# Notes on Stochastic Differential Equations *A synthesis of handwritten notes from MATH310 (F21/S22/F24), [video lectures](https://www.youtube.com/playlist?list=PLfzyv9XksyEfntM7sGdNDLp8kw06I5trG) by [Juan MR Parrondo](https://en.wikipedia.org/wiki/J._M._R._Parrondo), and parts of, [An Introduction to Stochastic Differential Equations](https://www.cmor-faculty.rice.edu/~cox/stoch/SDE.course.pdf) Version 1.2, by [Lawrence C. Evans](https://en.wikipedia.org/wiki/Lawrence_C._Evans) (2004)* --- ## Overview and Motivation These notes trace a path from discrete [random walks](https://en.wikipedia.org/wiki/Random_walk) to continuous [stochastic processes](https://en.wikipedia.org/wiki/Stochastic_process), ultimately arriving at [stochastic differential equations](https://en.wikipedia.org/wiki/Stochastic_differential_equation) and the modified calculus needed to work with them. The journey reveals how noise fundamentally changes the rules of calculus and why the [Itô correction](https://en.wikipedia.org/wiki/It%C3%B4%27s_lemma) is essential for properly modeling real-world phenomena. ### Document Structure - [[Random-Walks]] - Develops the symmetric random walk from first principles through the binomial distribution to the continuum limit. Shows how the scaling $\Delta y \propto \sqrt{\Delta t}$ naturally emerges and connects to diffusion. - [[Wiener-Process]] - Constructs the Wiener process as the continuum limit of random walks and explores its mathematical properties. Develops the autocorrelation structure and introduces white noise as the formal derivative, including the L² basis representation. - [[Ito-Formula]] - Explains why classical calculus fails for Brownian motion through the key example where "y returns the Wiener process itself." Derives Itô's lemma showing how $(dW)^2 = dt$ necessitates the extra second-derivative term. - [[SDE-Fundamentals]] - Introduces stochastic differential equations starting from noise-perturbed exponential decay. Works through the complete solution of the first-order linear SDE using Itô's lemma and connects to physical systems via the Langevin equation. --- ## 1. From Random Walks to Continuous Processes *See [[Random-Walks#1. Definition and Basic Structure]] for full development* ### The Discrete Foundation We begin with the [symmetric random walk](https://en.wikipedia.org/wiki/Random_walk#Symmetric_random_walk) - a particle taking steps of size $\pm\Delta y$ with equal probability. After $n = t/\Delta t$ steps: - **Mean position**: $\langle y(t) \rangle = 0$ - **Variance**: $\sigma_y^2 = n(\Delta y)^2 = \frac{t(\Delta y)^2}{\Delta t}$ The variance grows linearly with time - a signature of diffusive behavior. ### The Continuum Limit *See [[Random-Walks#4. The Continuum Limit]] for detailed derivation* To obtain a meaningful limit as $\Delta t \to 0$, we must scale the step size: $\Delta y = \sqrt{D \Delta t}$, where $D$ is the [diffusion coefficient](https://en.wikipedia.org/wiki/Mass_diffusivity). This yields: $\sigma_y^2 = Dt$ In this limit, the random walk converges to the **[Wiener process](https://en.wikipedia.org/wiki/Wiener_process)** $W(t)$ - the mathematical model of [Brownian motion](https://en.wikipedia.org/wiki/Brownian_motion). > [!info] Key Insight > The scaling $\Delta y \propto \sqrt{\Delta t}$ is crucial. It's why Brownian paths are continuous but nowhere differentiable - the particle moves infinitely fast at infinitesimal scales. --- ## 