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\section{A dynamic theory of the supply \& demand curves}%\label{sec:dynamics} curves}\label{sec:dynamics}  \subsection{Definitions} 
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In the Walrasian story, supply and demand pre-exist and the Walrasian auctioneer gropes ({\it t\^atonne}) to find the price $p_t^*$ that maximizes the amount of possible transactions. The auction then takes place at time $t$ and removes instantly all matched orders. Assuming that all the supply and demand intentions close   to the transaction price were revealed before the auction and were matched, the state of the MSD just after the auction is simple to describe, see Figures \ref{fig:SD} \& \ref{fig:scheme}:  \begin{equation}%\label{eq:after} \begin{equation}\label{eq:after}  \left\lbrace   \begin{array}{ccc}  \rho_S(p,t^+) &= &\rho_S(p,t^-) \quad (p > p_t^*)\\ 
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Our central assumptions are \emph{heterogeneity}, together with the hypothesis that idiosyncratic behaviours ``average out'' in the limit of a very large number of participants, i.e., no single agent accounts for a finite fraction   of the total supply or demand. While not strictly necessary, this assumption leads to a deterministic aggregate behaviour and allows one to gloss over some rather involved mathematics.  \subsection{The model in terms of optimizing agents}  %\label{sec:agent-opt} agents}\label{sec:agent-opt}  The above assumptions might appear obscure to those used to think in terms of rational optimizing agents and equilibria. Here we rephrase these assumptions in a language closer to standard economic intuition. 
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of some adequate statistical ensembles. For the sake of simplicity, we consider no interest rate and no risk of any kind.  At time $t$, each agent computes his optimal action $p_t^i, \theta_t^i$ as the result of the following optimisation program:  \begin{equation}%\label{opt} \begin{equation}\label{opt}  (p_t^i,\theta_t^i) = \underset{p,\theta}{\text{argmin}}~~\mathcal{U}_i(p,\theta|\widehat{p}_t^i,{\cal F}_t).  \end{equation}  Because of the random evolution of the outside world summarized by $\widehat{p}_t^i,{\cal F}_t$, the value of $\theta_t^i \in \{-1,0,+1\}$ can change between 
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no optimal strategic component that is missing from the above utility maximisation program.  \subsection{The ``free evolution'' equation for the MSD curves}%\label{sec:free} curves}\label{sec:free}  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}. 
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one finds \cite{donier2014fully}:\footnote{See also \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}.}  \begin{equation}%\label{eq:dynamics} \begin{equation}\label{eq:dynamics}  \left\lbrace   \begin{array}{ccc}  \partial_t \rho_D(y,t) &=& {\cal D} \partial^2_{yy} \rho_D(y,t) - \nu_+(y) \rho_D(y,t) + \omega_+(y); \\ 
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The above equations for $\rho_D(y,t)$ and $\rho_S(y,t)$ are linear and can be formally solved in the general case, starting from an arbitrary initial condition such as Equation (\ref{eq:after}), using a spectral decomposition of  the evolution operator. This general solution is however not very illuminating, and we rather focus here in the special case where $\nu_\pm(y) \equiv \nu$ does not depend on $y$ nor on the side of the latent order book. The general solution can then be written in a fairly transparent way, as:  \begin{equation}%\label{eq:gen-sol} \begin{equation}\label{eq:gen-sol}  \rho_{S,D}(y,t) = \int_{-\infty}^{+\infty} \frac{\d y'}{\sqrt{4 \pi \D t}} \, \rho_{S,D}(y',t=0^+) e^{-\frac{(y'-y)^2}{4 \D t} - \nu t}   + \int_0^t \d t' \int_{-\infty}^{+\infty} \frac{\d y'}{\sqrt{4 \pi \D (t-t')}} \, \omega_{\pm}(y') e^{-\frac{(y'-y)^2}{4 \D (t-t')} - \nu (t-t')},  \end{equation} 
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