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A Japanese mathematician published a result in 1944 that nobody understood for twenty years. Today it runs inside every options desk on Wall Street. Goldman pays $400K to quants who can derive it from scratch and explain why classical calculus gives the wrong answer without it. His name is...

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Once you swallow the fact that Brownian motion is continuous everywhere and differentiable nowhere, Itô’s quadratic variation is the next quiet shock waiting for you: Ordinary smooth curves have zero quadratic variation...if you chop time into tiny pieces and add up (increment)² over the interval, that sum goes to zero as your partition gets finer. But for Brownian motion that same sum doesn’t vanish, it locks onto the length of the time interval.😲 So, you have this path with no velocity anywhere, but its squared wiggles accumulate in a completely rigid, deterministic way. Over time T, the total quadratic variation is exactly T. 🫡 That’s the trade the universe makes here...you lose the classical derivative, but you gain a new calculus where the “dB times dB” term behaves like dt, and suddenly integrals against Brownian motion make sense and you can do stochastic calculus. 📝 Brownian motion refuses to be smooth, but its roughness is so structured that when you look at it through the lens of quadratic variation, the randomness averages out and you’re left with a clean clock ticking underneath the noise. #BrownianMotion #ProbabilityTheory #StochasticCalculus #RandomWalks #Ito #QuadraticVariation
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Once you swallow the fact that Brownian motion is continuous everywhere and differentiable nowhere, Itô’s quadratic variation is the next quiet shock waiting for you: Ordinary smooth curves have zero quadratic variation...if you chop time into tiny pieces and add up (increment)² over the interval, that sum goes to zero as your partition gets finer. But for Brownian motion that same sum doesn’t vanish, it locks onto the length of the time interval.😲 So, you have this path with no velocity anywhere, but its squared wiggles accumulate in a completely rigid, deterministic way. Over time T, the total quadratic variation is exactly T. 🫡 That’s the trade the universe makes here...you lose the classical derivative, but you gain a new calculus where the “dB times dB” term behaves like dt, and suddenly integrals against Brownian motion make sense and you can do stochastic calculus. 📝 Brownian motion refuses to be smooth, but its roughness is so structured that when you look at it through the lens of quadratic variation, the randomness averages out and you’re left with a clean clock ticking underneath the noise. #BrownianMotion #ProbabilityTheory #StochasticCalculus #RandomWalks #Ito #QuadraticVariation

Mathelirium

100,544 Aufrufe • vor 8 Monaten

An MIT professor explained the one idea funds pay $500K for, and he does it in the first hour of freshman calculus. MIT put the whole lecture online. it has been free for twenty years. almost nobody finishes it. his point: calculus is not a hundred formulas to memorize. it is one idea. take two points on a curve, slide them together until the gap vanishes, and the slope you are left with is the derivative. everything else is just consequences. he is David Jerison, and his 18.01 lectures have taught more people calculus than any room on earth. skip to the board, where the power rule, the product rule, the chain rule, the fifty things you were forced to memorize, all fall out of that one limit. no formula sheet. no tricks. one definition, reused. that same slope is velocity. it is marginal cost. it is the gradient that trains a neural network. one operation, hiding everywhere something moves. the math is free. what nobody can sell you is the patience to understand the one limit instead of cramming the fifty formulas that fall out of it. that hour is the foundation, and it is the only part that ever pays.
50:54

An MIT professor explained the one idea funds pay $500K for, and he does it in the first hour of freshman calculus. MIT put the whole lecture online. it has been free for twenty years. almost nobody finishes it. his point: calculus is not a hundred formulas to memorize. it is one idea. take two points on a curve, slide them together until the gap vanishes, and the slope you are left with is the derivative. everything else is just consequences. he is David Jerison, and his 18.01 lectures have taught more people calculus than any room on earth. skip to the board, where the power rule, the product rule, the chain rule, the fifty things you were forced to memorize, all fall out of that one limit. no formula sheet. no tricks. one definition, reused. that same slope is velocity. it is marginal cost. it is the gradient that trains a neural network. one operation, hiding everywhere something moves. the math is free. what nobody can sell you is the patience to understand the one limit instead of cramming the fifty formulas that fall out of it. that hour is the foundation, and it is the only part that ever pays.

Loran

1,075,833 Aufrufe • vor 1 Monat

Ten million people have watched an MIT professor teach a course whose first lecture is literally titled "What is a Derivative?" Almost none have written down the two-line answer. He filmed the lecture once in the fall of 2007 and it has been on YouTube ever since. Math tutors charge $200 an hour to teach a diluted version of what he covered in 50 minutes for free. His name is David Jerison. He is a professor of mathematics at MIT and the instructor of 18.01 Single Variable Calculus, one of the most-watched math courses in the history of the internet. The 50-minute clip in this video is Lecture 1, filmed at MIT in the fall of 2007. Jerison is deriving the definition of a derivative from a single tangent line. The whole framework fits on a napkin. A derivative is just how much y changes when x moves a tiny bit. Draw a tangent line to any curve at any point. The slope of that line is the derivative. Memorize one formula, the power rule, and you can differentiate every polynomial on earth in your head. Chain that with a handful of exceptions and you can differentiate almost every function humanity has ever written down. That single set of rules is what every neural network runs on gradient descent, what every rocket landing at SpaceX solves in real time, and what every options desk at Goldman Sachs is running behind every quote you see on the screen. "In mathematics you don't understand things. You just get used to them." That is John von Neumann, the mathematician who helped design the atomic bomb and invent the modern computer. Jerison returns to the same idea in every lecture. Almost no student giving up on calculus has heard von Neumann say it out loud. Every quant fund on Wall Street pays entry-level analysts $250,000 to know the same power rule Jerison derives on the board. Every AI bootcamp charges tens of thousands to teach a diluted version of the same equation on a laptop. The lecture is free on MIT OpenCourseWare. The textbook is under sixty dollars. Almost none of the millions who watched have ever taken the power rule and applied it to their own numbers on their own paper. The math is free. The willingness to actually take one derivative before your next model, trade, or engineering trade-off is the entire edge.
50:48

