What Maths Do You Need Before Stochastic Calculus?

Start with single-variable calculus and elementary probability. Get comfortable with derivatives, second-order Taylor expansions, expectation, variance, the normal distribution, conditional expectation, and independence. Then learn discrete-time random walks and the idea of a filtration before starting Brownian motion and Itô calculus. Learn basic matrix operations alongside this if your goal includes multi-asset modelling.

You can postpone formal measure theory for a first pass. A better readiness test is whether you can explain why tomorrow's fair coin toss still has mean zero after you learn today's result. That single distinction between unconditional and conditional averages will do more for you in the first weeks than memorising a list of distributions.

Work backwards from what you'll eventually do with Itô's formula. The Taylor expansion tells you which derivatives to compute. Conditional expectation tells you which parts of an increment are predictable. A random walk gives you a concrete model on which to practise both ideas before the continuous-time notation arrives.

What you should be able to do first

Work through the prerequisites in this order, using a question to check each one.

Calculus. Approximate $\log(1+x)$ near zero as $x-x^2/2$ . Can you explain when keeping the squared term improves the approximation?
Probability. If $Z$ is standard normal, explain why $a+bZ$ has mean $a$ and variance $b^2$ . Can you compute both from their definitions?
Conditional expectation. For a random walk, compute the expected value at step $n+1$ given every move up to step $n$ . Can you explain why independence matters?
Random walks and limits. Divide one unit of time into $n$ steps. Give each move size $1/\sqrt{n}$ and verify that the sum of $n$ independent moves has variance one.
Continuous time. Read the Brownian increment law and quadratic variation before tackling the statement of Itô's lemma.

For proofs in a graduate course, add sigma-algebras, measurability, integrability, and convergence in probability or in mean square. For a first pricing simulation, elementary versions of the ideas above are enough to begin. The two tracks can grow together as exercises demand more precision.

One random walk that tests three prerequisites

Take four independent symmetric steps of size $1/2$ . The total displacement $X_4$ has variance $4(1/2)^2=1$ . Given the first two steps, the expected final displacement is $X_2$ , because the remaining steps each have mean zero. The sum of squared moves is $4(1/2)^2=1$ on every path, even if the final displacement is zero.

Now try $n=100$ with step size $0.1$ . The variance at the final time stays one, and so does the sum of squared moves. Smaller individual moves can still produce a nonzero total of squared changes. That observation prepares you for Brownian quadratic variation, even though Brownian increments themselves are Gaussian.