A Question about Stochastic Integral

[hzzhangyu]

Homework Statement

How to prove that the Ito Integral int_0^t e^s dB_s is normally-distributed, for a given t?

Homework Equations

The Attempt at a Solution

This Ito Integral could be defined as a R-S Integral, and the Riemann Sum should be a linear function of normal r.v.s, thus the Riemann Sum is normal. However I don't know how to prove that the limit of a sequence of normal random variable is normal... Or is there another way to prove it?

Thanks!

 

[mighty2000]

Thank you for your question. The Ito Integral int_0^t e^s dB_s is indeed normally distributed, and this can be proven using the following steps:

1. First, recall that the Ito Integral is defined as the limit of Riemann sums, where each Riemann sum is a linear combination of normal random variables. This means that the Ito Integral itself is a linear combination of normal random variables, and therefore it is also normally distributed.

2. Next, we can use the properties of normal distributions to show that the Ito Integral is normally distributed with mean 0 and variance t. Recall that the mean of a linear combination of random variables is equal to the linear combination of their means. In this case, the mean of the Ito Integral is 0, since the mean of each Riemann sum is 0. Similarly, the variance of a linear combination of independent random variables is equal to the sum of their variances. Since the Riemann sums are independent, the variance of the Ito Integral is equal to the sum of the variances of the Riemann sums, which is t.

3. Finally, we can use the central limit theorem to show that the Ito Integral is normally distributed. The central limit theorem states that the sum of a large number of independent and identically distributed random variables will tend towards a normal distribution. Since the Riemann sums are independent and identically distributed, as the number of Riemann sums increases, the Ito Integral will tend towards a normal distribution.

I hope this helps to answer your question. If you have any further questions or concerns, please let me know. Best of luck with your research.