Deterministic in contrast to stochastic modeling

The first attempt to model a process is often a deterministic setup with differential equations. The existing stochastic influence is suppressed and hopefully negligible. However, sometimes the stochastic component is important. We demonstrate and clarify this for a growth process. The deterministic approach is given by Yn + 1 = Yn + g(Yn) or dYt = g(Yt)dt, Y0 = 1, g a positive function. The corresponding stochastic equation is Xn + 1 = Xn + g(Xn)(1 + xi n) or dXt = g(Xt)dt + f(Xt)dWt, xi some random variable, W the Brownian motion. We compare the asymptotic behavior of the deterministic solution versus the stochastic solution.