Ten million people have watched an MIT professor teach a course whose first lecture is literally titled "What is a Derivative?" Almost none have written down the two-line answer. He filmed the lecture once in the fall of 2007 and it has been on YouTube ever since. Math tutors charge $200 an hour to teach a diluted version of what he covered in 50 minutes for free. His name is David Jerison. He is a professor of mathematics at MIT and the instructor of 18.01 Single Variable Calculus, one of the most-watched math courses in the history of the internet. The 50-minute clip in this video is Lecture 1, filmed at MIT in the fall of 2007. Jerison is deriving the definition of a derivative from a single tangent line. The whole framework fits on a napkin. A derivative is just how much y changes when x moves a tiny bit. Draw a tangent line to any curve at any point. The slope of that line is the derivative. Memorize one formula, the power rule, and you can differentiate every polynomial on earth in your head. Chain that with a handful of exceptions and you can differentiate almost every function humanity has ever written down. That single set of rules is what every neural network runs on gradient descent, what every rocket landing at SpaceX solves in real time, and what every options desk at Goldman Sachs is running behind every quote you see on the screen. "In mathematics you don't understand things. You just get used to them." That is John von Neumann, the mathematician who helped design the atomic bomb and invent the modern computer. Jerison returns to the same idea in every lecture. Almost no student giving up on calculus has heard von Neumann say it out loud. Every quant fund on Wall Street pays entry-level analysts $250,000 to know the same power rule Jerison derives on the board. Every AI bootcamp charges tens of thousands to teach a diluted version of the same equation on a laptop. The lecture is free on MIT OpenCourseWare. The textbook is under sixty dollars. Almost none of the millions who watched have ever taken the power rule and applied it to their own numbers on their own paper. The math is free. The willingness to actually take one derivative before your next model, trade, or engineering trade-off is the entire edge.

Lumen

24,218 Aufrufe • vor 4 Tagen

Nassim Taleb breaks down the Black-Scholes formula - and the story he tells starts with a forgotten man. Louis Bachelier priced options 73 years before Black-Scholes - and modeled Brownian motion 5 years before Einstein. it was 1900. a 30-year-old Frenchman defends a thesis under Henri Poincaré that quietly invents two fields at once: the math of random markets, and the first real formula for pricing options. his reward? the committee marks it down - "too much finance." no top grade, no patron, no career. he spends his life applying for jobs one rank below what his work deserved, and dies forgotten in 1946. then 1973: Black, Scholes & Merton publish the option-pricing formula. it reshapes Wall Street and wins the 1997 Nobel - for two of them (Black had died, so missed it). almost nobody remembers the Frenchman who got there first. and here's Taleb's twist: traders never actually use Black-Scholes. what the market runs on is a version of Bachelier's original. the Nobel, he argues, wasn't even for the pricing - it was for dressing it up to fit the economic theory of the day. ~15-min lecture, free. the man who built modern finance — and watched the credit go to everyone but him ↓
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Nassim Taleb breaks down the Black-Scholes formula - and the story he tells starts with a forgotten man. Louis Bachelier priced options 73 years before Black-Scholes - and modeled Brownian motion 5 years before Einstein. it was 1900. a 30-year-old Frenchman defends a thesis under Henri Poincaré that quietly invents two fields at once: the math of random markets, and the first real formula for pricing options. his reward? the committee marks it down - "too much finance." no top grade, no patron, no career. he spends his life applying for jobs one rank below what his work deserved, and dies forgotten in 1946. then 1973: Black, Scholes & Merton publish the option-pricing formula. it reshapes Wall Street and wins the 1997 Nobel - for two of them (Black had died, so missed it). almost nobody remembers the Frenchman who got there first. and here's Taleb's twist: traders never actually use Black-Scholes. what the market runs on is a version of Bachelier's original. the Nobel, he argues, wasn't even for the pricing - it was for dressing it up to fit the economic theory of the day. ~15-min lecture, free. the man who built modern finance — and watched the credit go to everyone but him ↓

0xDipper

245,386 Aufrufe • vor 2 Monaten

The smartest man at Bell Labs found a way to make money from a stock that goes nowhere. up 40%, down 40%, flat on the year, and you still walk away richer. he barely wrote it down. it survived in a black-and-white MIT lecture from 1970. it has been free ever since. almost nobody has watched it. the idea: your money does not grow by adding, it grows by multiplying. and the second you switch from adding to multiplying, one function quietly runs everything: the logarithm. the man at the board is Herb Gross, MIT, whose 1970 calculus course taught more traders their own math than most finance degrees ever will. here is the part that pays. growth compounds along a curve that bends, and that bend is the whole secret. because it bends, a volatile stock that goes nowhere can be turned into real profit. hold it, rebalance it, buy low and sell high on autopilot, and the wiggle itself becomes the return. no ticker. no forecast. one curve, and the way it bends. Bell Labs called it Shannon's Demon: money out of a stock that never moved. it lives inside the logarithm Gross draws on the board. the stock did nothing. the math did everything. it has been free since 1970, and almost no one bothered to learn the one function that turns motion into money.
33:58

The smartest man at Bell Labs found a way to make money from a stock that goes nowhere. up 40%, down 40%, flat on the year, and you still walk away richer. he barely wrote it down. it survived in a black-and-white MIT lecture from 1970. it has been free ever since. almost nobody has watched it. the idea: your money does not grow by adding, it grows by multiplying. and the second you switch from adding to multiplying, one function quietly runs everything: the logarithm. the man at the board is Herb Gross, MIT, whose 1970 calculus course taught more traders their own math than most finance degrees ever will. here is the part that pays. growth compounds along a curve that bends, and that bend is the whole secret. because it bends, a volatile stock that goes nowhere can be turned into real profit. hold it, rebalance it, buy low and sell high on autopilot, and the wiggle itself becomes the return. no ticker. no forecast. one curve, and the way it bends. Bell Labs called it Shannon's Demon: money out of a stock that never moved. it lives inside the logarithm Gross draws on the board. the stock did nothing. the math did everything. it has been free since 1970, and almost no one bothered to learn the one function that turns motion into money.

