2026/07/26 by Rabee Tourky, Paulo Monteiro
Economics, Econometrics and Finance · Mathematics · #econ.EM #econ.TH #math.FA #math.PR #msc:60E15 #msc:60G35 #msc:62C10 #msc:91G15
20 pages. v2: Added author Monteiro. Main theorem slightly generalised to quadratic trading costs to include neighboring literature; the Kyle model is corner case. Title and exposition revised; no changes to the gamma = 0 result. Corrected an error in exposition of authors' 2017 paper. numerous streamlining of proof
arxiv created 2026/08/01 · arxiv updated 2026/08/04
Let V and U be independent standard normal random variables. For a Borel function ϕ: ℝ → ℝ, let Pϕ be a version of the inverse regression Pϕ(y) = E[V | ϕ(V)+U = y], and let Fϕ(x) = E[Pϕ(x+U)] be its Gaussian smoothing. We prove that ϕ(v) ∈ argmaxx \ xv - x Fϕ(x) \ for every real v if and only if ϕ= id, the identity function. This is the pointwise best-response condition of the normalised one-period Kyle insider trading model; consequently the affine equilibrium strategy of Kyle (1985) is unique amongst all strategies. This settles the uniqueness question for the one-period Gaussian Kyle model. The additional ingredient relative to the McLennan, Monteiro and Tourky (2017) analytic framework is probabilistic: an exchangeable pair, obtained by resampling the value from the market makers' posterior, whose balance identities, combined with a Gaussian inequality for monotone functions, make the posterior mean affine.