# Thread: Variance and mean of a deterministic process

1. ## Variance and mean of a deterministic process

Let $(f_t)_{t \geq 0}$ be a deterministic process (i.e. independent of $\omega$) such that $\int_{0}^{\infty }\int_{t}^{2}dt<\infty$. Then this process obviously belongs to $H$. Show that for each t the integral $\int_{0}^{t}f_sdW_s$ is a normal random variable with mean 0 and variance $\int_{0}^{t}f_s^2ds$

Any help?

Thanks

2. ## Re: Variance and mean of a deterministic process

Hello,

Use the basic properties of a stochastic integral!
The expectation of a stochastic integral is proved to be 0 (by using the core definition, dealing with sum of brownian increments).
The variance is obtained by Ito's isometry.