0
votes
1answer
31 views

convergence of Ito integral

Suppose there is a deterministic process $\phi$ in $L^2(R)$. Need to prove that $\int_0^n \phi_u dW_u$ converges in $L^2(P)$ to some $X\in L^2(P)$ as $n\rightarrow\infty$. Also need to show that ...
2
votes
3answers
235 views

Proof of Levy's zero-one law

Let $(\Omega, \mathcal{F},\mathbb P)$ be a probability space and let $X$ be a random variable in $L^1$. Let $(\mathcal{F}_k)_k$ be any filtration, and define $\mathcal{F}_{\infty}$ to be the minimal ...
1
vote
1answer
195 views

basic application of Strong Law of Large Numbers

In $$ \sum_{j=0}^q {q\choose j}{1\over n}\sum_{i=1}^n X_i^j(-\bar X)^{q-j} \quad \overrightarrow{a.s.} \quad \sum_{j=0}^q {q\choose j} \mathbb{E}(X^j) (-\mathbb{E}(X))^{q-j} $$ using the Strong ...