The notions of first-order variation and total variation of a function or a stochastic process are equated in this book.
However, I found their definitions different from two other sources:
- In Wikipedia the total variation of a function $f$ from time $0$ to time $T$ is defined as $ \sup_\Pi \sum_{i=0}^{n-1} | f(t_{i+1})-f(t_i) | . $
- In Stochastic Calculus for Finance II: Continuous-time Models by Steve Shreve, the first-order variation of a function or stochastic process $f$ from time $0$ to time $T$ is defined as: $ \lim_{||\Pi|| \to 0} \sum_{j=0}^{n-1} [f(t_{j+1}) - f(t_j)] $ where $\Pi=[t_0, t_1, \dots, t_n] \text{ and } 0 \leq t_0 < t_1 < \cdots < t_n = T$ and $||\Pi||= \max_{j=0,\dots, n-1} (t_{j+1} -t_j).$
Are they two different definitions, or equivalent? Thanks!