Loran

237,866 Aufrufe • vor 1 Monat

Brownian Motion Has a Hidden Speed Limit Once you’ve come to terms with Brownian motion having no derivative, acquiring a deterministic Quadratic Variation, and crossing zero infinitely often in every neighbourhood, the next surprise is the Law of The Iterated Logarithm. A Brownian path may look completely unconstrained, but its largest excursions obey an astonishingly sharp envelope. The right scale is not merely √t, but √[2t log log t], and that strange log log t term is exactly what sets the true boundary for how far Bₜ can wander. The theorem says that, almost surely, the path comes arbitrarily close to the upper and lower curves ±√[2t log log t] infinitely many times, yet does not genuinely live beyond them. Choose a smaller multiple and Bₜ will break through your barrier infinitely often. Choose a larger one and, beyond some point, it almost never reaches it again. #BrownianMotion #LawOfTheIteratedLogarithm #StochasticProcesses #RandomPaths #ProbabilityTheory #FractalBehaviour
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Brownian Motion Has a Hidden Speed Limit Once you’ve come to terms with Brownian motion having no derivative, acquiring a deterministic Quadratic Variation, and crossing zero infinitely often in every neighbourhood, the next surprise is the Law of The Iterated Logarithm. A Brownian path may look completely unconstrained, but its largest excursions obey an astonishingly sharp envelope. The right scale is not merely √t, but √[2t log log t], and that strange log log t term is exactly what sets the true boundary for how far Bₜ can wander. The theorem says that, almost surely, the path comes arbitrarily close to the upper and lower curves ±√[2t log log t] infinitely many times, yet does not genuinely live beyond them. Choose a smaller multiple and Bₜ will break through your barrier infinitely often. Choose a larger one and, beyond some point, it almost never reaches it again. #BrownianMotion #LawOfTheIteratedLogarithm #StochasticProcesses #RandomPaths #ProbabilityTheory #FractalBehaviour

Mathelirium

29,105 Aufrufe • vor 5 Monaten

The Curve That Forced Mathematicians To Invent A New Calculus There is a strange curve called Brownian path. It traces a continuous path without ever jumping, but it never settles into a well-defined direction at any point, so it’s continuous everywhere but differentiable nowhere. We’re used to the idea that if something is continuous, then if you zoom in far enough it should start to look like a straight line. A Brownian path refuses. Zoom in and you don’t reveal a hidden tangent. You reveal fresh roughness. This animation is a microscope on that fact. No matter how much we magnify, the path won’t straighten out. It keeps wrinkling at every scale. #BrownianMotion #StochasticProcesses #RandomWalks #ProbabilityTheory #Fractals #MathAnimation
1:00

The Curve That Forced Mathematicians To Invent A New Calculus There is a strange curve called Brownian path. It traces a continuous path without ever jumping, but it never settles into a well-defined direction at any point, so it’s continuous everywhere but differentiable nowhere. We’re used to the idea that if something is continuous, then if you zoom in far enough it should start to look like a straight line. A Brownian path refuses. Zoom in and you don’t reveal a hidden tangent. You reveal fresh roughness. This animation is a microscope on that fact. No matter how much we magnify, the path won’t straighten out. It keeps wrinkling at every scale. #BrownianMotion #StochasticProcesses #RandomWalks #ProbabilityTheory #Fractals #MathAnimation

Mathelirium

31,688 Aufrufe • vor 6 Monaten

Alan Oppenheim, MIT professor: "Jump Trading pays quants $650K to pull one real signal out of market noise. I wrote the textbook on that exact math and taught it for free." this free MIT lecture is the math the whole "quant edge" is built on, and almost nobody selling you a course will name it. the man teaching it wrote the signal-processing textbook every engineer on earth learned from, and he put the lectures online for nothing. here is the idea. a price chart is a signal buried in noise. most of what you see is random static, and hidden inside is a faint, repeating pattern with real mathematics for pulling it out. that is not a metaphor. jim simons was a codebreaker who did exactly this to enemy transmissions, then pointed the same math at the market and made a hundred billion dollars. none of it is new. oppenheim taught this in 1987, the math is older, and every lecture is free. same story i told in the article above. the edge was never a secret formula. it was knowing a market is a noisy signal, and having the tools to read it. here is what the gurus skip. the math to separate signal from noise is free and everyone has it. the edge is cleaner data than the next desk, and the discipline to trust a faint signal across thousands of trades instead of chasing the static. the filter is free. the patience to use it is not.
29:44

Alan Oppenheim, MIT professor: "Jump Trading pays quants $650K to pull one real signal out of market noise. I wrote the textbook on that exact math and taught it for free." this free MIT lecture is the math the whole "quant edge" is built on, and almost nobody selling you a course will name it. the man teaching it wrote the signal-processing textbook every engineer on earth learned from, and he put the lectures online for nothing. here is the idea. a price chart is a signal buried in noise. most of what you see is random static, and hidden inside is a faint, repeating pattern with real mathematics for pulling it out. that is not a metaphor. jim simons was a codebreaker who did exactly this to enemy transmissions, then pointed the same math at the market and made a hundred billion dollars. none of it is new. oppenheim taught this in 1987, the math is older, and every lecture is free. same story i told in the article above. the edge was never a secret formula. it was knowing a market is a noisy signal, and having the tools to read it. here is what the gurus skip. the math to separate signal from noise is free and everyone has it. the edge is cleaner data than the next desk, and the discipline to trust a faint signal across thousands of trades instead of chasing the static. the filter is free. the patience to use it is not.

