# Thread: convergence of random variables

1. ## convergence of random variables

Hey,
I've to show the following convergence.
$\displaystyle \lim\limits_{h \to 0} \mathbb{E}[e^{\gamma \sum\limits_{n=1}^{\frac{T}{h}} \log_e(1+\underline\alpha_{(n-1)h}R_n)}] = \mathbb{E}[e^{\gamma(\int\limits_0^T(\hat\mu_s -r) -\frac{1}{2} (\underline\alpha_s)^2\sigma^2 ds + \int\limits_0^T \underline\alpha_s \sigma dW_s)}]$,

$\displaystyle R_n=e^{(\hat\mu_s -r-\frac{1}{2}\sigma^2)h+\sigma(W_{nh}-W_{(n-1)h})}-1$,

whereas
$\displaystyle W$ is a standard Brownian motion
$\displaystyle \hat\mu_{nh}$ is a normal distributed random variable with continous paths
$\displaystyle \underline\alpha_{nh}$ is a random variable with values in the interval $\displaystyle [0,1]$, depends on $\displaystyle \hat\mu_{nh}$ and has continous paths
$\displaystyle 0<h<1$
$\displaystyle \sigma>0$ is a constant
$\displaystyle 0<\gamma<1$ is a constant

I started by using the taylor expansion of the exponential function for $\displaystyle R_n$
$\displaystyle (\hat\mu_s -r-\frac{1}{2}\sigma^2)h+\sigma(W_{nh}-W_{(n-1)h})+\frac{1}{2}((\hat\mu_s -r-\frac{1}{2}\sigma^2)h+\sigma(W_{nh}-W_{(n-1)h}))^2+...$
and also the taylor expansion for the logarithm:
$\displaystyle \log_e(1+\underline\alpha_{(n-1)h}R_n)]=\underline\alpha_{(n-1)h}R_n-\frac{1}{2}(\underline\alpha_{(n-1)h}R_n)^2+...$
But how can I approximate the remainder term?

Can I change the order of the expectation value and the limit?
I know that
$\displaystyle \mathbb{E}[e^{\gamma \sum\limits_{n=1}^{\frac{T}{h}} \log_e(1+\underline\alpha_{(n-1)h}R_n)}] \leq \mathbb{E}[e^{\gamma(\int\limits_0^T(\hat\mu_s -r) -\frac{1}{2} (\underline\alpha_s)^2\sigma^2 ds + \int\limits_0^T \underline\alpha_s \sigma dW_s)}], \; \forall h \in (0,1).$

The following equation holds due to the definition of the Itô integral. Am I right?
$\displaystyle \lim\limits_{h\to 0}\sum\limits_{n=1}^{\frac{T}{h}}\underline\alpha_ {(n-1)h} \sigma (W_{nh}-W_{(n-1)h})=\int\limits_0^T \underline\alpha_s \sigma dW_s }$
where the limit is taken in $\displaystyle L_2.$

Does anybody have an idea how to show this convergence?
Can anybody help me?
Thanks!

2. Unfortunately, there's a little mistake in the thread.
Below is the correct $\displaystyle R_n$

$\displaystyle R_n=e^{(\hat\mu_{(n-1)h} -r-\frac{1}{2}\sigma^2)h+\sigma(W_{nh}-W_{(n-1)h})}-1$
and the taylor expansion
$\displaystyle (\hat\mu_{(n-1)h-r-\frac{1}{2}\sigma^2)h+\sigma(W_{nh}-W_{(n-1)h})+\frac{1}{2}((\hat\mu_{(n-1)h} -r-\frac{1}{2}\sigma^2)h+\sigma(W_{nh}-W_{(n-1)h}))^2+...$