Take the 2-minute tour ×
Mathematics Stack Exchange is a question and answer site for people studying math at any level and professionals in related fields. It's 100% free, no registration required.

I have a lot of trouble in applying the functional monotone class theorem. Therefore I'm solving some exercises to get some experience. Maybe someone could help me with the following. Suppose I have shown that the following equality is true:

$$E[g f(W_{t+h}-W_t)]=E[g]E[f(W_{t+h}-W_t)]$$

for a bounded and measurable function $g$ and for a bounded and continuous function $f$ on $\mathbb{R}$. Now I should use the functional monotone convergence theorem to extend this to all f, which are bounded and measurable on $\mathbb{R}$. According to PlanetMath (Theorem 2), how would you choose $\mathcal{K}$ and $\mathcal{H}$ in this situation. Since I have trouble to apply the theorem, it would be appreciated if someone could help me to see a example of its use.

math

share|improve this question

1 Answer 1

up vote 3 down vote accepted

I think $\mathcal K = C_b(\mathbb R)$ and $$\mathcal H = \{f \text{ bounded measurable}\, ;\; \text{($\ast$) holds for all bounded measurable } g \}$$ where \begin{equation}\tag{$\ast$} E[gf(W_{t+h} - W_t)] = E[g]E[f(W_{t+h} - W_t)] \end{equation} would work, using monotone (or dominated) convergence to show closedness of $\mathcal H$ under monotone limits.

Note that $\mathcal K$ generates the Borel-$\sigma$-algebra, because any closed set $C$ is the zero set of a bounded continuous function, e.g. of $x \mapsto \min\{1,d(x,C)\}$, where $d(x,C) = \inf\{d(x,y)\mid y \in C\}$.

share|improve this answer
    
@ Sam L. thanks a lot! This was also my thought, but I was not quite sure about the generated $\sigma$-algebra. –  math Jun 24 '12 at 12:43

Your Answer

 
discard

By posting your answer, you agree to the privacy policy and terms of service.

Not the answer you're looking for? Browse other questions tagged or ask your own question.