= Dyadic increment chaining
{title2=$K_\alpha=2\sum_{n\geq0}2^{n\alpha}A_n$}
For a function on the <dyadic rationals> in $[0,1]$, let $A_n$ be its largest absolute adjacent increment on the level-$n$ grid. If $K_\alpha<\infty$ for $\alpha>0$, the function extends uniquely to a <Hölder continuous function> with
$$
|X_t-X_s|\leq K_\alpha|t-s|^\alpha.
$$
Choose $n$ with $2^{-n}\leq|t-s|<2^{1-n}$. The level-$n$ left approximations differ by at most two grid steps, and each finer approximation adds at most one level increment. Thus the difference is bounded by $2\sum_{j\geq n}A_j\leq2^{-n\alpha}K_\alpha$. For a <stochastic process> with $\|\xi_t-\xi_s\|_p\leq C|t-s|^\beta$, the bound $\|A_n\|_p\leq C2^{-n(\beta-1/p)}$ makes this series converge in the <Lp norm> for $0<\alpha<\beta-1/p$.
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