deletions | additions       diff --git a/section_A_dynamic_theory_of__.tex b/section_A_dynamic_theory_of__.tex  index 65d48ea..c4cf30d 100644  --- a/section_A_dynamic_theory_of__.tex  +++ b/section_A_dynamic_theory_of__.tex 
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\ee  supplemented by the boundary condition $\phi_\infty(u \gg 1) \approx \L \sqrt{\D} u$, where $\L$ is a constant to be determined below. [Note that the solution of Equation (\ref{eq:atkinson}) is determined up to a   multiplicative factor that must be fixed by some external condition].  \begin{figure}[t!]  \begin{center}  \includegraphics[height=7.5cm]{images/atkinson}  \end{center}  \caption{Graph of the normalized exact solution $\phi_\infty(u)$, and its affine approximation. The whole picture must be rescaled by a factor $\sqrt{\tau}$ to recover the order book when the inter-auction time is $\tau$. The hatched region corresponds to the volume to be executed, and therefore scales with $\tau$.}  \label{fig:atkinson}  \end{figure}  \begin{figure}[t!]  \begin{center}  \includegraphics[height=7.5cm]{images/books}  \end{center}  \caption{Left: Shape of the marginal supply curve immediately before the auctions, for different inter-auction times $\tau$, in the case $\omega_{\pm}(y) = \omega_{\pm}^0\mathbbm{1}_{\{y\overset{>}{\underset{<}{ }}0\}}$. Right: Shape of the MSD immediately after the auctions, again for different   inter-auction times $\tau$. Note that as $\tau \to 0$, the MSD acquires a characteristic V-shape.}  \label{fig:books}  \end{figure}  Equation (\ref{eq:atkinson}) is of the Wiener-Hopf type and its analytical solution can be found in \cite{atkinson1974wiener,boersma1974note}. We plot numerically this solution in Figure \ref{fig:atkinson}; it is seen to be   numerically very close to an affine function for $u > 0$: $\phi_\infty(u) \approx \L \sqrt{\D} (u + u_0)$ with $u_0 \approx 0.824$. In summary, the stationary shape $\rho_{S,\text{st.}}(y)$ of the marginal supply curve in the frequent auction limit $\tau \to 0$ and close to the transaction price ($y = O(\sqrt{\D \tau})$,   has a {\it universal shape}, independent of the detailed specification of the model (i.e., the functions $\nu_\pm(y)$ and $\omega_\pm(y)$). This supply curve is given by   $\sqrt{\tau} \phi_\infty(y/\sqrt{\D \tau})$, which can itself be approximated by a simple affine function that will fully suffice for the purpose of the present paper:  \be\label{eq:final}  \rho_{S,\text{st.}}(y \geq 0) \approx \L (y + y_0); \qquad y_0 = u_0 \sqrt{\D \tau}; \qquad (\tau \to 0),  \ee  and similarly for $\rho_{D,\text{st.}}(y)$, see Figure \ref{fig:books}. The detailed interpretation of this result -- in terms of market liquidity and price impact -- will be given below.   We however still need to find the value of $\L$. This is done by comparing with the stationary solution $\varphi_{\text{st.}}(y)$ of Equation (\ref{eq:dynamics}) that satisfies the boundary solution   $\varphi_{\text{st.}}(0)=0$ (valid for $\tau=0$). For $\nu_\pm(y) = \nu$, $\varphi_{\text{st.}}(y)$ can be computed explicitly and is given by:  \be\nonumber  \varphi_{\text{st.}}(y) = \frac{1}{\D} e^{-\sqrt{\nu/\D}\, y} \int_0^y {\rm d}y' e^{2\sqrt{\nu/\D} \,y'} \int_{y'}^\infty {\rm d}y'' e^{-\sqrt{\nu/\D} \,y''}\omega(y'').  \ee  Expanding $\varphi_{\text{st.}}(y)$ for small $y$ (but still much larger than $\sqrt{\D \tau}$) finally leads to:  \be\nonumber  \varphi_{\text{st.}}(y) \approx \L y; \qquad \L = \frac{1}{\D} \int_{0}^\infty {\rm d}y' e^{-\sqrt{\nu/\D} \,y'}\omega(y'),  \ee  where $\L$ can be seen as a measure of the market liquidity (see \cite{donier2014fully} and below). Again in the simple case $\omega_{\pm}(y) = \omega^0\mathbbm{1}_{\{y\overset{>}{\underset{<}{ }}0\}}$, one  finds:  \be\nonumber  \L = \frac{\omega_0}{\sqrt{\nu {\cal D}}}.  \ee  Therefore, liquidity increases with the order arrival rate and decreases with their cancellation rate, as above, but also decreases   with the diffusion constant ${\cal D}$ that can be loosely identified with market volatility (see the discussion above).