Understanding basic stochastic differential equations This is from a physics course in economics, the literature provides a bare minimum of mathematical explanations. I am trying to understand how to work with stochastic differential equations given in exercises. Any explanation of how to approach would be appreciated. I am assuming this is very easy, but since the given literature is unreadable for me, I have no idea. 

Assume that the time evolution of two stock prices $S_1$ and $S_2$ are
  described by the two following Wiener process, 
$$dS_1 = \sigma_1\epsilon\sqrt{dt}\\ dS_2  = \sigma_2\epsilon\sqrt{dt}
+ \sigma_0\epsilon_0\sqrt{dt},$$
where $\sigma_0, \sigma_1, \sigma_2$ are volatilities, $\epsilon_0$
  and $\epsilon$ are independent , normally distributed random numbers
  with variance one. Furthermore, assume that $S_1(0) = S_2(0) = 0$.
1. If $\sigma_2 = 0$, what is the correlation between S_1(t) and S_2(t)?
2. Calculate the variance and the correlation between the two following portfolios
$$F_1 = S_1 \\ F_2 = \sigma_2S_1 - \sigma_1S_2 $$



*

*Assuming $\sigma_2 = 0$ yields $dS_2 = \sigma_0\epsilon_0 \sqrt{dt}$. They provide no proper definition of the correlation, but from what I have seen in an example, it seems to be given by the moment $\langle S_1 S_2\rangle$. How is this integral derived from the given information? Do we compute


$$\langle dS_1dS_2 \rangle = \sigma_0\sigma_1\langle\epsilon_0\epsilon dt\rangle = \sigma_0\sigma_1\langle\epsilon_0\rangle\langle\epsilon\rangle dt ?$$
Where would we go from here?
2.. Itôs formula seems to be the key. Again, they do not provide a proper definition, but I'm guessing the approach is the following. Let $f(x,t) = x$ and define $F_1 = f(S_1, t)$ and $F_2 = \sigma_2f(S_1,t) - \sigma_1f(S_2, t)$. We should get the following
$$dF_1 = \sigma_1\epsilon\sqrt{dt} \\
dF_2 = \sigma_2dS_1 - \sigma_1dS_2 = -\sigma_0\sigma_1\epsilon_0\sqrt{dt}.$$
Any suggestions? As of writing I just got my hands on a copy of Oksendal's "Stochastic differential equations" which I hope will have an approach that I am more comfortable with. 
 A: This formulation of an SDE is more suited for the discretized SDE
\begin{align}
ΔS_1&=σ_1ϵ\sqrt{Δt},\\ ΔS_2&=σ_2ϵ\sqrt{Δt}+σ_0ϵ_0\sqrt{Δt}
\end{align}
which can be solved by simply summing up
\begin{align}
S_1(nΔt)&=\sum_{k=0}^{n-1}σ_1(kΔt)ϵ(kΔt)\sqrt{Δt}\\ 
S_2(nΔt)&=\sum_{k=0}^{n-1}\bigl(σ_2(kΔt)ϵ(kΔt)+σ_0(kΔt)ϵ_0(kΔt)\bigr)\sqrt{Δt}
\end{align}
and all of the $ϵ_0(kΔt)$, $ϵ(kΔt)$ are independend standard-normally distributed random variables, which makes the correlation computations rather easy.
A: 1) Well one maybe pedantic thing I would point out is that you dont want $\langle dS_1 dS_2 \rangle$ but rather $\langle S_1 S_2 \rangle$. In more complicated SDEs that would be a more substantial issue, here you can use properties of how independent Gaussian RVs sum to conclude that it's basically almost the same thing. The rest of your ideas are mostly correct, but I hope that you know what key property of $\epsilon_0$ and $\epsilon$ you are using to allow you to factor the expectation. Once it's factored like that what do you know about the mean of $\epsilon$ for example?
2) I wouldn't really say this even needs Ito's formula. You can just get it from the linearity of the Ito integral. If you try to compute a variance for $F_2$ (take $\langle F_2^2 \rangle$, plug in what it is in terms of $S_1$ and $S_2$ and expand) you'll get terms like $\langle \epsilon \epsilon_0 \rangle$, and $\langle \epsilon^2 \rangle$. Some of those will be zero. Some won't. Use what you know about $\epsilon$ and $\epsilon_0$.
