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given good (or asset), would the price be set to $p$. The equilibrium price $p^*$ is then such that $D(p^*) = S(p^*)$, which maximizes the amount of good exchanged among agents, given the set of preferences corresponding to the current supply and demand curves \cite{walras2013elements}. In reality, the full knowledge of $S(p)$ and $D(p)$ is problematic, and Walras envisioned his famous {\it t\^atonnement} process as a mean to observe the supply/demand curves. However, there is a whole aspect of the dynamics of markets that is totally absent in Walras' framework. While it describes how a pre-existing supply and demand would result in a clearing price, it does not tell  us anything about what happens \emph{after} the transaction has taken place. In this sense, the Walrasian price is of very limited scope, since the theory ceases to apply as soon as the price is discovered.  A practical solution to match supply and demand is the so-called ``order book'' \cite{harris1990liquidity, glosten1994electronic}, \cite{harris1990liquidity,glosten1994electronic},  where each agent posts the quantities s/he is willing to buy or sell as a function of the price $p$. $S(p)$ (resp. $D(p)$) is then the sum of all sell (buy) quantities posted at or above (below) price $p$. At each time step, the auctioneer can then clear the market by finding the (unique) price such that $D(p^*) = S(p^*)$. This is   in fact how most financial markets worked before the advent of electronic matching engines, when market makers played the role of ``active'' Walrasian auctioneers, in the sense that they would themselves contribute to the order book as to insure orderly trading and stable prices \cite{glosten1985bid,madhavan2000market}. 
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The availability of detailed data where all market events are recorded (i.e. trades, quotes, cancellations, etc.) has generated a flurry of empirical papers, describing many aspects of price formation at the microstructural level   (for a review see \cite{biais1995empirical,bouchaud2008markets}). Correspondingly, a host of stylized models of the order book have appeared, with different starting points and objectives.   For example, ``zero-intelligence'' models \cite{smith2003statistical, bouchaud2002statistical,farmer2005predictive,cont2010stochastic,gareche2013fokker} \cite{smith2003statistical,bouchaud2002statistical,farmer2005predictive,cont2010stochastic,gareche2013fokker}  form an important class of models of the order book, where one assumed that agents act mechanically (rather than strategically) leading to simple Poissonian statistics for the order flow. Although obviously too simple to account for what goes on in financial markets, such models reveal some interesting relationships between observables (spreads, volatility, activity, etc.) \cite{farmer2005predictive,cont2013price}. Much more elaborate models have also been developed, taking into account the heterogeneity, strategies and preferences of market participants \cite{foucault1999order, rocsu2009dynamic, rosu2014liquidity, maglaras2011multiclass,lachapelle2013efficiency}, \cite{foucault1999order,rocsu2009dynamic,rosu2014liquidity,maglaras2011multiclass,lachapelle2013efficiency},  some including the queues behind the best buy/sell prices \cite{huang2014simulating}. The present paper is clearly partly inspired by the above strand of papers on real limit order books, in particular \cite{smith2003statistical,bouchaud2002statistical}. However, we depart from these models on one very fundamental issue.   Instead of trying to describe the evolution of the \emph{visible} order book (where only a tiny fraction of the outstanding liquidity is revealed, and whose dynamics is dominated by highly strategic market-makers/HFT), we want to describe the much deeper and much slower ``latent'' order book, introduced in \cite{Toth:2011}, that contains all buy/sell {\it intentions}, whether displayed or not by market participants. In other words, we model the true underlying supply and demand curves that would materialize if the transaction price was to move closer to the reservation prices. The distinction between the visible limit order book   (which, as stated above, gives a very poor indication on liquidity at larger scales) and the true supply and demand curves is absolutely crucial for all that follows. The model described below is   a generalisation of the ideas introduced in \cite{Toth:2011,Iacopo:2013} and in \cite{donier2014fully}. It builds upon the intuition that   agents can revise their reservation prices in an heterogeneous manner, introduced long ago in \cite{bak1997price} and recently revisited in completely different contexts in \cite{lasry2007mean, lehalle2011high} \cite{lasry2007mean,lehalle2011high}  and in \cite{Toth:2011}. The motivation of the latter paper was to explain the universal concave (``square-root'') impact of directional trade sequences mentioned in the introduction, that deeply challenges standard equilibrium models.   In summary, the aim of the present paper is to reconcile the insights gained by the financial literature on price formation with a more Walrasian view of supply and demand that provides us with a macroscopic theory  
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Endowed with the above hypothesis, one can derive stochastic partial differential equations for the evolution of the marginal supply ($\rho_S(p,t)=\partial_p S(p,t)$) and the marginal   demand ($\rho_D(p,t)=-\partial_p D(p,t)$) in the absence of transactions \cite{donier2014fully}.  It turns out that, as expected, these equations take a simpler form in the reference frame of the (moving) fundamental price $\widehat p_t$. Introducing the shifted price $y = p - \widehat p_t$,   one finds \cite{donier2014fully}:\footnote{See also \cite{lasry2007mean, lehalle2011high} \cite{lasry2007mean,lehalle2011high}  for similar ideas in the context of mean-field games. Note that Equation (\ref{eq:dynamics}) is strictly valid when $\rho_S(p,t)$ and $\rho_D(p,t)$ are be interpreted as the marginal supply and demand curves averaged over the noise processes.   Otherwise some noisy component remains, see e.g. \cite{dean1996langevin}.}  \be\label{eq:dynamics}