approximating model that is either a controlled diffusion or a controlled measure over continuous functions of time. A perturbation is constrained in terms of its relative entropy. Several different two-player zero-sum games that yield robust decision rules and are related to one another, to the max-min expected utility theory of Gilboa and Schmeidler (1989), and to the recursive risk-sensitivity criterion described in discrete time by Hansen and Sargent (1995). To represent perturbed models, we use martingales on the probability space associated with the approximating model. Alternative sequential and non-sequential versions of robust control theory imply identical robust decision rules that are dynamically consistent in a useful sense.

Relative Entropy, Exponential Utility, and Robust Dynamic Pricing

“…Following this, we formulate the robust pricing problem as a two-player zero-sum stochastic differential game through the Girsanov Theorem for point processes. This section is intended to serve as an introduction to this general approach to formulating stochastic optimization problems when there is model uncertainty, and the reader is directed to Dai Pra et al (1996) and Petersen et al (2000) for more details, Charalambous et al (2004) and Ugrinovskii and Petersen (1999) for further work in this direction, and Hansen et al (2005) for finance applications. We would like to mention, however, that the primary focus of the literature has been discrete-time problems or continuoustime problems where uncertainty is modelled by Brownian motion.…”

“…Following Chen and Epstein (2002) (see also Gilboa andSchmeidler 1989 andHansen et al 2005), we use the terms ambiguity or model uncertainty (which will be used interchangeably in this paper) to describe the general situation where there is uncertainty concerning the accuracy of the underlying stochastic model. This contrasts to a risky situation (such as the toss of a fair coin) where the probability distribution on the set of possible outcomes is known (i.e., unambiguous).…”

“…Also, = 0 if and only if t = 1 (or equivalently, ≡ ). In this section, we show how model uncertainty can be expressed through a constraint on relative entropy, which, together with the Girsanov Theorem (Theorem 3.1), leads to a two-player zero-sum stochastic differential game that can be solved for the robust pricing policy (see also Dai Pra et al 1996, Hansen et al 2005, and Petersen et al 2000 for a similar construction).…”

“…The size of this constraint set represents our confidence in the nominal model, and in the special case where it is a singleton set, we recover the problem studied in Gallego and van Ryzin (1994). Similar approaches to the problem of dynamic optimization with uncertainty have been used in systems theory and electrical engineering (Charalambous et al 2004, Dai Pra et al 1996, Petersen et al 2000, Ugrinovskii and Petersen 1999, 2001) as well as economics and finance (Hansen et al 2005), although in these contexts, randomness is modelled using Brownian motion. One contribution of this paper is the application of this general relative entropy approach to the problem of dynamic revenue management where uncertainty is modelled by a point process.…”

“…Following this, we show in §3 how uncertainty in the arrival rate model for a single-product pricing problem can be represented using the notion of relative entropy. This formulation (an adaptation of ideas from Dai Pra et al 1996, Petersen et al 2000, and Hansen et al 2005) allows us to account for possible path dependence in the actual real-world arrival rate process and errors in the nominal model through a constraint on relative entropy. We show in §4 how the problem of optimal pricing when there is model uncertainty can be formulated as a stochastic differential game (see Dai Pra et al 1996, Petersen et al 2000, and solved through a version of the so-called Isaacs' equation, which is a generalization of the dynamic programming equation to two-player zero-sum stochastic differential games.…”