1
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1answer
307 views

Haar basis on $L^2(0,1)$ - proof?

I have the following problem. We defined $\mathbb{H}=\{f_0,\quad f_{j,n} \quad j=1,...,2^{n-1} \quad n=1,2,...\}$ where for all $t\in[0,1]$ we put $f_0(t)=1$ and setting $K=2j-1$, $$f_{j,n}(t)=\left\{ ...
1
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1answer
68 views

Covariation of Wiener processes, $\langle W_1,W_2\rangle_t = \rho t$.

I'm wondering why this is true: $\langle W_1,W_2\rangle_t = \rho t$. Where $W_1$ and $W_2$ are standard Brownian Motion. I know that $\langle W_1,W_2\rangle_t = 0.5\big[ \langle W_1 + W_2 \rangle - ...
2
votes
1answer
369 views

Expectation of exponential martingale and indicator function.

Let $W$ be a Wiener process, $r,\sigma \in \mathbb{R}_+$ and $S(T) = S(t)e^{(r-\frac12 \sigma^2)(T-t) + \sigma(W(T)-W(t))}$. I want to evaluate $$A:=E[e^{- \frac12 \sigma^2 (T-t) - ...
2
votes
0answers
40 views

Finite p-th variation implies zero-valued q-th variation.

The Question: Let $X$ be a continuous process, and suppose $0 < p < q$. Prove the case $V_t^p(X) < \infty \implies V_t^q(X) = 0$. Definitions: The standard setup. $\Pi := ...
1
vote
1answer
147 views

Demonstrate that every martingale is a local martingale.

The Original Question: Demonstrate that every martingale is a local martingale. Attempt at a Solution: Consider the standard setup of this problem: $\mathscr{F}_t$ is the filtration that satisfies ...