Finite-variation process

A process $A$ has finite variation on compact intervals when every sample path has finite total variation of a function there. Such a process can be integrated pathwise by the Lebesgue-Stieltjes integral.

For a path of finite variation, its total-variation process iswhere the supremum is over every partition of an interval of $[0,t]$. It is an increasing process, and the signed measure $dA$ satisfies $|dA|=dV(A)$.

If a càdlàg function has finite variation on compact intervals, its total-variation process is finite, nondecreasing, and càdlàg. Left limits follow from monotonicity. For right continuity, approximate the variation on $[t,s]$ by one finite partition. Changing its first endpoint from $t$ to $u\downarrow t$ changes the sum by at most $|X_u-X_t|$, which tends to zero. Additivity of variation then bounds $V(u)-V(t)$ by an arbitrarily small error. Its jump is the absolute jump of $X$. Without locally finite variation, right continuity can fail even as an extended-valued assertion: $X_t=t\sin (1/t)$ with $X_0=0$ has infinite variation on every interval starting at zero.