This paper presents the link between stochastic approximation and clinical trials based on randomized urn models investigated by Bai and Hu [Stochastic Process. Appl. 80 (1999) 87-101], Bai and Hu [Ann. Appl. Probab. 15 (2005) 914-940] and Bai, Hu and Shen [J. Multivariate Anal. 81 (2002) 1-18].We reformulate the dynamics of both the urn composition and the assigned treatments as standard stochastic approximation (SA) algorithms with remainder. Then, we derive the a.s. convergence and the asymptotic normality [central limit theorem (CLT)] of the normalized procedure under less stringent assumptions by calling upon the ODE and SDE methods. As a second step, we investigate a more involved family of models, known as multi-arm clinical trials, where the urn updating depends on the past performances of the treatments. By increasing the dimension of the state vector, our SA approach provides this time a new asymptotic normality result.
Fragmentation, the search for liquidity, and high-frequency traders: These are the realities of modern markets. Traditional models of market microstructure have studied the highly simplified interaction between an idealized market-maker or specialist and a stream of external orders that may come from noise traders or informed traders. In the modern marketplace, the market itself is replaced by a loosely coupled network of visible and hidden venues, linked together by high-frequency traders and by algorithmic strategies. The distinction between market-makers who post liquidity and directional traders who take liquidity no longer exists. All traders are searching for liquidity, which may be flickering across many different locations with varying latencies, fill probabilities, and costs. That is the world this book addresses, treating these issues as central and fundamental rather than unwelcome complexities on top of a simple framework.This market evolution is the farthest one in equity markets, thanks in large part to their size, social prominence as indicators of corporate value, and large variety of active traders from retail investors to sophisticated proprietary operations and large fundamental asset managers. Regulation has also been most active in equity markets, most importantly Reg NMS in the US and MiFiD in Europe. Other asset markets, such as foreign exchange, futures, and fixed income, are further back along this pathway, but it is clear that the direction of evolution is toward the landscape treated in this book rather than back to simpler times. Regulation will continue to shape further development of all these markets, and all market participants have an interest in increasing their as well as the regulators' broad understanding of the underlying issues. The central focus of the book is liquidity: Loosely speaking, the ease and efficiency with which large transactions can be performed. For any real user of the market, this is the primary concern, although academic researchers may focus on other aspects. Thus, fragmentation and high-frequency trading are addressed from this point of view. Throughout the book, the emphasis is on features of the marketplace that are of tangible and pressing concern to traders, investors, and regulators.The authors have extensive personal experience of the development of the European equity markets as traders and as participants in conversations with regulators and other interested parties. They bring this experience to bear on every aspect of the discussion as well as deep quantitative understanding. The resulting book is a unique mixture of real market knowledge and theoretical explanation. There is nothing else out there like it, and this book will be a central resource for many different market participants. Bertrand Patillet, Deputy Chief Executive Officer of CA Cheuvreux until April 2013MiFID I removed the freedom of national regulators to maintain the secular obligation to concentrate orders on historical markets. In this way, the regulation, without a doubt, lif...
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 a priori knows the executed quantities by every venues.
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.
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