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The limit order book of an exchange represents an information store of market participants' future aims and for many traders the information held in this store is of interest. However, information loss occurs between orders being entered into the exchange and limit order book data being sent out. We present an online algorithm which carries out Bayesian inference to replace information lost at the level of the exchange server and apply our proof of concept algorithm to real historical data from some of the world's most liquid futures contracts as traded on CME GLOBEX, EUREX and NYSE Liffe exchanges.

Prediction of future security returns is possible by decomposing a securities price into weighted superpositions of underlying basis states, given stationary distributions of the basis states. The (ensemble) Hilbert-Huang transform (HHT) is an empirical two-step online methodology which carries out such a decomposition from a multi-component noisy time series. HHT allows estimation of each component's instantaneous phase, period and amplitude. A hypothesis is presented where markets exist in the binary states of trend or cycle. Switching between states is based on phase-shifting in a dyadic filter bank. A trading algorithm is presented which exploits this model by combining intra-day predictions for trend and cycle components along with a much lower frequency drift component. The algorithm is simulated on e-mini S&P 500 futures data from CME GLOBEX at one minute sampling frequency. Results are presented which show a combined strategy Sharpe ratio in excess of 3.Biographical notes: Hugh L. Christensen received undergraduate and masters degrees from Oxford University and the PhD from the Signal Processing Laboratory, Cambridge University. He has worked in quantitative research for the UK Government and subsequently for algorithmic trading companies, applying techniques from signal processing and machine learning to high-frequency trading. His research interests include online methodologies for time series prediction, signal combination and the application of machine learning techniques to limit order book data.Simon J. Godsill is the Professor of Statistical Signal Processing at Cambridge University. He is an Associate Editor for IEEE Trans. Signal Processing and the journal Bayesian Analysis, and is a member of IEEE Signal Processing Theory and Prediction using Hilbert-Huang transform 373 Methods Committee. He has research interests in Bayesian and statistical methods for signal processing, Monte Carlo algorithms for Bayesian problems, modeling and enhancement of audio and musical signals, source separation, tracking and genomic signal processing. He has authored over 250 peer-reviewed journal and conference articles.

Abstract.A previously published and widely quoted paper, "Price Dynamics in a Markovian Limit Order Market," [SIAM J. Financial Math., 4 (2013), pp. 1-25] has provided analytic solutions for many interesting quantities for the price dynamics of the limit order book. However, for unbalanced order flow cases, one of its major results, the price increase probability conditional on the state of the limit order book, is found to be incorrect due to its citing an erroneous formula from another paper. We correct this error by proposing another analytical solution using the characteristic functions of probability distributions. The final analytical solution presented in this paper is of a simple form and easy to calculate.Key words. limit order book, random walk, price dynamics, hitting probability AMS subject classifications. 60J27, 60E10, 60K25DOI. 10.1137/16M10574371. Introduction. The limit order book (LOB) is a depository of resting limit orders that provides liquidity, allowing participants to trade in the market. Much effort has been put into trying to extract information from the LOB and predict the direction of price change in the near future.A recent paper [2] showed that several interesting conditional probabilities, including the direction of the next price change conditional on the state of the LOB, can be obtained analytically using a simplified model for the LOB based on a previous model proposed in [4]. This simplified model can capture the dynamics of market orders and limit orders and their influence on price dynamics in a more analytically tractable way. The paper brought new insights into the LOB, helped people understand its dynamics [1], [8], [6], and inspired applications based on it, for example, building optimal trading strategies [9].However, we find that one of its major results, Proposition 3, which provides the analytic solution to the conditional probability of price movement of the LOB when the order flow is unbalanced, is incorrect. This proposition itself is important because nowadays much attention has been focused on the study of the order flow imbalance (OFI) process, which is thought to be more sensitive to market information than price [3], [10]. Also, Proposition 3 gives the short term price dynamics of the LOB driven by this OFI process under the Markovian assumption in [2].A numerical and mathematical proof for this mistake is provided in this correction note,