It is know that the quadratic variation of Brownian motion $B_t$ is deterministic, $[B]_t=t$.
What other stochastic processes $X_t$ have a deterministic quadratic variation $f_X(t)$ ? Can we characterise this set ?
This set must include at least all process of the form $B_{g(t)}$ for any continuous $g: \mathbb R^+ \rightarrow \mathbb R^+$. What other elements are in this set ?
Best Answer
I hope the following is an answer in the way you wanted it to be. If there are any mistakes in it, please correct me.
As far as I know, the quadratic variation is only defiened for semimartingales, so we just consider this type of process. As $[X]_0=X_0^2$ we need $|X_0|$ to be deterministic. For simplicity let's focus first on continous processes. Since semimartingales are sums of local martingales and FV-processes, and since the continuous part of the quadratic variation is zero for FV-processes, we can always add a continuous FV-process to any stochastic process with a deterministic quadratic variation without destroying this property. As all contiuous local martingales are time-changed Brownian motions and since their quadratic variation is the change-of-time, the set contains all time-changed Brownian motions with a deterministic change-of-time (it might sound a bit silly, but this seems to be the answer). Now consider the discontinuous part. As the jumps of the quadratic variation are exactly the squared jumps of the process $\Delta [X]_t=(\Delta X_t)^2$ we need to have deterministic jumps. In summary the set (Hint: This is not unique decomposition.) consists of all processes $$X=M+A+J$$ (with a deterministic value $|X_0|$), where $M$ is a time-changed Brownian motions with a deterministic change-of-time, $A$ is a continuous FV-process and $J$ is a deterministic pure jump process. $\big($I'm not sure about this: As a non-constant deterministic pure jump process can't be a local martingale, the process $J$ can be included in the (noncontinuous) FV-part, thus $X$ is the sum of a a time-changed Brownian motions with a deterministic change-of-time and a FV-process with deterministic jumps.$\big)$
Another way to look at this is to consider purely discontinuous semimartingales. These are sums of FV-processes and purely discontinuous local martingales, which means that continuous part of the quadratic variation ist zero. As every semimartingale $X$ decomposes uniquely as $$X=X^c+X^d,$$ where $X^c$ is a continuous local martingale with ${X^c_0=0}$ and ${X^d}$ is a purely discontinuous semimartingale, the desired set consists of processes $X$ (with a deterministic value $|X_0|$) which are the sum of a purely discontinuous semimartingale with deterministic jumps and a time-changed Brownian motion with a deterministic change-of-time.