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Consider two Wiener processes: $$ \begin{aligned} X_a &\sim\mathcal N(0,a) \\ X_{a-b} &\sim\mathcal N(0,a-b) \end{aligned} $$ How do I show that: $$ X_a - X_{a-b} \sim\mathcal N(.,.) $$ That is, how do I show that $X_a - X_{a-b}$ is Gaussian?


I know that $E(X_a - X_{a-b}) = 0$

and $$ \begin{aligned} \text{Var}(X_a - X_{a-b}) &= E(X_a^2) - E(X_{a-b}^2) \\ &= \text{min}(a,a) -\text{min}(a-b,a-b) \\ &= a - a - b \\ &= b \end{aligned} $$


Note that both $X_a$ and $X_{a-b}$ have the following properties:

  • They are normally distributed with mean $0$.
  • Their sample paths are continuous.
  • Its covariance function is $\rho(s,t) := \text{Cov}(X_s,X_t)=E(X_sX_t)=\text{min}(s,t)$.
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What definition/construction of a Wiener process are you using? From that you should know much more about it than just the marginal distribution at a fixed time. –  cardinal Nov 7 '12 at 14:26
    
@cardinal Fixed. –  user14281 Nov 7 '12 at 14:30
    
Hi, Jase. Hmm. What is the source of this "characterization" that you've listed? (The properties are correct, but it is not a complete definition of a Wiener process.) –  cardinal Nov 7 '12 at 14:43
    
@cardinal It is presented in my lecture slides for "one-dimensional standard scalar BM". I left out the stuff about the filtration satisfying the usual conditions and the process $X$ being adapted to this filtration. –  user14281 Nov 7 '12 at 15:03
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1 Answer

$X_a = X_{a-b} + \int_{a-b}^a dX_s$

$\Rightarrow X_a - X_{a-b} = X_{a-b} + \int_{a-b}^a dX_s - X_{a-b}$

$ = \int_{a-b}^a dX_s$

$ = X_{a - (a-b)} = X_b$.

Since $\{X_t\}_{t \in [0,T]}$ is a Wiener process and $b \in [0,T]$, then $X_b$ is Gaussian by the fact that $X_b \sim N(0,b)$ (from the definition of a Wiener process).

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Unfortunately, this is not correct, as written. –  cardinal Nov 7 '12 at 14:41
    
@cardinal Which line is wrong? I stole the first line from the bottom of page 7 here: http://www.maxmatsuda.com/Papers/Intro/Intro%20to%20Brownian%20Matsuda.pdf, then the rest seems to follow. –  user14281 Nov 7 '12 at 15:02
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