How to transform an exact master's equation to a continous approximation Fokker-Planck equation? Given a simple stochastic process like: $A + A \rightarrow 0$, which has a Master's equation of 
\begin{equation}
\dot{P}_n = \frac{(n+2)(n+1)}{2}P_{n+2} - \frac{n(n-1)}{2}P_n
\end{equation}
which describes the time evolution of $P_n$, the probability of the state with $n$ individuals.
Assuming $n\gg 1$, you can take a continuous approximation and get to a Fokker-Planck equation.  But how?
 A: Apparently, this is how it's done:
Replace ${P}_n$ with $P(x,t)$ and $n +1$ with $x + \Delta x$.  This gives you:
\begin{align}
\dot{P}(x,t) &= \frac{(x + 2\Delta x)(x + \Delta x)}{2}P(x+2\Delta x,t) - \frac{x(x - \Delta x)}{2} P(x,t)\\
&= \frac{x^2}{2}\left(P(x+2\Delta x,t) - P(x,t)\right) + O(\Delta x)\\
&= \frac{x^2}{2}\left(P(x+2\Delta x,t) - 2P(x+\Delta x,t) + P(x,t)\right) + x^2(P(x+\Delta x, t) -P(x,t))  +  O(\Delta x)\\
&\approx \frac{x^2}{2}\frac{d^2P(x,t)}{dt^2} + x^2 \frac{dP}{dx}
\end{align} 
Is that right?
When you multiply by $x$ and integrate over all $x$ you get
\begin{equation}
\dot{\langle x \rangle} = 3( \langle x \rangle - \langle x^2 \rangle)
\end{equation}
which, except for the factor of 3 is the same as you get directly from the masters equation if you multiply by $n$ and sum over all $n$. Ie,
\begin{equation}
\dot{\langle n \rangle} = \langle n \rangle - \langle n^2 \rangle
\end{equation}
I can accept a constant factor as an artifact of the approximation, that somehow the limit should be taken more carefully or the implicit parameter of the problem (1) is modified by a constant factor under the continuous approximation.
If someone has a better answer, I'll gladly unaccept my own for it but for now, this will do.
