3
$\begingroup$

I learned that Brownian motion is any stochastic process $(W_{t})_{t\geq0}$ that satisfies four well-known properties. But I still don't understand how these four properties uniquely determine Brownian Motion.

The four properties are

  1. $W_0=0$,

  2. for times $ 0 \leq t_1 \leq t_2 <...< t_n the random variables $W_{t_2} - W_{t_1}$, $W_{t_3}-W_{t_2}$,...,$W_{t_n}-W_{t_{n-1}}$ are independent.

  3. For any $0\leq s \leq t the increment $W_{t} - W_{s}$ has the Gaussian distribution with mean 0 and variance $t-s$.

  4. For all $\omega$ in a set of probability one, $B_t (w)$ is a continuous function of $t$.

More precisely,

Let $(W_{t})_{t\geq0}$ and $(W_{t}')_{t\geq0}$ be two stochastic processes that satisfy the four properties of a Brownian motion. Prove that these stochastic processes must then have the same finite dimensional distributions, that is, $E[f(W_{t_{1}},...,W_{t_{n}})]=E[f(W_{t_{1}}^{'},...,W_{t_{n}}^{'})]$ for any $n\geq1$ , $t_{1},...,t_{n}\geq0$ , and bounded measurable $f:\,\mathbb{R}^{n}\rightarrow\mathbb{R}$ .

  1. How do I go about showing this....I am having trouble with multivariate gaussians. 2. Does the Kolmogorov's Extension theorem play some sort of a role here?
  • 0
    There are several very different ways to construct Brownian motion. For example, Paul Lévy viewed it through a sequence of dyadic continuous gaussian linear interpolating approximations. Hence you might want to write down the *four properties* you have in mind.2012-02-22
  • 0
    I understand. But, my primary difficulty here is computational: how does one even show E[f(Wt1,...,Wtn)]=E[f(W′t1,...,W′tn)]2012-02-22
  • 0
    In some constructions, the value of these marginals is one of the defining properties.2012-02-22
  • 0
    [Finite dimensional distributions](http://en.wikipedia.org/wiki/Finite-dimensional_distribution) are defined as $$ \nu_{t_1,\dots,t_n}(B_1,\dots,B_n) = \mathsf P\{X_{t_1}\in B_1,\dots,X_{t_n}\in B_n\} $$ and you may find it easier to work with such formulations. However it depends on four properties you have in mind - please write them in your question.2012-02-22
  • 0
    Ok i made those properties more explicit2012-02-22

2 Answers 2