Rossst.03

281,905 Aufrufe • vor 1 Monat

In 2013 an MIT mathematician wrote one minus sign on a chalkboard and explained the exact trick hedge funds farm billions from in 2026. MIT filmed it. it has been free for 13 years. it has fewer views than a phone unboxing. his point: volatility is not a risk you survive. it is a crop you harvest. every "volatility harvesting" thread charging you $500 in 2026 is reselling the one line Choongbum Lee derives for nothing. he is a mathematician, not a guru. no thumbnail, no promise. just the term that makes a shaking portfolio beat a calm one. skip to where he writes d(log S). out falls minus one-half sigma squared. that tiny term is the "$12.50 from nowhere" in the post above. no black box. no signal. one piece of chalk. a quant I know made his whole desk watch minute 40 before they could touch the rebalancer. you are 13 years late. it is still free.
1:17:34

In 2013 an MIT mathematician wrote one minus sign on a chalkboard and explained the exact trick hedge funds farm billions from in 2026. MIT filmed it. it has been free for 13 years. it has fewer views than a phone unboxing. his point: volatility is not a risk you survive. it is a crop you harvest. every "volatility harvesting" thread charging you $500 in 2026 is reselling the one line Choongbum Lee derives for nothing. he is a mathematician, not a guru. no thumbnail, no promise. just the term that makes a shaking portfolio beat a calm one. skip to where he writes d(log S). out falls minus one-half sigma squared. that tiny term is the "$12.50 from nowhere" in the post above. no black box. no signal. one piece of chalk. a quant I know made his whole desk watch minute 40 before they could touch the rebalancer. you are 13 years late. it is still free.

Voltex

32,751 Aufrufe • vor 1 Monat

Millions have watched an MIT professor accidentally destroy the American sports betting industry in a free 12-lecture undergraduate poker course. MIT charges $85,000 a year to sit in that classroom. He posted every lecture on OpenCourseWare for nothing. Almost no one who has ever placed a DraftKings same-game parlay has finished all twelve. His name is Kevin Desmond. He is an MIT alum, a professional poker player, and the instructor of 15.S50 Poker Theory and Analytics, which MIT gave undergraduates college credit for taking during January of 2015. The 43-minute clip in this video is one lecture from that course, filmed at MIT that same month. The chart on the screen behind him looks like a poker graph. It is the exact math that decides whether a Wall Street quant clears $500,000 a year, whether a FanDuel bettor loses their rent money on a Sunday afternoon, and whether a Silicon Valley founder can walk into a term sheet negotiation without being taken apart in the room. Desmond compresses the mathematical foundation of every adversarial decision on earth into five ideas. Ranges. You never know your opponent's exact hand. You know a distribution of hands weighted by probability. Every FanDuel bettor picking a parlay on a hunch is playing without a range. Every retail trader guessing a competitor's next move is guessing blind. Pot odds. The equation that tells you when a call has positive expected value. Every VC term sheet and every insurance premium reduces to it. Every same-game parlay on DraftKings violates it in ways the app is legally allowed to hide from you. Expected value. Sum every outcome weighted by probability. Casinos are built on it. Poker pros live on it. Sports bettors violate it every time they chase a loss hoping for a hot Sunday. Game theory optimal. The Nash equilibrium of poker. The strategy no opponent can exploit no matter how well they read you. Quant funds pay $500,000 bonuses for one senior who can solve for it under pressure. Exploitative play. When to deviate from GTO to punish a specific mistake. What every senior desk on Wall Street does against retail order flow, every trading session, every day. Every quant fund on Wall Street runs a hiring pipeline that starts with this material. Every prop trading desk drills it into juniors before their first live session. The MIT professor who filmed the whole course posted it on OpenCourseWare for the price of an internet connection. "Every time you play a hand differently from the way you would have played it if you could see all your opponent's cards, they gain." That is David Sklansky's Fundamental Theorem of Poker. Desmond opens the course with it. It is also the exact statement of information asymmetry that every trading floor, casino, and DraftKings promo card on earth is built to exploit. The lectures are free on MIT OpenCourseWare. The problem sets are online. Every equation Desmond derives fits on one page. The math is free. The willingness to spend 43 minutes on one lecture before opening a sportsbook app, placing a parlay, or entering a negotiation is a much rarer commodity than the confidence to walk in without it.
43:06

