Rösler U
Math Biosci. 1991 Dec;107(2):155-60. doi: 10.1016/0025-5564(91)90002-z.
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.
对一个过程进行建模的首次尝试通常是采用带有微分方程的确定性设置。现有的随机影响被抑制，并且有望可以忽略不计。然而，有时随机成分很重要。我们针对一个增长过程来证明并阐明这一点。确定性方法由(Y_{n + 1} = Y_n + g(Y_n))或(dY_t = g(Y_t)dt)给出，(Y_0 = 1)，(g)是一个正函数。相应的随机方程是(X_{n + 1} = X_n + g(X_n)(1 + \xi_n))或(dX_t = g(X_t)dt + f(X_t)dW_t)，(\xi)是某个随机变量，(W)是布朗运动。我们比较确定性解与随机解的渐近行为。
