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

$$F_1 = S_1 \\ F_2 = \sigma_2S_1 - \sigma_1S_2 $$

  1. 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.

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    $\begingroup$ They really define $S_1$ and $S_2$ using only two random variables $\epsilon$ and $\epsilon_0$? This seems spectacularly wrong, even in a context where the rigor is a bit lax. $\endgroup$ – Did Mar 11 '14 at 6:50
  • $\begingroup$ Yes, I guess so. Not sure why this is problematic? $\endgroup$ – guest Mar 11 '14 at 15:00
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    $\begingroup$ Why? Maybe because this is similar to somebody enumerating the positive integers as 1, 2, 2, 2, 2, 2, 2, ... $\endgroup$ – Did Mar 11 '14 at 15:34

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.

  • $\begingroup$ I think he's saying that he is not allowed to presume that they are standard normal. I think they want him to take a limit and use CLT etc. It's kindof hard to tell from the question. $\endgroup$ – Kai Sikorski Mar 12 '14 at 22:11
  • $\begingroup$ It's in the task description, see the, as of yet, unedited question, the $ϵ$ are i.i.d. with standard normal distribution. $\endgroup$ – LutzL Mar 12 '14 at 23:37
  • $\begingroup$ Oh sorry, he said something about CLT in the comment to my answer so I thought he wasn't given that its normal $\endgroup$ – Kai Sikorski Mar 13 '14 at 2:50

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 $\epsilon_0$.

  • $\begingroup$ Yes, my question whether $\langle S_1S_2\rangle$ could be obtained using $\langle dS_1ds_2 \rangle$ and in that case, how this would be done. I'm not sure what you mean by key property you mean, that they are independent? Regarding the meanof $\epsilon$, it is not specified? They tend to assume normalization $\langle \epsilon \rangle = 0$, otherwise we at least know it is finite. Apparently, they use some version of CLT to show that a process $dP = \mu dt + \sigma \epsilon \sqrt{dt}$ is gaussian with mean $S(t_0)+\mu(t-t_0)^2$ and variance $\sigma^2(t-t_0)+(S(t_0)+\mu(t-t_0))^2$? $\endgroup$ – guest Mar 11 '14 at 12:41
  • $\begingroup$ Yes independence is what you need there. Look at the sums that define S and compute their properties. Indeed you will need clt $\endgroup$ – Kai Sikorski Mar 11 '14 at 16:07

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