How do you do calculus on a curve that has no slope? Brownian motion is continuous, but it's so jagged that classical calculus completely breaks down. Enter Kiyosi Itô—the quiet genius who built a calculus for roughness. 🧵 #Mathematics#DataScience#DataGuy
Ordinary calculus assumes curves flatten out when you zoom in. Brownian paths refuse: zoom in infinitely, and they remain rough. Because increments scale as dW∼dt, squaring them yields (dW)2∼dt. Those wiggles don't vanish; they accumulate into time. 🧠
In 1944, Itô published a 6-page paper defining a new integral. Instead of smoothing the noise, he embraced it. He evaluated integrand values at the left endpoint of each step, baking a strict discipline into the math: you cannot peek into the future to make decisions today. 🛡️
In 1951, he introduced "Itô’s Lemma"—the stochastic chain rule. Because squared wiggles matter, nonlinear functions of random processes require a curvature correction term: 21σ2∂x2∂2f This is the math behind Wall Street option pricing and ML diffusion models. 📉🤖
Itô's correction explains "volatility drag" in finance. If an asset evolves geometrically: dSt=μStdt+σStdWt The log price grows at μ−21σ2. Volatility drag is the mathematical price of randomness. 💸
Tweet 6: Itô didn't smooth randomness into calculus; he rebuilt calculus so randomness could remain rough. Read the full story of the quiet mathematician who made uncertainty calculable on https://t.co/aziAtnQVDE. Read the full post here: https://t.co/f1WvXT8Yln
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1/5 Ever heard of Louis Bachelier? In 1900, his thesis introduced the concept that stock price changes are random—a revelation that sat unnoticed for decades. 🌟
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