deletions | additions
diff --git a/Coming_back_to_Equation_ref__.tex b/Coming_back_to_Equation_ref__.tex
index efcfc81..5163a71 100644
--- a/Coming_back_to_Equation_ref__.tex
+++ b/Coming_back_to_Equation_ref__.tex
...
The last item we need is the shape of the supply curve {\it below} the transaction price just before the next auction, that gives the amount of supply/demand on the ``wrong'' side of the price, i.e. precisely
the volume exchanged at the auction. Using the simple affine approximation of Equation (\ref{eq:final}), one finds:
\be\nonumber \begin{equation}\nonumber
\rho_{S,\text{st.}}(y < 0) \approx \mathcal{L} \int_0^{+\infty} \frac{{\rm d} y'}{\sqrt{4 \pi \mathcal{D} \tau}} \, (y' + y_0) e^{-\frac{(y'-y)^2}{4 \mathcal{D} \tau}},
\ee \end{equation}
or, again setting $y = -u \sqrt{\mathcal{D} \tau}$ and $y' = w\sqrt{\mathcal{D} \tau}$,
\be\label{eq:final2} \begin{equation}\label{eq:final2}
\rho_{S,\text{st.}}(y < 0) \approx \mathcal{L} \sqrt{\mathcal{D} \tau} \int_0^{+\infty} \frac{{\rm d} w}{\sqrt{4 \pi}} \, (w + u_0) e^{-\frac{(w+u)^2}{4}} =
\mathcal{L} \sqrt{\mathcal{D} \tau} \left[\frac{e^{-u^2/4}}{\sqrt{\pi}}+ \frac12 (u_0 -u) (1 - \text{Erf}(u/2))\right].
\ee \end{equation}
From this expression, the total volume $v^*$ exchanged during each auction is found to be:
\be\nonumber \begin{equation}\nonumber
v^* = \int_{-\infty}^0 {\rm d} y \rho_{S,\text{st.}}(y) = \mathcal{L} {\mathcal{D} \tau} \left[\frac12 + \frac{u_0}{\sqrt{\pi}} \right] \approx 0.965 \mathcal{L} {\mathcal{D} \tau},
\ee \end{equation}
whereas the exact result (that can be obtained directly from the diffusion equation in the $\tau \to 0$ limit) is $v^* = \mathcal{L} {\mathcal{D} \tau}$. The error induced by our
simple affine approximation is thus only a few percents. Interestingly, one sees that the total transacted volume $V$ in a finite time interval $T$, given by $V=v^* T/\tau$,
remains finite when $\tau \to 0$, and equal to $V= \mathcal{L} \mathcal{D} T$. This observation should be put in perspective with the recent evolution of financial markets, where the time between transactions
...
From the shape of the MSD close to transaction price given by Equation (\ref{eq:final}), it is immediate to compute the supply and demand curves just before an auction when the inter-auction time $\tau$
tends to $0$. Denoting again as $y$ the difference between the price level $p$ and the fundamental price $\widehat p_t$, one finds:
\be\label{eq:SD-final} \begin{equation}\label{eq:SD-final}
\begin{aligned}
S(p \geq \widehat p_t) &= \mathcal{L} (y_0 y + \frac12 y^2) + v^*\\
D(p \leq \widehat p_t) &= \mathcal{L} (-y_0 y + \frac12 y^2) + v^*
\end{aligned}
\ee \end{equation}
where, as found in the previous section, $y_0 \equiv u_0 \sqrt{\mathcal{D} \tau} \approx 0.824 \sqrt{\mathcal{D} \tau}$. From Equation (\ref{eq:final2}) above, it is readily seen that the supply (resp. demand) curve below (resp. above) $\widehat p_t$
can be written as $v^* F(y/\sqrt{\mathcal{D} \tau})$, where $F(u)$ is a certain function that goes from $F(0)=1$ to $F(\infty)=0$.\footnote{This function reads, explicitly:
\be\nonumber \begin{equation}\nonumber
\left[\frac12 + \frac{u_0}{\sqrt{\pi}} \right] F(u) = \frac12 (1 - \text{Erf}(u/2))(\frac{u^2}{2} - u_0 u + 1) - \frac{e^{-u^2/4}}{\sqrt{\pi}} (\frac{u}{2}-u_0).
\ee \end{equation}
}
The above equation Equation (\ref{eq:SD-final}) immediately allows us to compute the impact ${\cal I}(Q)\equiv y^*$ of an extra buy quantity $Q$, as the solution of $\mathcal{L} (y_0 y^* + \frac12 y^{*2}) + v^* = Q + v^* F(y^*/\sqrt{\mathcal{D} \tau})$.
It is clear that the solution can be written as $y^* = \sqrt{\mathcal{D} \tau} Y(Q/\mathcal{L} \mathcal{D} \tau)$, where $Y(q)$ obeys $u_0 Y + \frac12 Y^2 + (1 - F(Y)) = q.$ The limiting behaviours of $Y$ in the limits $q \ll 1$ and $q \gg 1$ are easy to compute, and read:
\be\nonumber \begin{equation}\nonumber
Y(q) \approx_{q \ll 1} 0.555 q; \qquad Y(q) \approx_{q \gg 1} \sqrt{2q}.
\ee \end{equation}
