Pillar
The book pillar. Intrinsic time, event-driven filters, RMT, network causality, Ricci curvature, TDA, optimal transport, Kuramoto, entropy, chaos, and Ising/LPPL crash precursors.
Prices are the residue of people arguing about value. Fractals, multifractals, sentiment, and topology all say the market is social, fat-tailed, and memory-laden. They describe wildness; they don't forecast it.
After a crash the violent days keep coming, dense then fading along the same power-law curve geologists use for aftershocks. Independent returns can't do that, so size down for weeks, not days.
Bachelier's normal-price model lets prices go negative; putting the walk on log price fixes it. The twist: raw price fits a log-normal badly, but price divided by volume fits one cleanly.
Market "long memory" is mostly short-range autocorrelation in disguise: correct for it and the Hurst signal collapses toward random. Kill a striking statistic with the boring explanation first.
The market's randomness itself drifts: competition arbitrages structure away, so recent data is more random than old and your backtest edge is an upper bound that decays, not a stable estimate.
Chop price into up/down "words", histogram them, and measure their entropy: near 1 the market is choppy noise, near 0 it keeps rhyming. Same formula as the indicator-quality score, opposite meaning.
Mutual information measures whether the next move is connected to the last few or just noise. Volatility can't see the difference; MI gates your system on when structure is present and off when it drains away.
Stochastic calculus is one twist on the integral you know. Riemann weights rectangles by equal width, Riemann-Stieltjes by a function g, Itô by Brownian motion. Random weights make the integral a process Y(t).