学习随机微积分（Stochastic Calculus）是否需先掌握随机过程（Stochastic Processes）？

Great question—this is a common point of confusion for folks diving into advanced math topics for independent study. Let’s break this down clearly.

• First, let’s address the overlap claim: While it’s true that stochastic calculus builds on specific branches of stochastic processes, the "low overlap" idea is a bit misleading. Stochastic calculus (think Ito calculus, which is the core for most applications) relies heavily on understanding Brownian motion—a fundamental stochastic process. That said, you don’t need to master the entire breadth of stochastic processes (like Markov chains, Poisson processes, etc.) before starting.

• Now, can you start stochastic calculus without formal stochastic processes training? The short answer is yes, but with caveats:

  • You’ll need to learn the basics of Brownian motion on the fly—things like its definition, properties (stationarity, independent increments, Gaussian distribution), and how it differs from deterministic functions. Most introductory stochastic calculus texts dedicate early chapters to this, so you can pick it up as you go.
  • You should have a solid foundation in calculus (single-variable, multivariable, and some real analysis—like limits, continuity, Lebesgue integration basics) and probability theory (random variables, expectation, variance, conditional probability, distributions). These are non-negotiable prerequisites.
  • Avoid skipping the foundational stochastic concepts introduced in your calculus textbook. Even if you haven’t taken a full stochastic processes course, treating those early sections as your crash course will prevent confusion later when you get to Ito’s lemma or stochastic differential equations.

• My recommendation for your independent study:

  • Pick an introductory stochastic calculus book that’s self-contained. Look for one that starts with a thorough overview of Brownian motion and the necessary stochastic basics before diving into the calculus.
  • If you hit a wall with a concept tied to broader stochastic processes, pause and do a deep dive into that specific topic (e.g., if you struggle with martingales, learn the basics of martingales specifically) instead of trying to learn the entire field.

内容的提问来源于stack exchange，提问作者Saira Rizvi