From Forecast to Position: Order-Book Signals, Market Impact, and Causal Trading

An order-book forecast becomes a trading decision through costs, constraints, and the information available when the position is chosen. This paper develops loss identities and certificates for that conversion. Positive curvature yields a convex-conjugate Bregman loss and a quadratic error bound, with a sharp refinement for bounded positions. Purely proportional costs create threshold decisions with linear worst-case sensitivity and sharp margin rates. For scalar forecast intervals, a conjugate-secant formula gives the exact minimax position, including nonsmooth costs and position bounds. Queue depletion attains the corresponding quadratic and linear minimax losses. A finite marked-event book model transfers generator, price-mark, and posterior errors into clock-time forecasts, with individually sharp horizon coefficients. A finite subspace test characterizes features that preserve every conditional-mean price forecast; such features need not define a Markov aggregate. A price-specific reduction residual sharpens the transition-norm comparison. Positive likelihood and future-increment enclosures give an alternative interval certificate before the position is chosen. Information unavailable to the trader has a separate decision cost. An exact convex decomposition separates this observation loss from implementation loss. For delayed book observations, a jump-variance identity prices the unobserved state changes; it distinguishes their first-order latency cost from the smaller bias of failing to propagate a stale state. A two-regime example has an explicit delay beyond which trading optimally stops. For trajectories, an adapted-space residual combines forecast, impact, and optimization errors under convex constraints, with exact affine information decompositions and a verified stationary causal tracking specialization. All results concern declared exogenous book laws and objectives. Synthetic calculations check the identities and rigorous enclosures without empirical profitability claims.