Stochastic calculus of Gaussian white noise

Question

I see a strange stochastic integral and don't know how to proceed?

Denote $\xi$ a Gaussian white noise,

Denote

$$I(n)=\int_t^{t+n\Delta t}ds\int_t^sd\xi$$ How to prove that $<I(1)I(1)>= \frac{1}{3}\Delta t^3$ and $<I(1)I(2)>= \frac{5}{6}\Delta t^3$?

Specifically, can anyone explain how to deal with these kind of integrals generally? Routine stochastic integral often only deals with Brownian motion. Any references and books are also greatly appreciated.

These equations are taken from a theoretical physics paper.

Many thanks!

Answer 1

Welcome to MSE: may you know that $(\Delta B(t))^2\sim \Delta t$ when $B(t)$ is brownian motion.I wish It helps you. so $$\int_t^{t+n\Delta t}1ds=n\Delta t=o(\Delta t)$$and $$\int_t^sd\xi\sim \sum \Delta B(t)=o(\sqrt{\Delta t})$$so It seems that $$I(n)=o(\sqrt{\Delta t}).o(\Delta t)=o(\Delta t^{\frac32})$$ can you proceed now ?