Adaptive Robust Control in Continuous Time

Abstract

We propose a continuous-time version of the adaptive robust methodology introduced in Bielecki et al. ( 2019). An agent solves a stochastic control problem where the underlying uncertainty follows a jump-diffusion process and the agent does not know the drift parameters of the process. The agent considers a set of alternative measures to make the control problem robust to model misspecification, and employs a continuous-time estimator to learn the value of the unknown parameters to make the control problem adaptive to the arrival of new information. We use measurable selection theorems to prove the dynamic programming principle of the adaptive robust problem and show that the value function of the agent is characterised by a non-linear partial differential equation. As an example, we derive the optimal adaptive robust strategy for an agent who acquires a large amount of shares in an order driven market and illustrate the financial performance of the execution strategy.

Rigorous multi-asset optimal execution with Bayesian learning of the drift

“…In order to solve the dynamic optimization problem (12), we use the tools of stochastic optimal control. Let us introduce the value function associated with the above problem w :…”

“…The HJB equation associated with our problem (12), and which we expect the value function to solve, is ∂ t u (t, x, q, S) + G(t, S) ⊺ Σ∇ S u (t, x, q, S) + 1 2 Tr ΣD 2 SS u (t, x, q, S) + sup…”

“…[17] also uses a similar approach for deriving optimal strategies with learning of posterior latent state distribution upon which the evolution of prices depend. More recently and similar to the problem addressed in this paper, [12] proposes a general approach for adaptive control which is robust to model misspecification, where the agent continuously learns the drift and her uncertainty follows a jump-diffusion. We also note that combining learning with optimization in mathematical finance is more explored in the context of the optimal portfolio choice literature.…”

Nonparametric Adaptive Robust Control Under Model Uncertainty

“…To the best of our knowledge, [BCC + 19] is the first work that incorporates the idea of online learning into the robust control paradigm. A follow-up work in [BC21] is an attempt to extend the adaptive robust control to the continuous time setup.…”

“…Note that the methods in [BCC + 19] and [BC21] are parametric and the practical usage of such methods relies on the assumption that the family of the unknown probability law of the underlying stochastic process is known to the controller. Some researchers have realized this drawback and adopts nonparametric statistical methods by assuming uncertainty for the family of parametric models.…”