2. The Wiener Process and White Noise *Full treatment in [[Wiener-Process#3. Definition of the Wiener Process]]* ### Properties of the Wiener Process The Wiener process $W(t)$ is characterized by: 1. $W(0) = 0$ 2. Independent increments 3. $W(t) - W(s) \sim \mathcal{N}(0, |t-s|)$ for $t > s$ 4. Continuous paths (almost surely) ### The Autocorrelation Structure *Detailed derivation in [[Wiener-Process#Autocorrelation Structure]]* For the Wiener process: $E[W(t_1)W(t_2)] = \sigma^2 \min\{t_1, t_2\}$ This leads to a profound result when we consider the formal derivative $\varepsilon(t) = dW/dt$: $E[\varepsilon(t_1)\varepsilon(t_2)] = \sigma^2 \delta(t_1 - t_2)$ [White noise](https://en.wikipedia.org/wiki/White_noise) is perfectly uncorrelated - a mathematical idealization that captures maximal randomness. ### L² Basis Representation *See [[Wiener-Process#4. L² Basis Construction]] for complete construction* We can construct white noise through orthogonal expansions: $\varepsilon(t) = \sum_{n=1}^{\infty} A_n \phi_n(t)$ where $\{\phi_n\}$ is an orthonormal basis and $A_n$ are independent Gaussian random variables with $E[A_n] = 0$ and $E[A_n^2] = 1$. --- ## 3. Stochastic Differential Equations *Introduction and examples in [[SDE-Fundamentals#1. Introduction and Motivation]]* ### The Fundamental Problem Consider adding noise to exponential decay: $\frac{dy}{dt} = -(\alpha + \xi(t))y$ where $\xi(t)$ is white noise. Since white noise isn't a proper function, we rewrite this using differentials: $dy = -\alpha y \, dt - y \, dW$ ### The Stochastic Integral Challenge *The key example explored in [[Ito-Formula#1. Why the Classical Chain Rule Fails]] and [[SDE-Fundamentals#3. The Stochastic Integral]]* When $y$ returns the Wiener process itself, we must compute $\int_0^t W(s) \, dW(s)$. The result depends on how we interpret the integral: - **[Itô](https://en.wikipedia.org/wiki/It%C3%B4_calculus)** (left endpoint): $E[\int W \, dW] = 0$ - **[Stratonovich](https://en.wikipedia.org/wiki/Stratonovich_integral)** (midpoint): $E[\int W \, dW] = \frac{\sigma^2 t}{2}$ Classical calculus suggests $\frac{W(t)^2}{2}$, which agrees with Stratonovich. But for discrete-time limits, Itô is required. --- ## 4. Itô's Formula: The Modified Chain Rule *Complete derivation in [[Ito-Formula#1. Why the Classical Chain Rule Fails]]* ### Why Classical Calculus Fails The key insight: $(dW)^2 = dt$ is not negligible! For Brownian motion: $W(t)\Delta W = \frac{1}{2}\Delta(W^2) - \frac{1}{2}(\Delta W)^2$ In the limit: $W \, dW = \frac{d(W^2)}{2} - \frac{dt}{2}$ The second-order differential of the Wiener process is first-order in time. ### [Itô's Formula](https://en.wikipedia.org/wiki/It%C3%B4%27s_lemma) *See [[Ito-Formula#3. Itô's Formula: The General Rule]] for full statement and [[Ito-Formula#Heuristic Derivation]] for intuition* For $Y = u(X,t)$ where $dX = b \, dt + \sigma \, dW$: $dY = \left(\frac{\partial u}{\partial t} + b\frac{\partial u}{\partial x} + \frac{1}{2}\sigma^2\frac{\partial^2 u}{\partial x^2}\right)dt + \sigma\frac{\partial u}{\partial x}dW$ The extra term $\frac{1}{2}\sigma^2\frac{\partial^2 u}{\partial x^2}dt$ is the **Itô correction**. --- ## 5. Solving the Linear SDE *Complete solution in [[SDE-Fundamentals#6. Solving the First-Order Linear SDE]]* ### The Problem $\frac{dy}{dt} = -(\alpha + \xi(t))y, \quad y(0) = y_0$ ### Solution via [Itô's Lemma](https://en.wikipedia.org/wiki/It%C3%B4%27s_lemma) *Step-by-step