Given a 2-dimensional Wiener process $(W_{1},W_{2})$ with correlation $\rho$.

Let \begin{equation*} X(t):= F(t) + \int_{0}^{t} f(s) dW_{1}(s) + \int_{0}^{t} g(s) dW_{2}(s)\end{equation*} for some nice enough deterministic functions $F$, $f$ and $g$. Let now $0<t_{1}<t_{2}<\ldots < t_{n+1}$ and $k\in\{1,2,\ldots,n+1 \}$. We define $X_{i}:=X(t_{i})$ and \begin{equation*} Y_{k}:= (X_{k}-X_{i})_{i\neq k}=(X_{k}-X_{1},\ldots,\widehat{X_{k}-X_{k}},\ldots,X_{k}-X_{n+1})\in\mathbb{R}^{n}. \end{equation*} I know would like to understand the following claim:

$Y_{k}$ has a multivariate normal distribution.

Any help or references would be appreciated very much.

As a follow-up:

I figured that the definition of 2-dimensional Wiener process $(W_{1},W_{2})$ with correlation $\rho$ is not quite clear to me. I assume that $W_{1}$ and $W_{2}$ being one-dimensional Wiener processes and $corr(W_{1}(t),W_{2}(t))=\rho$ is not enough, isn't it!?

I would assume that we have to assume that $(W_{1}(t),W_{2}(t))$ has a two-dimensional normal distribution with mean 0.

Generally, is there a standard definition for a n-dimensional Wiener process with correlation? If so, I would be happy to get some references.

Else, my guess would be that a n-dimensional stochastic process $W=(W_{1},\ldots,W_{n})$ is a n-dimensional process with:

  1. $W(0)=0$ a.s.
  2. $W_{t}$ a.s.-continuous
  3. increments are independent
  4. $W_{t}-W_{s}\sim N(0,\Sigma)$, for $t>s$


$\Sigma$ is a positive-definite and symmetric matrix with diagonal elements equal to $t-s$

  • $\begingroup$ $(\xi_1, \ldots, \xi_n)$ is multi-normal iff any combinations $\sum_{i=1}^n a_i \xi_i$ is normal. $\endgroup$ – Gordon Sep 10 '17 at 14:06

As noted above, the random vector $Y_k$ is multi-normal if for any combinations \begin{align*} \sum_{i\ne k} a_i (X_k-X_i) \tag{1} \end{align*} is normal. WLOG, we assume that $1<k<n+1$. Note that \begin{align*} \sum_{i\ne k} a_i (X_k-X_i) &=-\sum_{i\ne k}a_i X_i +X_k \sum_{i\ne k} a_i\\ &=-a_{n+1}(X_{n+1}-X_n)\\ &\quad -(a_{n+1}+a_n)(X_n-X_{n-1})\\ &\quad - \cdots \\ &\quad -\sum_{i=k+1}^{n+1} a_i(X_{k+1}-X_k)\\ &\quad +\sum_{i=1}^{k-1}a_i(K_k-X_{k-1})\\ &\quad +\cdots\\ &\quad +a_1(X_2-X_1). \end{align*} Since $(W_1, W_2)$ is a 2-dimensional Brownian motion, by Cholesky decomposition, \begin{align*} W_1(t) &= B_1(t),\\ W_2(t) &= \rho B_1(t) + \sqrt{1-\rho^2}B_2(t), \end{align*} where $B_1$ and $B_2$ are two independent Brownian motions. Then, for $i=2,\ldots, n+1$, \begin{align*} X_i-X_{i-1} &= F(t_i)-F(t_{i-1}) + \int_{t_{i-1}}^{t_i}f(s)dW_1(s) + \int_{t_{i-1}}^{t_i}g(s)dW_2(s)\\ &= F(t_i)-F(t_{i-1}) + \int_{t_{i-1}}^{t_i}(f(s)+\rho g(s))dB_1(s) + \int_{t_{i-1}}^{t_i}\sqrt{1-\rho^2}g(s)dB_2(s) \end{align*} is normal. Moreover, since \begin{align*} (X_2-X_1),\, \ldots, \, (X_{n+1}-X_n) \end{align*} are independent, their combinations \begin{align*} \sum_{i\ne k} a_i (X_k-X_i) \end{align*} is also normal. That is, $Y_k$ is multi-normal.

  • $\begingroup$ In fact, I was considering the case where $W_{1}$ and $W_{2}$ are not necessarily indenpendent, but have correlation $\rho$. Can one then adapt your argument? $\endgroup$ – Strickland Sep 10 '17 at 18:55
  • $\begingroup$ In that case you can use that $W_2(t) \sim \rho W_1(t) + \sqrt{1 - \rho^2} W_3(t)$ where $W_3$ is another independent Brownian motion and where $\sim$ denotes equality in distribution. $\endgroup$ – LocalVolatility Sep 11 '17 at 7:05
  • $\begingroup$ I actually have a follow-up question: see above $\endgroup$ – Strickland Sep 15 '17 at 15:47
  • $\begingroup$ See Appendix A of the book Stochastic Differential Equations by Oksendal. $\endgroup$ – Gordon Sep 15 '17 at 16:21
  • $\begingroup$ The reference just seems to treat multidim. normally distributed random variables and not really what I was looking for (see above) $\endgroup$ – Strickland Sep 15 '17 at 17:04

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.