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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}.}  \begin{equation}\label{eq:dynamics}