brownian noise and stochastic differential equations

Question

Consider the SDE

$$dx=3x(t)dt+dW(t)$$

Where we're dealing with Brownian noise. Now, dW comes from

$$dW(t)=\int_0^{dt}ds\ \eta (s)$$

As far as I understood, $\eta$ is the noise distribution (usually Gaussian). Now, where comes the kind of noise into the equation (white noise, Brownian noise etc.)? Does the autocorrelation function of $\eta$ have something to do with it?

EDIT for clarity: I would like to know how to model the type of noise, i.e. what mathematical properties $\eta$ needs to have in order for the noise to be white, Brownian etc. The fact that it is Gaussian distributed does not account for that.

Answer 1

Although Brownian paths are not differentiable point wise, we may interpret their time derivative in a distributional sense to get a generalized stochastic process called white noise. We denote it by $$\eta (t,\omega )=\overset{\centerdot }{\mathop{B}}\,(t,\omega )$$ We also use the notation $$d\eta=dB_t$$ The term white noise arises from the spectral theory of stationary random processes, according to which white noise has a at power spectrum that is uniformly distributed over all frequencies (like white light).This can be observed from the Fourier representation of Brownian motion in $$B(t)=\frac{1}{\sqrt{\pi }}\left( {{a}_{0}}t+2\sum\limits_{n=1}^{\infty }{{{a}_{n}}\frac{\sin nt}{n}} \right)$$ where a formal term-by-term differentiation yields a Fourier series all of whose coefficients are Gaussian random variables with same variance.

Answer 2

Unfortunately, I have never really found an explanation for basic stochastic calculus which is both mathematically accessible and technically correct. I will give the usual mathematical treatment, but I concede that it is not particularly accessible.

$W$ is a process on $[0,\infty)$ with the following basic properties:

• $W(0)=0$
• $W$ is a Gaussian process.
• $W(t)$ has mean zero and variance $t$.
• $E[W(s)W(t)]=\min \{ s,t \}$
• Whenever $t_0 \leq t_1<t_2 \leq t_3$, $W_{t_3}-W_{t_2}$ is independent of $W_{t_1}-W_{t_0}$. We say that $W$ has independent increments.

Then $dW$ is just notation. Specifically, it is a notational shorthand for a so-called stochastic integral. The stochastic integral of a non-random function is basically defined the way you would expect from Riemann-Stieltjes integration:

$$\int_0^t f(s) dW(s) = \lim_{n \to \infty} \sum_{i=1}^{n-1} f(t_i^*) (W(t_{i+1})-W(t_i))$$

where $t_1<t_2<\dots<t_n$ is a partition of $[0,t]$ and $t_i^*$ is any point between $t_i$ and $t_{i+1}$. It turns out that this is well-defined, even though it is not an ordinary Riemann-Stieltjes integral, because $W$ turns out to not have bounded variation.

When $f$ becomes random (so we have $f(t,\omega)$ instead of just $f(t)$), now the result surprisingly depends on the choice of the evaluation point $t_i^*$! The most commonly used convention is to take $t_i^*=t_i$, which results in the so-called Ito integral. This has the advantage that $\int_0^t f(s,\omega) dW(s)$ is truly a "noise" term in that it has mean zero for all times (for "reasonable" $f$). The other commonly used convention is to take $t_i^*=\frac{t_i+t_{i+1}}{2}$, which results in the so-called Stratonovich integral. This has the advantage that the rules of the resulting stochastic calculus are essentially the same as in the classical setting, whereas they are not in the Ito setting.

In view of this ambiguity, there is some reason to avoid trying to talk about "$dW$" as its own process from a mathematicians' perspective. Nevertheless, the basic properties above tell us that if you want a "physicist's representation" of $dW$, its autocorrelation function will have to be $C(t,t')=\delta(t-t')$. If your autocorrelation is something other than a multiple of this one, then you don't have white noise. Still, there can't really be any such thing as white noise (as a bona fide stochastic process). For one thing, if there were such a thing, its sample paths would have to be almost surely non-Lebesgue measurable. ZF set theory is consistent with the statement that no such functions even exist, and even in ZFC set theory (where they do exist), none can be explicitly constructed.