Statistical Learning with Sublinear Regret of Propagator Models
This work addresses optimal trading strategies under uncertainty for financial agents, representing an incremental advance in combining non-parametric estimation with stochastic control.
The paper tackles the problem of an agent liquidating a risky asset with unknown transient and temporary price impacts, by developing a trading algorithm that alternates between exploration and exploitation phases to achieve sublinear regret with high probability, with convergence rates characterized by the singularity of the propagator.
We consider a class of learning problems in which an agent liquidates a risky asset while creating both transient price impact driven by an unknown convolution propagator and linear temporary price impact with an unknown parameter. We characterize the trader's performance as maximization of a revenue-risk functional, where the trader also exploits available information on a price predicting signal. We present a trading algorithm that alternates between exploration and exploitation phases and achieves sublinear regrets with high probability. For the exploration phase we propose a novel approach for non-parametric estimation of the price impact kernel by observing only the visible price process and derive sharp bounds on the convergence rate, which are characterised by the singularity of the propagator. These kernel estimation methods extend existing methods from the area of Tikhonov regularisation for inverse problems and are of independent interest. The bound on the regret in the exploitation phase is obtained by deriving stability results for the optimizer and value function of the associated class of infinite-dimensional stochastic control problems. As a complementary result we propose a regression-based algorithm to estimate the conditional expectation of non-Markovian signals and derive its convergence rate.