uniformly integrable

**Juju** · February 5th 2011, 08:04 AM

Hallo!
Is the following sequence of integrable random variables $(X_h)_{h \in [0,1]}$ also uniformly integrable?

$X_h=\prod\limits_{n=1}^{\frac{T}{h}}(1+\alpha_{(n-1)h}(\exp\{\mu_{(n-1)h}h+ \sigma (W_{nh}-W_{(n-1)h})\} -1))^\gamma$
whereas
$W$ is a standard Brownian motion,
$\sigma$ a constant $\in \matbb{R}_+,$
$\alpha_t$ is a random variable with values in [0,1],
$\mu_t$ is a standard-normal distributed random variable and $\mu_t$ is continous in $t$
[0,T] is the time interval and it is required that $N:=\frac{T}{h} \in \matbb{N}$ and
$\gamma$ is a constant $\in (0,1)$

I also know that $X_h$ converges in probability for $h \to 0$ to a integrable random variable $X$ .
I've no idea how to show it.
Can anybody help me?

I found out that $\mathbb{E}[X_h]\leq \mathbb{E}[X]<\infty \; \forall h \in [0,1] \quad$ and $X_h$ converges in probability to this random variable $X.$

Can I now conclude that $(X_h)_{h \in [0,1]}$ is uniformly integrable?

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