Note on the HJB equation in ‘Dynamic Pricing with a Prior on Market Response’

A π,z J(z) = lim t>0,t→0 e −αt E z,π [J(z(t))] − J(z) t Lemma 1. Let J ∈ J satisfy J(0, a, b) = 0. Let τ = inf{t : J(z t) = 0}. E τ 0 e −αt e −π(z)/r a b π(z) + A π,z J(z) dt = J π (z 0) − J(z 0) Proof: Define H π J(z) = e −π(z)/r a b π(z) + A π,z J(z) Lemma 6 later verifies that this definition is in agreement with our previous definition (in [2]) provided J ∈ J. Let τ be a stopping time of the filtration σ(z t). We then have: E τ 0 e −αt H π J(z t)dt = E τ 0 e −αt e −π(zt)/r a b π(z t) + A π,z J(z t) dt = J π (z 0) + E z 0 e −ατ J(z τ) − J(z 0) = J π (z 0) − J(z 0) where the second equality follows from Dynkin's formula. 2 Our proofs to both Theorems 1 and 2 in [2] will rely on showing the existence of a bounded solution to the HJB Equation (HJ)(z) = 0 for z ∈ S ˜ x,˜ a, ˜ b .