First, I'm not majoring mathematics. I'm studying economics and during reading a thesis I can't understand the 'wiener process' well.
I read some books about it and understand the main idea and standard formula, but the thesis says $\frac{dS}{S}=\mu dt+\sigma_1 dz_1$
$\frac{dD}{D}=r dt+\sigma_2 dz_2$
where $S,D$ are geometric brownian motion, and
$z_1,z_2$ are standard brownian motion (wiener process), then
$dz_1 dz_2=\rho dt$
where $\rho$ denotes the correlation coefficient between the two Wiener processes.
I want to know how to derive the last one-correlation coefficient- becuase the books that I read there were no information about $\rho$.
Can anyone help me to understand?

  • $\begingroup$ There are entire courses dedicated to this. You arent going to understand SDE's from a single answer on a QA site... $\endgroup$ – rmh52 Oct 25 '13 at 15:24
  • $\begingroup$ @rmh52 Then can you recommend me any book to study that part? In my book, it covers $var$ and $cov$ but not $corr$... $\endgroup$ – time Oct 25 '13 at 15:29
  • $\begingroup$ If you know about variance and covariance, then the correlation $\rho$ of two random variables $X,Y$ is just $\rho = \operatorname{Cov}(X,Y)/\sqrt{\operatorname{Var}(X)\operatorname{Var}(Y)}$. See en.wikipedia.org/wiki/… $\endgroup$ – Nate Eldredge Oct 25 '13 at 16:05
  • $\begingroup$ amazon.com/gp/aw/d/3540047581 $\endgroup$ – rmh52 Nov 2 '13 at 20:25
  • $\begingroup$ The solutions for the exercises in this text are almost all available online (sorry I don't have a link) which makes it great for self study or supplementing another text. $\endgroup$ – rmh52 Nov 2 '13 at 20:26

Although @rmh52 is mostly correct, I'll take a stab at it. Viewing stochastic processes such as Geometric Brownian Motion (GBM) as limits of discrete time processes helps to build intuition about all of its components. For example, as you mention above GBM satisfies the SDE: $$dS_t = \mu S_t dt +\sigma S_t dW_t$$ which you can think about approximating by starting with $S_0$ at time $0$ and adding, at time $t$, for each increment $\Delta t$ in time, a small amount $\Delta S_t=\mu S_t+ \sigma S_t \Delta W_t$, where the change $\Delta W_t$ in the Wiener process is normally distributed with variance $\Delta t$ and independent of the other increments to the Wiener process you have performed thus far.

Taking this one step further is to think about a two-dimensional system of GBM - which you have described above as: $$dS_t = \mu D_t dt +\sigma_1 D_t dZ_t$$ $$dD_t = rD_t dt +\sigma_2 S_t dW_t$$ How would you simulate this system, in particular how would you generate the increments to the Wiener processes $W_t,D_t$? These should both be normally distributed with variance $\Delta t$, but they do not necessarily have to be independent - you can correlate them with any constant $-1\leq \rho\leq 1$ to introduce correlation between the two Brownian motions!

This is applicable, for example, when applying multi-dimensional GBM models to stock prices. You may want to assume that the stock prices for, say, Ford and GM are GBMs - but you want the movements of these two companies to be (to a certain extent) correlated - that is, when Ford moves up GM generally does also, and vice versa. This is possible with correlation!


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