Context
There are two different Wiener processes $W_t$ and $V_t$. It's known that they are independent. Additionally, we are given with third Wiener process $B_t$ that is given by the formula $$B_t = aW_t+bV_t, \quad \quad a^2 + b^2= 1.$$
Problem
Find the limit in $L^2$ of $$S_n = \sum_{i=1}^n\left[B_{it/n} - B_{(i-1)t/n}\right]\left[V_{it/n} - V_{(i-1)t/n}\right]$$
as $n$ tends to infinity.
My ideas
I assume that this is the type of task where we need to calculate the expected value and the variance. As the latter tends to $0$ (it should), we can say that the desired limit is the expected value. The issue is that it's very overextended work to calculate the $E(S_n)$.
Let $W_{it/n} = X_i$ and $V_{it/n} = Y_i$. We have
$$\Bbb E(S_n) = \Bbb E\sum_{i=1}^n [aX_i + bY_i - aX_{i-1} - bY_{i-1}][Y_{i} - Y_{i-1}]$$ which can be written as $$\sum \Bbb E\bigg( aX_iY_i + bY_i^2 - aX_{i-1}Y_i - bY_{i-1}Y_i - aX_iY_{i-1} - bY_iY_{i-1} + aX_{i-1}Y_{i-1} + bY_{i-1}^2\bigg).$$ Next calculations confuse me (what is $\Bbb E(X_i Y_{i-1})$?) Is it zero? And the main question how to calculate the variance?
If I'm not mistaken, $\Bbb E(S_n) = nb \to \infty$, so we don't need variance. Am I right?
from Hot Weekly Questions - Mathematics Stack Exchange
student
Post a Comment