“ Adaptive Execution : Exploration and Learning of Price Impact ” by Beomsoo Park and Benjamin

Theorem 1. For any θ * ∈ Θ, there exists a unique symmetric solution P to (5) that satisfies R + B P B > 0 and ρ sr (A + BL) < 1 where L = −(R + B P B) −1 (S + B P A) and ρ sr (·) denotes a spectral radius. Moreover, a policy π = (π 1 , π 2 ,. . .) with π t (I t−1) = Lz t−1 is an optimal policy among admissible policies that attains minimum expected average cost tr(P ˜ Ω). Proof. First, we will show the existence of a solution P to (5) that satisfies R + B P B > 0 and ρ sr (A + BL) < 1. One challenge is that the matrix pair (A, B) is not controllable because return-predictive factors f t evolve over time independently of actions u t. This autonomous factor dynamics leads us into a useful observation: When we reexpress (5) in terms of P =        P xx