Millions have watched an MIT professor accidentally destroy the American sports betting industry in a free 12-lecture undergraduate poker course. MIT charges $85,000 a year to sit in that classroom. He posted every lecture on OpenCourseWare for nothing. Almost no one who has ever placed a DraftKings same-game parlay has finished all twelve. His name is Kevin Desmond. He is an MIT alum, a professional poker player, and the instructor of 15.S50 Poker Theory and Analytics, which MIT gave undergraduates college credit for taking during January of 2015. The 43-minute clip in this video is one lecture from that course, filmed at MIT that same month. The chart on the screen behind him looks like a poker graph. It is the exact math that decides whether a Wall Street quant clears $500,000 a year, whether a FanDuel bettor loses their rent money on a Sunday afternoon, and whether a Silicon Valley founder can walk into a term sheet negotiation without being taken apart in the room. Desmond compresses the mathematical foundation of every adversarial decision on earth into five ideas. Ranges. You never know your opponent's exact hand. You know a distribution of hands weighted by probability. Every FanDuel bettor picking a parlay on a hunch is playing without a range. Every retail trader guessing a competitor's next move is guessing blind. Pot odds. The equation that tells you when a call has positive expected value. Every VC term sheet and every insurance premium reduces to it. Every same-game parlay on DraftKings violates it in ways the app is legally allowed to hide from you. Expected value. Sum every outcome weighted by probability. Casinos are built on it. Poker pros live on it. Sports bettors violate it every time they chase a loss hoping for a hot Sunday. Game theory optimal. The Nash equilibrium of poker. The strategy no opponent can exploit no matter how well they read you. Quant funds pay $500,000 bonuses for one senior who can solve for it under pressure. Exploitative play. When to deviate from GTO to punish a specific mistake. What every senior desk on Wall Street does against retail order flow, every trading session, every day. Every quant fund on Wall Street runs a hiring pipeline that starts with this material. Every prop trading desk drills it into juniors before their first live session. The MIT professor who filmed the whole course posted it on OpenCourseWare for the price of an internet connection. "Every time you play a hand differently from the way you would have played it if you could see all your opponent's cards, they gain." That is David Sklansky's Fundamental Theorem of Poker. Desmond opens the course with it. It is also the exact statement of information asymmetry that every trading floor, casino, and DraftKings promo card on earth is built to exploit. The lectures are free on MIT OpenCourseWare. The problem sets are online. Every equation Desmond derives fits on one page. The math is free. The willingness to spend 43 minutes on one lecture before opening a sportsbook app, placing a parlay, or entering a negotiation is a much rarer commodity than the confidence to walk in without it.

Lumen

57,583 Aufrufe • vor 14 Tagen

Once you’ve accepted that Brownian motion doesn’t have a classical dX/dt but does have this rigid quadratic variation, you’re ready for the first real Itô vs chain rule punch: take X_t = B_t and look at f(x) = x². In ordinary calculus you’d write d(B_t²) = 2 B_t dB_t and move on, because the only thing that survives is f′(x)·dx. Itô’s formula says that’s not enough for Brownian motion: the infinitely many tiny (dB_t)² terms don’t wash out, they add up to dt, so the correct identity is d(B_t²) = 2 B_t dB_t + dt (see insert formula). In the animation we fix one Brownian path, draw B_t² as the pale truth, then build the naive path that accumulates only the 2 B_t dB_t term and watch it drift away from B_t², while the corrected path that accumulates 2 B_t dB_t plus an extra linear drift t hugs B_t² almost perfectly. The whole point is visible on screen: the missing +t isn’t a technicality, it’s the deterministic shadow of quadratic variation correcting the chain rule so calculus survives contact with noise. #ItosFormula #BrownianMotion #StochasticCalculus #QuadraticVariation #RandomPaths #ProbabilityTheory
1:00

Once you’ve accepted that Brownian motion doesn’t have a classical dX/dt but does have this rigid quadratic variation, you’re ready for the first real Itô vs chain rule punch: take X_t = B_t and look at f(x) = x². In ordinary calculus you’d write d(B_t²) = 2 B_t dB_t and move on, because the only thing that survives is f′(x)·dx. Itô’s formula says that’s not enough for Brownian motion: the infinitely many tiny (dB_t)² terms don’t wash out, they add up to dt, so the correct identity is d(B_t²) = 2 B_t dB_t + dt (see insert formula). In the animation we fix one Brownian path, draw B_t² as the pale truth, then build the naive path that accumulates only the 2 B_t dB_t term and watch it drift away from B_t², while the corrected path that accumulates 2 B_t dB_t plus an extra linear drift t hugs B_t² almost perfectly. The whole point is visible on screen: the missing +t isn’t a technicality, it’s the deterministic shadow of quadratic variation correcting the chain rule so calculus survives contact with noise. #ItosFormula #BrownianMotion #StochasticCalculus #QuadraticVariation #RandomPaths #ProbabilityTheory

Mathelirium

31,380 Aufrufe • vor 8 Monaten

This man teaches at a community college in California. His salary: around $800,000 a year. The engineers who passed calculus because of him: $1,800,000 to start. He has more calculus students than Harvard, MIT, and Stanford combined. This is Professor Leonard's Calculus 2, Lecture 6.2. Free on YouTube. Professor Leonard has taught calculus on YouTube for over a decade. His channel has millions of subscribers across 150 countries. Every major university has students who watch him the night before their exam. Then the concept. An inverse function is a machine that undoes another machine. If a function takes 2 and gives you 8, the inverse takes 8 and gives you back 2. Finding an inverse means switching every x and y in the equation and solving for y again. The graph flips across the line y = x like a mirror. Then the problem. Sometimes it is easy to find the inverse. Sometimes it is impossible to write it explicitly. A function like 3π sin x + sin x cannot be solved for x with algebra. You have to think. What angle makes the whole thing equal to 1? You work backwards through the unit circle until the answer appears. Then the shortcut. If you want the derivative of an inverse at a point, you do not need the inverse itself. You only need the derivative of the original function. The formula: the derivative of the inverse at a point equals 1 divided by the derivative of the original function evaluated at the switched point. The inverse flips the coordinates, so you flip where you plug in. Watch the moment he shows why G prime of 8 equals 1/12 without ever writing the inverse function. Every engineering student memorizes the derivative rules. Professor Leonard's lecture is the one that shows why the inverse derivative formula is just those same rules run backwards. A software engineer at a semiconductor company in Austin said Professor Leonard's channel is the reason she passed Calculus 2 on her second attempt. She graduated, joined the company, and now makes $165,000 a year. Bookmark this and watch later - after this lecture every inverse problem on your exam will feel like a question you already answered.
43:46

