Derivation of the master equation for any stochastic process

In this trivial example, ci = k. It is clear that the average X is equal to the solution of the continuous-deterministic reaction rate law equation 

Of course, the continuous-deterministic solution does not afford for calculation of variances and fluctuations around the mean. Furthermore, for non-linear systems the solution of the ordinary differential solution may not even accurately capture the average concentration, resulting in incorrect results. 

Unfortunately, there is no tractable solution of the master equation for more complex systems of higher order reactions. In such cases, one can resort to numerical simulations that sample the probability distribution. We discuss these in Chapter 18. 

We can now look beyond chemical reacting systems and consider any arbitrary stochastic process Xft), indexed by time t, as the sequence of random variables, Xih), X(t2),X(tn), observed at times h t2 

Define the joint probability that the variable attains specific values X(ti) = XuX(t2) = X2. X tn) = Xn aS 

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