I need to verify that a particular statement from this paper is correct https://www.sciencedirect.com/science/article/pii/S0895717703901476
Lets assume that a species growth is given by the following deterministic differential equation $\frac{dx}{dt}=\alpha x$
Now if instead $\alpha$ is a stochastic parameter such that $\alpha_t=\alpha+\sigma \epsilon_t$ where $\epsilon_t$ is Gaussian white noise. Then the authors claim that the deterministic ODE can be replaced the following SDE $$dx(t)=\alpha x dt+x\sigma dW(t)$$ where $W(t)$ is wiener process. In my opinion this is wrong because $\lim_{dt\rightarrow 0}\sqrt{dt}N(0,1)=dW(t)$. But here we have $\lim_{dt\rightarrow 0}dtN(0,1)$. Does this matter, or am I missing something?