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TL;DR

An equation of the form \( dX_t = \mu(X_t, t)dt + \sigma(X_t, t)dW_t \), describing a process driven by both deterministic drift and random Brownian noise.

By Valenke Exam Prep Team·Last updated 2026-06-03

Stochastic Differential Equation

An equation of the form \( dX_t = \mu(X_t, t)dt + \sigma(X_t, t)dW_t \), describing a process driven by both deterministic drift and random Brownian noise.

Why it matters for interviews

SDEs are the language of continuous-time finance. GBM, Ornstein-Uhlenbeck, CIR, Heston -- all fundamental models are SDEs. Quant interviews test the ability to set up, solve, and interpret SDEs.

Definition and Mathematical Foundation

An equation of the form \( dX_t = \mu(X_t, t)dt + \sigma(X_t, t)dW_t \), describing a process driven by both deterministic drift and random Brownian noise.

Application in Quantitative Finance

SDEs are the language of continuous-time finance. GBM, Ornstein-Uhlenbeck, CIR, Heston -- all fundamental models are SDEs. Quant interviews test the ability to set up, solve, and interpret SDEs.

Related Concepts

Brownian Motion (Wiener Process)

Related Terms

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Frequently Asked Questions

What does it mean to solve an SDE?▼
Finding the explicit expression for \( X_t \) in terms of \( W_t \) and t. For GBM, the solution is \( S_t = S_0 e^{(\mu-\sigma^2/2)t + \sigma W_t} \). Many SDEs do not have closed-form solutions and require numerical methods.
What is the Ornstein-Uhlenbeck process?▼
Defined by \( dX_t = \theta(\mu - X_t)dt + \sigma dW_t \). It is mean-reverting: X is pulled toward \( \mu \) at rate \( \theta \). Used to model interest rates, pairs trading spreads, and volatility.
How are SDEs simulated numerically?▼
The Euler-Maruyama scheme discretizes: \( X_{t+\Delta t} = X_t + \mu \Delta t + \sigma \sqrt{\Delta t} Z \) where \( Z \sim N(0,1) \). Higher-order schemes like Milstein improve convergence.