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Is it possible to translate a small system of ODEs to a discrete stochastic model? If so, can someone show me how?

I am trying to obtain the transition rate equations of the continuous time markov chain and then construct the infinitesimal generator matrix.

Equations: $$ \frac{dx}{dt} = Q_x-\lambda_x \cdot x - \beta_x \cdot x \cdot z \\ \frac{dy}{dt} = Q_y -\lambda_y \cdot y - \beta_y \cdot y \cdot z \\ \frac{dz}{dt} = Q_z - \lambda_z \cdot z - \frac{\beta_z}{n} \cdot y \cdot z $$

where the $Q$'s, $\lambda$'s, and $\beta$'s are parameters in $\mathbb{R}^{+}$ and $n$ takes on positive integer values.

Thanks in advance for any help.

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    Do you mean essentially writing down the "reactions" that characterize your law of mass action? I'm not sure what you're going for. Your system is essentially injection of each of "substances" $x,y,z$ at certain rates; first order decay of each of them (into something else that you're not tracking) at certain rates; and second order consumption of each of them, through interactions of $x$ with $z$ and $y$ with $z$. With that in mind, are you perhaps trying to think of the new stochastic system as having integer-valued $x,y,z$, some sort of "number of atoms" rather than "concentration"? – 2017-01-17
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    @Ian Yes, I want to start by writing the reactions. And integer-valued instead of concentration. – 2017-01-17
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    OK, well, assuming the parameters are just fixed, your reactions are like I said. Your transition rates are based on those reactions; for example, you can go from $(n_x,n_y,n_z)$ to $(n_x-1,n_y,n_z)$ with rate $\lambda_x$. But your infinitesimal generator matrix is infinite, unless you enforce some cap on the numbers. – 2017-01-17
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    @Ian Ok - how can I write a second order consumption? Say, like $(n_x,n_y,n_z) \rightarrow (n_x-1,n_y,n_z)$ with rate $\lambda_x + \beta_x \cdot n_x \cdot n_z$? – 2017-01-17
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    For $x$, sure. For $y$ and $z$, the choice of reactions depends on whether $\beta_y$ and $\beta_z$ actually correspond to different processes. Usually they don't. In this case, at the ODE level you have a stoichiometric relationship between the rates (so that $\beta_y$ and $\beta_z/n$ are not independent parameters). At the stochastic level, the change simply occurs through a single reaction, such as $(n_x,n_y,n_z) \to (n_x,n_y-1,n_z-1)$. – 2017-01-17

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