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# Exactly solvable model for the diffusive price-dynamics paradox under long-range correlated market-order flow


# Exactly solvable model for the diffusive price-dynamics paradox under long-range correlated market-order flow









We develop an exactly solvable nonlinear time-series model by incorporating the square-root price-impact law into the Lillo--Mike--Farmer (LMF) model to resolve the diffusive price-dynamics paradox under predictable market-order flow. In financial market microstructure, it is well established that the price dynamics are approximately described by Brownian motion at long times. However, it is also well-known that market-order flow is clearly predictable due to long-range correlations, as mathematically formulated by the LMF model. Since market orders have a positive price impact in general, predictable market-order flow seems to contradict Brownian price dynamics. In this work, we resolve this diffusive price-dynamics paradox by developing nonlinear time-series models that generalize the LMF model based on the square-root price-impact law. Our time-series models can be mathematically mapped onto the Lévy-walk framework---an exactly solvable class of non-Markovian stochastic processes developed in statistical physics. We prove that the price dynamics are diffusive at long times under the square-root law even under predictable market-order flow. Our work highlights the crucial practical importance of the square-root law in understanding the microstructural foundation of the Efficient Market Hypothesis.