deletions | additions       diff --git a/section_A_dynamic_theory_of__.tex b/section_A_dynamic_theory_of__.tex  index 1859224..e5bacff 100644  --- a/section_A_dynamic_theory_of__.tex  +++ b/section_A_dynamic_theory_of__.tex 
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\end{itemize}  We will furthermore assume that the ``news'' term ${\rm d}\xi_t$ is a Wiener noise of variance $\sigma^2 \d {\rm d}  t$, corresponding to a Brownian motion for the fundamental price $\widehat p_t = \int^t {\rm d}\xi_{t'}$ with volatility $\sigma$.   Normalising the mean of the $\beta^i$'s to unity thus corresponds to the assumption that agents are on average unbiased in their interpretation of the news -- i.e. their intentions remain  centred around the fundamental price $\widehat p_t$ in the course of time -- but see the expanded discussion of this point in Section \ref{sec:discussion}.  
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(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  $t$ and $t + \d {\rm d}  t$. For the sake of simplicity, we assume that the change of the state of the world in time $\d ${\rm d}  t$ is never so large as to induce direct transitions from $\mp 1 \to \pm 1$ without pausing at $0$. Hence, between $t$ and $t + \d {\rm d}  t$, the following transitions (or absence thereof) are possible: \begin{itemize}  \item $0 \to 0$: this clearly induces no change in the MSD curves;  \item $0 \to \pm 1$: in this case, agent $i$ previously absent from the market becomes either a buyer or a seller, with reservation price $p_t^i$ given by 
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has changed. Writing $p_t^i = f_i(\widehat{p}_t^i,t)$, where $f$ is a regular function if $\mathcal{U}_i$ is regular enough, and applying It\^{o}'s lemma, one finds:  \begin{equation}\nonumber  \begin{aligned}  \d {\rm d}  p_t^i &= \frac{\partial f_i}{\partial t} \d {\rm d}  t + \frac{\partial f_i}{\partial p} \d {\rm d}  \widehat{p}_t^i + \frac{\sigma_i^2}{2} \frac{\partial^2 f_i}{\partial p^2} \d {\rm d}  t \\ &= \alpha_t^i \d {\rm d}  \widehat{p}_t^i + \gamma_t^i \d {\rm d}  t. \end{aligned}  \end{equation}  The drift term $\gamma_t^i$ will play little role in the following (see previous footnote), and we neglect it henceforth. In order to recover the specification  of the above section, we further decompose the price revision $\d ${\rm d}  p_t^i = \alpha_t^i \d {\rm d}  \widehat{p}_t^i$ into a \emph{common} component $\beta^i {\rm d}\xi_t$ and an \emph{idiosyncratic} component ${\rm d}W_{i,t}$ as above.  \end{itemize} 
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\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); \\  \partial_t \rho_S(y,t) &=& \underbrace{\D \underbrace{{\rm d}  \partial^2_{yy} \rho_S(y,t)}_{\text{Updates}} - \underbrace{\nu_-(y) \rho_S(y,t)}_{\text{Cancellations}} + \underbrace{\omega_-(y)}_{\text{New orders}},\\  \end{array}  \right. 
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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}  \rho_{S,D}(y,t) = \int_{-\infty}^{+\infty} \frac{\d \frac{{\rm d}  y'}{\sqrt{4 \pi \D {\rm d}  t}} \, \rho_{S,D}(y',t=0^+) e^{-\frac{(y'-y)^2}{4 \D {\rm d}  t} - \nu t} + \int_0^t \d {\rm d}  t' \int_{-\infty}^{+\infty} \frac{\d \frac{{\rm d}  y'}{\sqrt{4 \pi \D {\rm d}  (t-t')}} \, \omega_{\pm}(y') e^{-\frac{(y'-y)^2}{4 \D {\rm d}  (t-t')} - \nu (t-t')}, \end{equation}  where $\rho_{S,D}(y,t=0^+)$ is the initial condition, i.e. just after the last auction.  
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