This man teaches at a community college in California. His salary: around $800,000 a year. The engineers who passed calculus because of him: $1,800,000 to start. He has more calculus students than Harvard, MIT, and Stanford combined. This is Professor Leonard's Calculus 2, Lecture 6.2. Free on YouTube. Professor Leonard has taught calculus on YouTube for over a decade. His channel has millions of subscribers across 150 countries. Every major university has students who watch him the night before their exam. Then the concept. An inverse function is a machine that undoes another machine. If a function takes 2 and gives you 8, the inverse takes 8 and gives you back 2. Finding an inverse means switching every x and y in the equation and solving for y again. The graph flips across the line y = x like a mirror. Then the problem. Sometimes it is easy to find the inverse. Sometimes it is impossible to write it explicitly. A function like 3π sin x + sin x cannot be solved for x with algebra. You have to think. What angle makes the whole thing equal to 1? You work backwards through the unit circle until the answer appears. Then the shortcut. If you want the derivative of an inverse at a point, you do not need the inverse itself. You only need the derivative of the original function. The formula: the derivative of the inverse at a point equals 1 divided by the derivative of the original function evaluated at the switched point. The inverse flips the coordinates, so you flip where you plug in. Watch the moment he shows why G prime of 8 equals 1/12 without ever writing the inverse function. Every engineering student memorizes the derivative rules. Professor Leonard's lecture is the one that shows why the inverse derivative formula is just those same rules run backwards. A software engineer at a semiconductor company in Austin said Professor Leonard's channel is the reason she passed Calculus 2 on her second attempt. She graduated, joined the company, and now makes $165,000 a year. Bookmark this and watch later - after this lecture every inverse problem on your exam will feel like a question you already answered.

Lupen

528,432 Aufrufe • vor 1 Tag

MIT filmed a mathematician proving the theorem behind every equation Maxwell ever wrote. Most engineers use it daily without understanding why any surface gives the same answer - and the ones who can explain it earn $350K at aerospace firms. Stokes' theorem is the reason Maxwell could write four equations instead of forty. Every result in electromagnetism, fluid mechanics, and differential geometry sits on top of it. Understanding it changes how you see every circulation and flux problem in physics. This is Denis Auroux. MIT, 18.02, Multivariable Calculus, Fall 2007. Lecture 31. It covers Stokes' theorem - the result that unifies line integrals, surface integrals, and curl into one statement. He opens with one confession. Then the statement. The work done by a vector field along any closed curve equals the flux of its curl through any surface bounded by that curve. Not a specific surface. Any surface you want. You pick whichever one makes the calculation easier. Then Green's theorem disappears. The result from the plane that converts line integrals to double integrals turns out to be Stokes' theorem in disguise. Same formula, restricted to a flat surface in the xy-plane. What looked like two separate theorems is one. Then orientation. Walk along the curve with the surface to your left and the normal vector points up. This rule is not chosen for elegance. It is the price of having a consistent coordinate system in three dimensions. Then the proof. Cut your surface into tiny flat tiles. Apply Green's theorem to each one. Sum everything up. All interior boundary edges cancel. Only the outer edge survives. The sum of tiny Green's theorems gives you Stokes' theorem for any surface. Watch the moment he computes the same line integral two ways. First directly - answer is pi. Then through a paraboloid instead of the obvious flat disk. Longer calculation, stranger setup, same answer. The theorem is not an abstraction. You can watch it work. A fluid dynamics engineer I know kept this lecture open during their first week modeling atmospheric circulation. Said it was the first time Stokes felt like a tool rather than a formula to verify on exams. Free on YouTube, MIT OpenCourseWare, 18.02. bookmark this and watch later - after this lecture every circulation, every flux, and every curl you encounter will feel like three views of the same object
47:56

MIT filmed a mathematician proving the theorem behind every equation Maxwell ever wrote. Most engineers use it daily without understanding why any surface gives the same answer - and the ones who can explain it earn $350K at aerospace firms. Stokes' theorem is the reason Maxwell could write four equations instead of forty. Every result in electromagnetism, fluid mechanics, and differential geometry sits on top of it. Understanding it changes how you see every circulation and flux problem in physics. This is Denis Auroux. MIT, 18.02, Multivariable Calculus, Fall 2007. Lecture 31. It covers Stokes' theorem - the result that unifies line integrals, surface integrals, and curl into one statement. He opens with one confession. Then the statement. The work done by a vector field along any closed curve equals the flux of its curl through any surface bounded by that curve. Not a specific surface. Any surface you want. You pick whichever one makes the calculation easier. Then Green's theorem disappears. The result from the plane that converts line integrals to double integrals turns out to be Stokes' theorem in disguise. Same formula, restricted to a flat surface in the xy-plane. What looked like two separate theorems is one. Then orientation. Walk along the curve with the surface to your left and the normal vector points up. This rule is not chosen for elegance. It is the price of having a consistent coordinate system in three dimensions. Then the proof. Cut your surface into tiny flat tiles. Apply Green's theorem to each one. Sum everything up. All interior boundary edges cancel. Only the outer edge survives. The sum of tiny Green's theorems gives you Stokes' theorem for any surface. Watch the moment he computes the same line integral two ways. First directly - answer is pi. Then through a paraboloid instead of the obvious flat disk. Longer calculation, stranger setup, same answer. The theorem is not an abstraction. You can watch it work. A fluid dynamics engineer I know kept this lecture open during their first week modeling atmospheric circulation. Said it was the first time Stokes felt like a tool rather than a formula to verify on exams. Free on YouTube, MIT OpenCourseWare, 18.02. bookmark this and watch later - after this lecture every circulation, every flux, and every curl you encounter will feel like three views of the same object

