# Covariance of Ito Integrals

Let $$f_t$$ and $$f_t$$ be two stochastic processes adapted to a filtration $$\mathcal F_t$$ such that

$\int_0^\infty \mathbf E (f_t^2) dt < \infty \qquad \text{and} \qquad \int_0^\infty \mathbf E (g_t^2) dt < \infty$

Let $$W_t$$ be a standard brownian motion  also adapted to the filtration $$\mathcal F_t$$ and define the stochastic processes

$X_t =\int_0^t f_s dW_s \qquad \text{and} \qquad Y_t=\int_0^t g_s dW_s$

Calculate the following:

1. $$\mathbf E (X_t X_s )$$
2. $$\mathbf E (X_t Y_t )$$
Hint: You know how to compute $$\mathbf E (X_t^2 )$$ and $$\mathbf E (Y_t^2 )$$. Use the fact that $$(a+b)^2 = a^2 +2ab + b^2$$ to answer the question. Simplify the result to get a compact expression for the answer.
3. Show that if $$f_t=\sin(2\pi t)$$ and $$g_t=\cos(2\pi t)$$ then $$X_1$$ and $$Y_1$$ are independent random variables.(Hint: use the result here  to deduce that $$X_1$$ and $$Y_1$$ are mean zero gaussian random variables. Now use the above results to show that the covariance of $$X_1$$ and $$Y_1$$ is zero. Combining these two facts implies that the random variables are independent.)