Key Concepts in Stochastic Differential Equations

Why This Matters

Stochastic differential equations sit at the intersection of probability theory, calculus, and real-world modeling—making them one of the most powerful tools you'll encounter in this course. When you're tested on SDEs, you're really being tested on your ability to understand how randomness interacts with continuous change, whether that's a stock price fluctuating over time, a particle diffusing through a medium, or an interest rate reverting to its long-term average. The core principles here—drift versus diffusion, measure changes, mean reversion, and martingale properties—show up repeatedly in both theoretical proofs and applied problems.

Don't just memorize the names and formulas. For each concept, ask yourself: What type of randomness does this capture? What real-world behavior does it model? How does it connect to other SDEs? If you can explain why geometric Brownian motion differs from an Ornstein-Uhlenbeck process, or why Girsanov's theorem matters for option pricing, you're thinking at the level the exam demands. These connections are what separate strong answers from mediocre ones.

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Foundational Building Blocks

These concepts form the mathematical foundation upon which all SDEs are built. Without understanding continuous-time random processes and how to differentiate functions of them, you cannot work with any SDE.

Brownian Motion and Wiener Processes

• Continuous-time random process—Brownian motion models random movement where paths are continuous but nowhere differentiable, making it the canonical "noise" in SDEs
• Independent, stationary increments with $W(t) - W(s) \sim N(0, t-s)$—the change over any interval depends only on the interval's length, not its location
• Variance grows linearly with time ($\text{Var}(W(t)) = t$), which is why longer time horizons mean greater uncertainty in diffusion-driven models

Itô's Formula

• Stochastic chain rule—extends ordinary calculus to functions of Brownian motion, but includes an extra second-derivative term due to the $dt$-scale variance of $dW$
• The correction term $\frac{1}{2}\frac{\partial^2 f}{\partial x^2}(dX)^2$ arises because $(dW)^2 = dt$ in the Itô calculus framework
• Essential for deriving SDEs—you'll use this to transform processes, verify solutions, and derive pricing equations like Black-Scholes

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Asset Price Models

These SDEs describe how prices evolve over time, balancing deterministic trends (drift) with random fluctuations (diffusion). The key distinction is whether prices can go negative and whether they exhibit mean reversion.

Geometric Brownian Motion

• Exponential structure ensures prices stay positive—the SDE $dS = \mu S \, dt + \sigma S \, dW$ models multiplicative (percentage) changes rather than additive ones
• Constant drift $\mu$ and volatility $\sigma$ produce log-normal price distributions, meaning $\ln(S_t)$ follows a normal distribution
• Foundation of Black-Scholes—this model's tractability makes it the standard assumption for equity prices, though it ignores fat tails and volatility clustering

Ornstein-Uhlenbeck Process

• Mean-reverting dynamics—the SDE $dX = \theta(\mu - X) \, dt + \sigma \, dW$ pulls the process toward long-term mean $\mu$ with speed $\theta$
• Can go negative, making it unsuitable for prices but ideal for interest rates, spreads, and volatility that fluctuate around equilibrium values
• Stationary distribution exists—unlike Brownian motion, the variance stabilizes at $\frac{\sigma^2}{2\theta}$ rather than growing indefinitely

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Probability and Measure Theory

These concepts address how we describe and transform the probability distributions governing stochastic processes. Mastering them is essential for derivative pricing and risk analysis.

Martingales and the Martingale Representation Theorem

• Fair game property—a martingale satisfies $E[X_t | \mathcal{F}_s] = X_s$, meaning the best prediction of future values is the current value (no drift)
• Representation theorem states any square-integrable martingale adapted to Brownian filtration can be written as $M_t = M_0 + \int_0^t H_s \, dW_s$ for some process $H$
• Critical for hedging—this theorem guarantees that derivative payoffs can be replicated by trading the underlying asset, enabling no-arbitrage pricing

Girsanov Theorem

• Changes the probability measure to transform a process with drift into a martingale (or vice versa) by adjusting the "reference frame"
• Risk-neutral pricing relies on this—under the risk-neutral measure $\mathbb{Q}$, discounted asset prices become martingales, simplifying option valuation
• The Radon-Nikodym derivative $\frac{d\mathbb{Q}}{d\mathbb{P}}$ quantifies how likely events are under the new measure relative to the original

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Evolution Equations

These equations describe how probability distributions or prices evolve over time, connecting SDEs to partial differential equations.

Fokker-Planck Equation

• Governs probability density evolution—given an SDE, the Fokker-Planck (or forward Kolmogorov) equation describes how $p(x,t)$ changes over time
• Drift and diffusion terms appear explicitly as $\frac{\partial p}{\partial t} = -\frac{\partial}{\partial x}[\mu(x)p] + \frac{1}{2}\frac{\partial^2}{\partial x^2}[\sigma^2(x)p]$
• Essential in statistical mechanics for analyzing equilibrium distributions and relaxation times of physical systems

Black-Scholes Equation

• PDE for option prices—derived by constructing a riskless hedge and applying Itô's formula, yielding $\frac{\partial V}{\partial t} + \frac{1}{2}\sigma^2 S^2 \frac{\partial^2 V}{\partial S^2} + rS\frac{\partial V}{\partial S} - rV = 0$
• Assumes GBM dynamics for the underlying asset, constant volatility, and continuous trading without transaction costs
• Closed-form solution exists for European options, giving the famous Black-Scholes formula that revolutionized derivatives markets

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Computational Methods and Applications

When analytical solutions don't exist, numerical methods become essential. Understanding when and why to simulate is as important as knowing the theory.

Numerical Methods for SDEs (Euler-Maruyama)

• Discretizes the SDE via $X_{n+1} = X_n + \mu(X_n)\Delta t + \sigma(X_n)\Delta W_n$, where $\Delta W_n \sim N(0, \Delta t)$
• Strong vs. weak convergence—Euler-Maruyama has strong order 0.5 (pathwise accuracy) and weak order 1.0 (distributional accuracy)
• Higher-order schemes like Milstein add correction terms ($\frac{1}{2}\sigma\sigma'[(\Delta W)^2 - \Delta t]$) to improve strong convergence to order 1.0

Applications in Finance and Physics

• Finance applications include option pricing, portfolio optimization, interest rate modeling (Vasicek, CIR), and credit risk analysis
• Physics applications span particle diffusion, thermal fluctuations, population dynamics, and noise-driven phase transitions
• Common thread is modeling systems where deterministic dynamics are perturbed by continuous random forcing—the SDE framework unifies these diverse phenomena

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Quick Reference Table

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Self-Check Questions

1. Both geometric Brownian motion and Ornstein-Uhlenbeck are driven by Brownian motion—what fundamental property distinguishes their long-term behavior, and which would you choose to model a mean-reverting interest rate?

2. Explain why Itô's formula includes a second-derivative term that doesn't appear in the ordinary chain rule. What property of Brownian motion causes this?

3. Compare and contrast the Fokker-Planck equation and the Black-Scholes equation: what does each one describe, and how do their "directions" in time differ?

4. If you're asked to price a derivative under the risk-neutral measure, which theorem justifies changing from the real-world measure $\mathbb{P}$ to $\mathbb{Q}$? What happens to the drift of the underlying asset under this transformation?

5. When would you use the Euler-Maruyama method instead of a closed-form solution, and what is its strong convergence order? How does the Milstein scheme improve upon it?