Zyphor

10,999 Aufrufe • vor 5 Tagen

In 2013 a Wall Street risk boss walked up to an MIT chalkboard and gave away the exact math desks charge $400,000 a year to run. MIT filmed it. it has been free for 13 years. 7.9 million people pressed play and almost none of them changed a thing. his point: the Wall Street edge was never a secret. it is a first lecture. options, risk, how a payout is actually built. every prediction-market trader hunting an edge in 2026 is chasing what Jake Xia put on a board in hour one of a free class. he ran risk for a living, not a channel. the video outdrew most blockbusters and still moved almost no one, because watching is not doing. skip to where he prices risk. he takes the thing quant threads sell for $500 and builds it from zero, slow enough for a beginner to follow. no paywall. no NDA. one board and a marker. a quant I know sends this link to every junior who whines the math is too hard. you are 13 years late. it is still free.
1:00:07

In 2013 a Wall Street risk boss walked up to an MIT chalkboard and gave away the exact math desks charge $400,000 a year to run. MIT filmed it. it has been free for 13 years. 7.9 million people pressed play and almost none of them changed a thing. his point: the Wall Street edge was never a secret. it is a first lecture. options, risk, how a payout is actually built. every prediction-market trader hunting an edge in 2026 is chasing what Jake Xia put on a board in hour one of a free class. he ran risk for a living, not a channel. the video outdrew most blockbusters and still moved almost no one, because watching is not doing. skip to where he prices risk. he takes the thing quant threads sell for $500 and builds it from zero, slow enough for a beginner to follow. no paywall. no NDA. one board and a marker. a quant I know sends this link to every junior who whines the math is too hard. you are 13 years late. it is still free.

Loran

151,219 Aufrufe • vor 1 Monat

An Indian physicist put the entire secret behind Renaissance's $100 billion on one chalkboard: a coin that barely beats 50/50. he is V. Balakrishnan, and his lecture went viral years after he recorded it. it has been free the whole time. almost nobody who trades has watched it. lecture one is the humblest thing in all of finance. a coin that lands your way a little more than half the time. that is the entire edge. that is all Renaissance ever had. the post above is the one equation for what to do with that coin. not what to bet. how much. bet too little and you die of old age. bet too much and one flip ends you. that same coin is every trade, every hand, every prediction market. a tiny edge, flipped a million times. the whole machine fits on his board, and he charged nothing for it. no slides. no notes. one coin and a stick of chalk. a quant I know says this old Indian lecture taught him probability better than his entire degree. the coin is free. the equation is free. the discipline to size it is the only thing left to earn.
1:01:55

An Indian physicist put the entire secret behind Renaissance's $100 billion on one chalkboard: a coin that barely beats 50/50. he is V. Balakrishnan, and his lecture went viral years after he recorded it. it has been free the whole time. almost nobody who trades has watched it. lecture one is the humblest thing in all of finance. a coin that lands your way a little more than half the time. that is the entire edge. that is all Renaissance ever had. the post above is the one equation for what to do with that coin. not what to bet. how much. bet too little and you die of old age. bet too much and one flip ends you. that same coin is every trade, every hand, every prediction market. a tiny edge, flipped a million times. the whole machine fits on his board, and he charged nothing for it. no slides. no notes. one coin and a stick of chalk. a quant I know says this old Indian lecture taught him probability better than his entire degree. the coin is free. the equation is free. the discipline to size it is the only thing left to earn.

Rossst.03

159,548 Aufrufe • vor 26 Tagen

IN 1986 MIT FILMED THE LECTURE WHERE CODE STOPPED BEING CODE AND BECAME DATA 43 minutes from Gerald Sussman, in the most legendary intro programming course ever recorded. -> The idea that lands: to a program, your code is just data it can read and rewrite. He builds a program that does real calculus -- not by crunching numbers, but by reading the math as a list and transforming it piece by piece. Then he shows the trick hiding under all of it: in this language, code and data are the same material. Forty years later that is exactly what an AI coding agent is -- a program that reads your code as data, rewrites it, and hands it back. Sussman drew the whole idea on a blackboard in 1986. Writing code was never the deepest skill -- understanding that code itself is data a program can manipulate is. This is where you learn it. Most people think AI writing code is brand new. The ones who watch this saw the blueprint 40 years ago. Bookmark & Watch it. This one's a legend ↓
43:30

IN 1986 MIT FILMED THE LECTURE WHERE CODE STOPPED BEING CODE AND BECAME DATA 43 minutes from Gerald Sussman, in the most legendary intro programming course ever recorded. -> The idea that lands: to a program, your code is just data it can read and rewrite. He builds a program that does real calculus -- not by crunching numbers, but by reading the math as a list and transforming it piece by piece. Then he shows the trick hiding under all of it: in this language, code and data are the same material. Forty years later that is exactly what an AI coding agent is -- a program that reads your code as data, rewrites it, and hands it back. Sussman drew the whole idea on a blackboard in 1986. Writing code was never the deepest skill -- understanding that code itself is data a program can manipulate is. This is where you learn it. Most people think AI writing code is brand new. The ones who watch this saw the blueprint 40 years ago. Bookmark & Watch it. This one's a legend ↓

slash1s

22,107 Aufrufe • vor 2 Monaten

Prabha Sharma, one of the sharpest women mathematicians India has produced: "Optiver puts every applicant through a probability gauntlet, then pays the ones who pass $500,000 a year for one skill: counting the ways an outcome can happen before anyone else does. that is exactly where this professor opens her course, and it is free" this free lecture is the foundation under every prediction market and every price you have ever seen. strip away the models and the AI and one question is left. in how many ways can this happen, and how many of them pay you. get that count right and the probability falls out on its own. get it wrong and every model built on top is wrong with it. Sharma builds it from zero, on a blackboard, in notation a schoolkid can follow. permutations, combinations, the n choose r sitting under every set of odds. Wall Street rents this out as a $500,000 edge. an IIT professor hands you the whole foundation for free.
51:57

Prabha Sharma, one of the sharpest women mathematicians India has produced: "Optiver puts every applicant through a probability gauntlet, then pays the ones who pass $500,000 a year for one skill: counting the ways an outcome can happen before anyone else does. that is exactly where this professor opens her course, and it is free" this free lecture is the foundation under every prediction market and every price you have ever seen. strip away the models and the AI and one question is left. in how many ways can this happen, and how many of them pay you. get that count right and the probability falls out on its own. get it wrong and every model built on top is wrong with it. Sharma builds it from zero, on a blackboard, in notation a schoolkid can follow. permutations, combinations, the n choose r sitting under every set of odds. Wall Street rents this out as a $500,000 edge. an IIT professor hands you the whole foundation for free.

