What is the expression $E[X_t dt]$ using stochastic calculus?

Question

I am trying to finish a calculation using Itô calculus but do not know the final step.

• What is the expression $E[X_t dt]$ equal to?
• Note that the $X_t$ is drifted Brownian motion $X_t = \mu_x t + \sigma_x W_t^x$.

Answer 1

Well, $dt$ is "constant", so it is independent of $X_t$ and can come out of the expectation. Thus you get

$$\mathbb{E}[X_t dt] = \mathbb{E}[X_t]dt $$

Assuming $\mu_x$ and $\sigma_x$ are deterministic, and that $W_t^x$ is a standard Brownian Motion, the expectation of $X_t$ is given by

$$\mathbb{E}[X_t] = \mathbb{E}[\mu_x t] + \mathbb{E}[\sigma_x W_t^x] = \mathbb{E}[\mu_x t] + 0 = \mu_x t $$

And finally

$$\mathbb{E}[X_tdY_t] = \mu_y \mu_x tdt$$