Stochastic Approximation with Averaging Innovation Applied to Finance

The aim of the paper is to establish a convergence theorem for multi-dimensional stochastic approximation when the "innovations" satisfy some "light" averaging properties in the presence of a pathwise Lyapunov function. These averaging assumptions allow us to unify apparently remote frameworks where the innovations are simulated (possibly deterministic like in Quasi-Monte Carlo simulation) or exogenous (like market data) with ergodic properties. We propose several fields of applications and illustrate our results on five examples mainly motivated by Finance

Top-down effects on early visual processing in humans: a predictive coding framework

An increasing number of human electroencephalography (EEG) studies examining the earliest component of the visual evoked potential, the so-called C1, have cast doubts on the previously prevalent notion that this component is impermeable to top-down effects. This article reviews the original studies that (i) described the C1, (ii) linked it to primary visual cortex (V1) activity, and (iii) suggested that its electrophysiological characteristics are exclusively determined by low-level stimulus attributes, particularly the spatial position of the stimulus within the visual field. We then describe conflicting evidence from animal studies and human neuroimaging experiments and provide an overview of recent EEG and magnetoencephalography (MEG) work showing that initial V1 activity in humans may be strongly modulated by higher-level cognitive factors. Finally, we formulate a theoretical framework for understanding top-down effects on early visual processing in terms of predictive coding

Optimal posting price of limit orders: learning by trading

Considering that a trader or a trading algorithm interacting with markets during continuous auctions can be modeled by an iterating procedure adjusting the price at which he posts orders at a given rhythm, this paper proposes a procedure minimizing his costs. We prove the a.s. convergence of the algorithm under assumptions on the cost function and give some practical criteria on model parameters to ensure that the conditions to use the algorithm are fulfilled (using notably the co-monotony principle). We illustrate our results with numerical experiments on both simulated data and using a financial market dataset

Optimal split of orders across liquidity pools: a stochastic algorithm approach

Evolutions of the trading landscape lead to the capability to exchange the same financial instrument on different venues. Because of liquidity issues, the trading firms split large orders across several trading destinations to optimize their execution. To solve this problem we devised two stochastic recursive learning procedures which adjust the proportions of the order to be sent to the different venues, one based on an optimization principle, the other on some reinforcement ideas. Both procedures are investigated from a theoretical point of view: we prove a.s. convergence of the optimization algorithm under some light ergodic (or "averaging") assumption on the input data process. No Markov property is needed. When the inputs are i.i.d. we show that the convergence rate is ruled by a Central Limit Theorem. Finally, the mutual performances of both algorithms are compared on simulated and real data with respect to an "oracle" strategy devised by an "insider" who knows a priori the executed quantities by every venues

Optimal split of orders across liquidity pools: a stochastic algorithm approach

Evolutions of the trading landscape lead to the capability to exchange the same financial instrument on different venues. Because of liquidity issues, the trading firms split large orders across several trading destinations to optimize their execution. To solve this problem we devised two stochastic recursive learning procedures which adjust the proportions of the order to be sent to the different venues, one based on an optimization principle, the other on some reinforcement ideas. Both procedures are investigated from a theoretical point of view: we prove a.s. convergence of the optimization algorithm under some light ergodic (or "averaging") assumption on the input data process. No Markov property is needed. When the inputs are i.i.d. we show that the convergence rate is ruled by a Central Limit Theorem. Finally, the mutual performances of both algorithms are compared on simulated and real data with respect to an "oracle" strategy devised by an "insider" who knows a priori the executed quantities by every venues.Asset allocation, Stochastic Lagrangian algorithm, reinforcement principle, monotone dynamic system

Assesing the frequency and clauses of out-of-stock events through store scanner data

This paper aims to provide an answer to the question of out-of-stock events (OOS), their frequency, the sales losses they generate, and their causes. The authors provide two contributions. They describe a new sales-based measure of OOS computed on the basis of store-level scanner data and identify several of the main determinants of OOS. They also introduce a significant distinction between complete and partial OOSout-of-stock events; store-level scanner data; assortment; retailing; marketing metrics