meaning of differentiation of stochastic process

Question

Let $X_t,t\in T $ continuous time stochastic process. What is the meaning of $dX_t$ which is differentiation of $X_t$? Does that mean $X_{t+dt}$ and $X_t$ are random variables so $dX_t \approx X_{t+dt}-X_t$ is also random variable?

Answer 1

In the context of stochastic differential equations, the meaning of $dX_t$ is "derived" from the definition of corresponding integrals $\mathcal I[f](\omega)=\int_S^Tf(t,\omega)dX_t(\omega)$. The integral is defined by suitable approximation of $f$ by elementary functions $\phi$ of the form $\phi(t,\omega)=\sum_j e_j(\omega)\cdot\chi_{[t_j,t_{j+1})}(t)$.

However, the problem is that the actually used suitable approximation strongly influences the definition of the resulting integral and its properties. The Ito integral corresponds more or less to using $e_j(\omega)=f(t_j,\omega)$, while using $e_j(\omega)=f(\frac{t_j+t_{j+1}}{2},\omega)$ leads to the Stratonovich integral.

If I try to translate these definitions back into your language/notation, the Ito integral would more or less correspond to your suggestion $dX_t \approx X_{t+dt}-X_t$, while the Stratonovich integral would correspond more or less to $dX_t \approx X_{t+dt/2}-X_{t-dt/2}$.