derivation in [[SDE-Fundamentals#6. Solving the First-Order Linear SDE]] and [[Ito-Formula#4. Key Examples]]* Let $u = \ln(y)$. Applying Itô's formula: $du = -\alpha dt - dW - \frac{\sigma^2}{2}dt = -\left(\alpha + \frac{\sigma^2}{2}\right)dt - dW$ Integrating: $y(t) = y_0 \exp\left[-\left(\alpha + \frac{\sigma^2}{2}\right)t - W(t)\right]$ ### Key Insights 1. The Itô correction appears as $\sigma^2/2$ in the drift 2. Despite the noise, $E[y(t)] = y_0 e^{-\alpha t}$ - decay is preserved on average 3. In finance, this becomes geometric Brownian motion: $S(t) = S_0 \exp[\sigma W(t) + (\mu - \sigma^2/2)t]$ --- ## 6. Physical Context *Historical development in [[Wiener-Process#1. Historical Development]] and [[SDE-Fundamentals#5. Physical Interpretations]]* ### Einstein's Brownian Motion The [Langevin equation](https://en.wikipedia.org/wiki/Langevin_equation) describes a particle subject to friction and thermal noise: $m\ddot{x} = -\gamma \dot{x} - \nabla V(x) + \sqrt{2\gamma k_B T} \xi(t)$ The fluctuation-dissipation theorem relates noise intensity to temperature: $\langle \eta(t)\eta(t') \rangle = 2\gamma k_B T \delta(t-t')$ ### Connection to PDEs *See [[SDE-Fundamentals#7. Connection to Partial Differential Equations]] for derivation* The [Fokker-Planck equation](https://en.wikipedia.org/wiki/Fokker%E2%80%93Planck_equation) describes the evolution of the probability density: $\frac{\partial p}{\partial t} = -\frac{\partial}{\partial x}[f(x)p] + \frac{1}{2}\frac{\partial^2}{\partial x^2}[g(x)^2 p]$ This bridges between individual stochastic trajectories and ensemble behavior. --- ## Key Takeaways 1. **Random walks converge to Brownian motion** with the crucial scaling $\Delta y \propto \sqrt{\Delta t}$ 2. **White noise** is the formal derivative of the Wiener process - infinitely rough but mathematically tractable 3. **Stochastic integrals** depend on interpretation (Itô vs Stratonovich) 4. **Itô's formula** modifies the chain rule: $(dW)^2 = dt$ changes everything 5. **The Itô correction** $\sigma^2/2$ appears whenever we transform stochastic processes 6. **Physical systems** naturally lead to SDEs through thermal fluctuations --- ## Cross-References - [[Random-Walks]] - Detailed development of the discrete foundation - [[Wiener-Process]] - Mathematical properties and construction - [[Ito-Formula]] - The modified chain rule and applications - [[SDE-Fundamentals]] - Introduction and first-order linear SDE --- ## Video Lectures ### Parrondo's Six-Part Series - [Part 1: Overview and Motivation](https://youtu.be/vcGpD7nZ3UM) - [Part 2: The Wiener Process](https://youtu.be/vRsfjcXBGuI) - [Part 3: Stochastic Integrals](https://youtu.be/9zfw_CoPYNE) - [Part 4: Itô Calculus](https://youtu.be/h1eNpKDOa2c) - [Part 5: Applications](https://youtu.be/7J82tcLynaU) --- ## References ### Course Materials - MATH310 F21 Handwritten Notes: Stochastic Differential Equations - MATH310 F24: Random Walks lecture notes - Evans, L.C. *An Introduction to Stochastic Differential Equations* ### Historical Sources - [Einstein, A.](https://en.wikipedia.org/wiki/Albert_Einstein) (1905). "Über die von der molekularkinetischen Theorie der Wärme geforderte Bewegung von in ruhenden Flüssigkeiten suspendierten Teilchen" - [Itô, Kiyosi](https://en.wikipedia.org/wiki/Kiyosi_It%C3%B4) (1944). "Stochastic Integral" - [Bachelier, L.](https://en.wikipedia.org/wiki/Louis_Bachelier) (1900). "Théorie de la spéculation"