Rossst.03

60,309 Aufrufe • vor 8 Tagen

Mrityunjoy Chakraborty, IIT Kharagpur professor: "Jane Street pays a 22 year old $500,000 a year for one skill: turning uncertainty into a number you can bet on. that number is the first thing this professor teaches, and it is free" this free lecture is the foundation every quant bot and hedge fund model is secretly built on. before the AI, before the strategy, there is one question. what is the chance this happens, exactly. get that number right and every bet after it is just math. get it wrong and no model, no bot, no amount of compute can save you. Chakraborty builds probability from the axioms up, on a blackboard, in notation a 15 year old can follow. Wall Street sells this as a $500,000 edge. an IIT professor hands you the source code for nothing.
58:21

Mrityunjoy Chakraborty, IIT Kharagpur professor: "Jane Street pays a 22 year old $500,000 a year for one skill: turning uncertainty into a number you can bet on. that number is the first thing this professor teaches, and it is free" this free lecture is the foundation every quant bot and hedge fund model is secretly built on. before the AI, before the strategy, there is one question. what is the chance this happens, exactly. get that number right and every bet after it is just math. get it wrong and no model, no bot, no amount of compute can save you. Chakraborty builds probability from the axioms up, on a blackboard, in notation a 15 year old can follow. Wall Street sells this as a $500,000 edge. an IIT professor hands you the source code for nothing.

Rossst.03

697,469 Aufrufe • vor 11 Tagen

An MIT professor solved the equation Wall Street pays $500K for, in a lecture about how to find the top of a hill. MIT put the whole thing online. it has been free for years. almost nobody finishes it. his point: finding the best possible answer to almost anything is one move. write down what you want to grow, take the derivative, and find the exact spot where it stops climbing. that spot is the answer. he is Denis Auroux, and his 18.02 lectures have taught more people calculus than most universities have students. now the equation from the post above, the one that tells you exactly how much to bet. it is not a secret formula. it is this. the growth of your money is a curve. bet too little and you are low on the hill. bet too much and you have gone over the top and started falling. Kelly's number is just the peak. no magic. no fund. one hill, and the single point where it stops going up. a quant I know rewatches this exact lecture before he sizes a single position. you were handed the formula and told to trust it. Auroux shows you where it actually comes from, for free, and it has been sitting there the whole time.
38:17

An MIT professor solved the equation Wall Street pays $500K for, in a lecture about how to find the top of a hill. MIT put the whole thing online. it has been free for years. almost nobody finishes it. his point: finding the best possible answer to almost anything is one move. write down what you want to grow, take the derivative, and find the exact spot where it stops climbing. that spot is the answer. he is Denis Auroux, and his 18.02 lectures have taught more people calculus than most universities have students. now the equation from the post above, the one that tells you exactly how much to bet. it is not a secret formula. it is this. the growth of your money is a curve. bet too little and you are low on the hill. bet too much and you have gone over the top and started falling. Kelly's number is just the peak. no magic. no fund. one hill, and the single point where it stops going up. a quant I know rewatches this exact lecture before he sizes a single position. you were handed the formula and told to trust it. Auroux shows you where it actually comes from, for free, and it has been sitting there the whole time.

Loran

95,357 Aufrufe • vor 1 Monat

IN 1999 MIT FILMED A MATH LECTURE THAT QUIETLY BECAME THE FOUNDATION OF EVERY AI MODEL YOU'VE EVER USED AND ALMOST NO ONE WAS TAUGHT TO SEE IT THAT WAY 39 minutes from Gilbert Strang, who taught this at MIT for over 60 years -- the linear algebra course an entire generation of engineers and data scientists grew up on. -> The shift it creates: you stop seeing matrices as boring grids of numbers and start seeing them as the language of space, data, and motion itself. School drilled you to crunch matrices by hand and never told you why. Strang shows you what they actually mean. Every neural net, every embedding, every model you prompt is linear algebra running underneath. The math you skipped is the engine of the thing you use all day. Memorizing the steps was never the skill -> seeing what the numbers do is. This is where it finally clicks. Most people fear linear algebra and move on. The ones who watched this see straight into how AI actually works. Bookmark & Watch it today, this one's a legend ↓
39:31

IN 1999 MIT FILMED A MATH LECTURE THAT QUIETLY BECAME THE FOUNDATION OF EVERY AI MODEL YOU'VE EVER USED AND ALMOST NO ONE WAS TAUGHT TO SEE IT THAT WAY 39 minutes from Gilbert Strang, who taught this at MIT for over 60 years -- the linear algebra course an entire generation of engineers and data scientists grew up on. -> The shift it creates: you stop seeing matrices as boring grids of numbers and start seeing them as the language of space, data, and motion itself. School drilled you to crunch matrices by hand and never told you why. Strang shows you what they actually mean. Every neural net, every embedding, every model you prompt is linear algebra running underneath. The math you skipped is the engine of the thing you use all day. Memorizing the steps was never the skill -> seeing what the numbers do is. This is where it finally clicks. Most people fear linear algebra and move on. The ones who watched this see straight into how AI actually works. Bookmark & Watch it today, this one's a legend ↓

slash1s

433,431 Aufrufe • vor 2 Monaten