Articles of quadratic variation

Quadratic Variation and Semimartingales

It is clear that every (I am particularly interested in continuous) semimartingale has a well defined quadratic variation process. However, what can be said about processes that have a well defined quadratic variation process? Is there a statement that says what exactly is needed in addition to existence of the quadratic variation in order for […]

Quadratic variation – Semimartingale

We know that any Semimartingale has Quadratic variation. I am interested to know if the converse is also true i.e. if a process has quadratic variation then it is semimartingale. Can some one prove/disprove it. Thank you very much.

Quadratic Variation of Brownian Motion

Quadratic Variation of a Brownian motion $B$ over the interval $[0,t]$ is defined as the limit in probability of any sequence of partitions $\Pi_n([0,t])=\{0=t^n_0<\cdots<t^n_{k(n)}=t\}$ of the interval $[0,t]$ such that $\lim_{n\to \infty}\max_{i=1,\cdots,k(n)}|t^n_i-t^n_{i-1}|=0$, of the functional $$V([0,t],\Pi_n)(B_.)=\sum_{i=1}^{k(n)}(B_{t_{i-1}}-B_{t_i})^2.$$ And for any such sequence of partition we have then $[B]_t=P-\lim_{n\to \infty} V([0,t],\Pi_n)(B_.)=t$. Nevertheless when you take the sup […]

Quadratic variation of Brownian motion and almost-sure convergence

Say that $W(t)$ is a Brownian motion. The quadratic variation $[W,W](t)$ is defined in terms of a partition $\Pi = \{0 = t_0 < t_1 < \cdots < t_n = t\}$ by $$ \begin{split} [W,W](t) &= \lim_{|\Pi|\to 0} \sum_{j=0}^{n-1} \Big( W(t_{j+1}) – W(t_j) \Big)^2\\ &= \lim_{|\Pi|\to 0} Q_n \end{split} $$ Here $|\Pi|\to 0$ means that […]

Angle bracket and sharp bracket for discontinuous processes

The question is quite simple actually. I am trying to understand the differences between the angle bracket $\left<X,Y\right>$ of two processes with jumps $X,Y$, and the sharp bracket of $[X,Y]$. I am aware that they are equivalent in the continuous case, but not for jump processes. The literature does not seem to be too explicit […]