distribution of a product of lognormal distributed random variables

**Juju** · June 24th 2011, 11:49 AM

I'm sure, this question is very trivial. But I'm a little bit confused right now.

$B=(B_t^1,\ldots,B_t^d)$ is a d-dimensional Brownian motion.

$R_n:=\exp\left\{\mu + \sum\limits_{j=1}^d \sigma (B_{nh}^j-B^j_{(n-1)h})\right\} \quad n=1,2,\ldots,N$
$R:=\exp\left\{\mu + \sum\limits_{j=1}^d \sigma B^j_h\right\}$
$\mu, \sigma$ are just constants.
Am I right, that $R_n$ are independent for $n=1,\ldots, N$ and $R_n\stackrel{d}{=}R$ ?
whrere $d$ denotes equality in distribution.

Thanks in advance!

**Moo** · June 25th 2011, 02:21 PM

Hello,

I think you're right on both points.

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