# stochastic Birth model simulation vs deterministic exponential growth not equal

I am trying to simulate a simple birth model, where each birth event increase population by 1. Birth rate is $\lambda$. In equations it looks like this

${dN \over dt}=\lambda N$, then $N=N_0e^{\lambda t}$

I try to simulate the system with stochastic model, I sample from exponential distribution with rate parameter ${1 \over \lambda}$, then each time a birth event occurs, the rate increase by $\lambda$, then sample from the new distribution... and so on.

I thought these two processes should be equivalent. but when I plog N vs t, the stochastic simulator is always a little slow but now by much, but it is definitely different. it is always slower(even if I do over 1000 trials and take the average).

Is there proof that stochastic simulation can simulate exponential growth exactly? If not, are we sure it is a good approximation? Is there anywhere I can find how good this approximation is?

Thank you

A basic result of the theory of branching processes is that what you call stochastic simulation produces a random process $(N_t)$ such that $N_t\mathrm e^{-\lambda t}\to W$ almost surely when $t\to\infty$, where $W$ is random, almost surely positive and almost surely finite. The strange thing in your description is that the theory also indicates that, starting from one individual at time $0$, one gets $E[W]=1$, which indicates that for some trials, one should observe that $N_t\lt\mathrm e^{\lambda t}$ when $t$ is large, and for some others, that $N_t\gt\mathrm e^{\lambda t}$ when $t$ is large. To explain this apparent paradox, one would need to know what it is your simulations are doing exactly and to see the result of some of the trials.
A further result is that in the binary division you describe, $W$ is exponentially distributed with parameter $1$. Thus $E[W]=1$, as in the general case, but $P[W\gt1]=1/\mathrm e$ hence roughly $63\%$ of the trials should be such that $N_t\lt\mathrm e^{\lambda t}$ when $t$ is large and only $37\%$ such that $N_t\gt\mathrm e^{\lambda t}$ when $t$ is large.
It sounds like there should be some discrepancy between the model and simulations. The rate equation you wrote for your model assumes a constant $\lambda$, but in your simulations its time-varying.