Definition of quadratic variation of a stochastic process.

Asked Oct 30 '19 at 20:44

Active Oct 31 '19 at 07:41

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Let $(X_t)$ a stochastic process. Let $\pi_n: 0\leq t_0^n<...<t_m^n=1$ s.t. $|\pi_n|\to 0$. We define the quadratic variation as $$L^2-\lim_{n\to \infty }\sum_{i=0}^{m-1}|X_{t_{i+1}^n}-X_{t_i^n}|^2=:var_2(X,[0,t]).$$
I also no that if $(X_t)$ is a Martingale, we define the Quadratic variation of $(X_t)$ but the unique increasing and predictable process $\left<X\right>_t$ in the Doob decomposition : $$X_t^2=M_t+\left<X\right>_t.$$

Question

Is there any connection between this two process except the fact that they have the same name ? For example, does $var_2(f,[0,t])=\left<X\right>_t$ (I know for example that for Brownian motion it's true, but maybe is it a coincidence ?)

edited Oct 31 '19 at 07:41

asked Oct 30 '19 at 20:44

user657324

1,953

I believe you left off the superscript $n$ from your $X_{t_{i+1}}-X_{t_i}$ in the first displayed equation. – Math1000 Oct 31 '19 at 04:03
@Math1000: yes thank you – user657324 Oct 31 '19 at 07:41
2

You might want to take a look at this – saz Oct 31 '19 at 09:39

Definition of quadratic variation of a stochastic process.

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