# Reading Track. Stochastic Differential Equations A sequence of five notes that walk from a coin-flip random walk to a differential equation with noise in it, and the modified calculus that noise forces on you. They were synthesized from MATH310 handwritten notes (F21, S22, F24), Juan M. R. Parrondo's video lectures, and Lawrence Evans's *An Introduction to Stochastic Differential Equations*. Read them in order. Each one ends with a short "related notes" list, and the hub note links every section. > [!info]- Who this is for > Anyone whose project has randomness in time. Prices and futures, a decision that accumulates evidence toward a threshold, agents jostling in a crowd, a population with demographic noise. If your model has the form "deterministic drift plus something random happening every instant," this is the mathematics under it. Prerequisites are calculus, a little probability (mean, variance, the normal distribution), and the willingness to accept that $(dW)^2 = dt$ before it is fully explained. ## The sequence 1. [[Random-Walks|Random Walks. From Discrete Steps to Continuous Processes]]. The symmetric walk, its mean and variance, the binomial distribution, and the two roads to the continuum. The central limit theorem gives a normal distribution, and the master equation with a Taylor expansion gives the diffusion equation. The key line is the scaling $\Delta y \propto \sqrt{\Delta t}$, which is why nothing after this behaves like ordinary calculus. 2. [[Wiener-Process|The Wiener Process]]. The continuum limit as an object in its own right. Definition, autocorrelation, white noise as its formal derivative (built on an $L^2$ basis), path properties (continuous everywhere, differentiable nowhere), the martingale property, and quadratic variation. 3. [[Ito-Formula|Itô's Formula. The Modified Chain Rule]]. Why the chain rule fails when the integrand is the Wiener process itself, the Itô versus Stratonovich choice of evaluation point, the missing term, and the general formula. Examples include powers of Brownian motion, the exponential martingale, noisy decay, and geometric Brownian motion, which is the stock-price model. 4. [[SDE-Fundamentals|Stochastic Differential Equations. Fundamentals]]. The first SDE, noise-perturbed exponential decay, solved completely. What the stochastic integral means, physical interpretations, and the connection to partial differential equations (Fokker-Planck). 5. [[Notes-on-Stochastic-Differential-Equations|Notes on Stochastic Differential Equations]]. The hub. A condensed pass through all of the above with section links into the four notes. Read this last as a review, or first if you want the map before the territory. A companion from the lecture sequence. [[Dirac-Delta-Distribution|The Dirac Delta as a Distribution]] is the object white noise is built from, and it is the same delta from the Day 5 and Day 6 lectures. ## Where it touches the course - The random walk in note 1 is the same object as the Yahtzee and dot-in-a-square activities, seen as a process in time. - The continuum limit in note 1 is the same move as the finite differences of Day 4, run in the opposite direction. - White noise in note 2 needs the delta from Day 5 and the Fourier ideas from Days 7 through 10. - Geometric Brownian motion in note 3 is the bridge to the [[MATH310F26 - Reading Track - Mathematical Finance|Mathematical Finance track]]. The binomial tree there is the random walk here, in log price. <!--