0.1 \(p\)-th Variation
If \(X_t(\cdot):\Omega \to \mathbb{R}\) is a continuous-time [[knowledge-mathematics-analysis-probability-theory-stochastic-processes-stochastic-processes|stochastic process]], then for \(p>0\) the \(p\)-th variation process of \(X_{t}\) is defined by \[ \left< X,X \right>_{t}^{(p)}(\omega) = \lim_{\Delta t_{k} \to 0}\sum_{t_{k}\leq t}\lvert X_{t_{k+1}}(\omega)-X_{t_{k}}(\omega) \rvert^p \] where \(0=t_{1}<t_{2}<\dots<t_{n}\) and \(\Delta t_{k}-t_{k+1}-t_{k}\).
In particular:
- \(p=1\) this is known as total variation; and
- \(p=2\) this is known as quadratic variation.