Pathwise stochastic control with applications to robust filtering

Abstract

We study the problem of pathwise stochastic optimal control, where the optimization is performed for each fixed realisation of the driving noise, by phrasing the problem in terms of the optimal control of rough differential equations. We investigate the degeneracy phenomenon induced by directly controlling the coefficient of the noise term, and propose a simple procedure to resolve this degeneracy whilst retaining dynamic programming. As an application, we use pathwise stochastic control in the context of stochastic filtering to construct filters which are robust to parameter uncertainty, demonstrating an original application of rough path theory to statistics.

Robust filtering and propagation of uncertainty in hidden Markov models

“…In fact, we have formulated our problem in terms of the optimal control of a rough differential equation, which we wish to perform for an arbitrary geometric rough path Y ∈ V 0,p g . Control problems of this type were first studied by Diehl et al [19], and subsequently by Allan and Cohen [3].…”

“…The first use of rough path theory in uncertainty-robust filtering was presented in Section 4 of Allan and Cohen [3], allowing to extend the results of [2], but remaining in the Kalman-Bucy setting. We highlight the results of Crisan, Diehl, Friz and Oberhauser [17] as the first use of rough paths to establish continuity of stochastic filters with respect to the (enhanced) observation path.…”

“…Another early application of rough paths to robust statistics was exhibited by Diehl, Friz and Mai [20], who show that continuity of maximum likelihood estimators for diffusion processes with unknown parameters is recovered upon lifting the observation to a rough path. The collection of works [3,17,20] thus provides three different but important notions of robustness in the statistics of observed diffusion processes.…”

“…One of the key steps in our approach is the derivation of a pathwise optimal control problem driven by the observation path t → Y t (ω). As was demonstrated in Diehl, Friz and Gassiat [19] and subsequently in [3], rough path theory provides a convenient framework for pathwise control problems, in which they are formulated as the optimal control of a rough differential equation (RDE). In the current work we also provide new results in this direction, particularly on the regularity of the corresponding (rough) value function.…”

“…In the current work we also provide new results in this direction, particularly on the regularity of the corresponding (rough) value function. A particular difficulty which we face, not encountered in [19] or [3], is that the spatial domain of our value function is actually time-dependent (and indeed rough), and thus requires delicate analysis. Moreover, and of independent interest, we also provide a new growth estimate for rough integrals depending on unconstrained parameters, which not only improves upon the corresponding estimate in [3], but in fact turns out to be sharp (see Lemma 2.5 below).…”