# Finding the mean of $X_t = \int_0^t sW_sdW_s$

For the stochastic integral, where $W_t$ is a Wiener process, I am trying to find the mean of $X_t = \int_0^t sW_sdW_s$. I have read before that any stochastic integral with $dWt$ has mean zero, but I dont know if it extends to cases where I have a random variable in the integrand as well. My approach is to decompose the integral $X_t$ into:

$$\int_0^t sW_sdW_s = \lim_{n \to \infty}\sum_{j=0}^{n-1}t_jW_{t_i}(W_{t_{i+1}}-W_{t_i})$$

Then,

$$E\left(\int_0^t sW_sdW_s\right) = \lim_{n \to \infty}\sum_{j=0}^{n-1}E\left(t_jW_{t_i}(W_{t_{i+1}}-W_{t_i})\right)$$

I believe that I can separate the terms in the expectation into:

$$E\left(t_jW_{t_i}(W_{t_{i+1}}-W_{t_i})\right) = E\left(t_jW_{t_i}\right)E\left(W_{t_{i+1}}-W_{t_i}\right)$$

I am not sure how to find $E\left(t_jW_{t_i}\right)$, although I know it is finite so it doesn't matter because $E\left(W_{t_{i+1}}-W_{t_i}\right) = 0$

• Once the integrand is adapted, the expectation is zero. – zhoraster Apr 8 '16 at 20:08
• Also you simply have $E[t_jW_{t_i}] = t_jE[W_{t_i}] = 0$ – BGM Apr 9 '16 at 2:32