Quadratic and $p^\mathrm{th}$ variation of stochastic processes through Schauder expansions

We present a class of stochastic processes which admit a unique quadratic variation along any sequence of partitions $(\pi^n)_{n\geq 1}$ with $\sum_{n\geq 1}|\pi^n|<\infty$, which generalizes the previous results for finitely refining partitions. This class of processes contains some signed Takagi-Landsberg functions with random coefficients and standard Brownian motions, and these processes admit $\frac{1}{4}$-Hölder continuous version. We study the quadratic and $p^\mathrm{th}$ variation of signed Takagi-Landsberg functions with random coefficients. Finally, we seek some generalizations and applications of our results.