Controlling random deviation in a birth-death process I have a process in which $X$ is a molecule going through birth ($x \rightarrow x + 1$) and death $(x \rightarrow x - 1)$. Rate of both reactions, namely synthesis rate for birth and elimination rate for death reaction, are proportional to number of molecules of $X$ (i.e. $[X]$) at any given time.
In this system, assume that synthesis rate itself increases with $[X]$; now I want to control the "random deviation" in $[X]$. Now I can't write down the expression which can tell me at what rate my elimination rate should increase to check the random deviation?
 A: Caveat: I don't know anything about chemistry, so I might be wrong.
Typically, these types of phenomena are characterized by a stochastic matrix. We can represent molecules as existing in two states: alive and dead. At time $t$, the probability of a molecule transitioning states at time $t+1$ is given by
$$
A=\begin{bmatrix} 
1-q & q \\
p & 1-p
\end{bmatrix}
$$ 
where the present state is represented in rows and the next state in columns. We can denote $P(\text{alive} \to \text{dead})$ as $q$ and the reverse as $p$. The diagonal entries are the probabilities of the molecule not transitioning states. If $x_0$ is our initial state, the state at time $k$ is $x_k=A^kx_0$. Recalling linear algebra, the diagonal factorization $A=PDP^{-1}$ makes computation of this matrix exponent much simpler because $A^k=PD^kP^{-1}$.
You write that these rates $p$ and $q$ depend on $[X]$. So we can denote $p=f([X])$ and $q=g([X])$. For any choice of functions, then, the transition matrix becomes
$$
A=\begin{bmatrix} 
1-g([X]) & g([X]) \\
f([X]) & 1-f([X])
\end{bmatrix}
.$$
I can't say much more about how to go about solving this problem, since these functions entirely hinge on the physical process of the chemistry. However, perhaps a type of linear model is appropriate, such as $\text{logit}(p)=\beta_0+\beta_1[X]$, where $\text{logit}(x)=\log(x)-\log(1-x)$?
