Economic Modelling · Stochastic models for security prices

Stochastic Differential Equations and Ito's Lemma Explained

Updated 11 October 2026 · Fact-checked

A stochastic differential equation (SDE) describes how a process changes over a tiny time step as a drift term plus a volatility term driven by Brownian motion. Ito's lemma gives the SDE of a function f(t, X). Expand to second order, then use (dW)² = dt, dt·dW = 0 and (dt)² = 0.

Understand Stochastic Differential Equations and Ito's Lemma

A normal differential equation describes a smooth path. Share prices are not smooth. They jump around randomly, so we add a random term driven by Brownian motion W(t). Brownian motion starts at 0, has independent increments, and W(t) − W(s) is normal with mean 0 and variance t − s for t > s.

An Ito process X(t) is written as an SDE: dX = μ(t, X) dt + σ(t, X) dW. The term μ is the drift. It is the expected change per unit time. The term σ is the volatility (diffusion coefficient). It scales the random shock. Over a small step dt, dW is normal with mean 0 and variance dt, so its size is about √dt, not dt.

This is why ordinary calculus fails. In normal calculus, (dx)² is negligible. Here (dW)² is of order dt, so it cannot be dropped. This is the key idea. When you expand a function f(t, X) with a Taylor series, you must keep the second-order term in dX.

Ito's lemma does that expansion for you. It gives df as a drift part plus a volatility part, both driven by the same dW. The extra term ½ σ² ∂²f/∂x² is the Ito correction. It has no counterpart in ordinary calculus.

In the IAI CM2 syllabus, the standard use is geometric Brownian motion, dS = μS dt + σS dW. Ito's lemma applied to ln S shows that ln S is normal, so S is lognormal. This is the base of the Black-Scholes model.

Key rules to remember

Ito process (SDE): dX(t) = μ(t, X) dt + σ(t, X) dW(t); μ is the drift, σ is the volatility. W(t) is standard Brownian motion.
Multiplication rules: (dW)² = dt; dt · dW = 0; (dt)² = 0; These let you simplify (dX)² to σ² dt.
Ito's lemma (function of t and X): df = (∂f/∂t + μ ∂f/∂x + ½ σ² ∂²f/∂x²) dt + σ ∂f/∂x dW; Partial derivatives are evaluated at (t, X(t)). Here dX = μ dt + σ dW.
Ito's lemma (function of X only): df = (μ f′(X) + ½ σ² f″(X)) dt + σ f′(X) dW; Special case when f has no explicit t.
Geometric Brownian motion: dS = μS dt + σS dW; Solution: S(t) = S(0) exp{(μ − ½σ²)t + σW(t)}.
Log of GBM: d(ln S) = (μ − ½σ²) dt + σ dW; So ln S(t) is normal with mean ln S(0) + (μ − ½σ²)t and variance σ²t.
Brownian motion with drift: X(t) = X(0) + μt + σW(t); From dX = μ dt + σ dW with constant μ and σ.

How to solve Stochastic Differential Equations and Ito's Lemma questions

Use this method for any question that asks for the SDE of a function of a stochastic process.

1Write the given SDE for X in the form dX = μ dt + σ dW. Identify μ and σ clearly, including any dependence on X and t.
2Write the function f(t, X) you need the dynamics of.
3Compute the three partial derivatives: ∂f/∂t, ∂f/∂x and ∂²f/∂x². Do each one separately.
4Substitute into Ito's lemma: df = (∂f/∂t + μ ∂f/∂x + ½σ² ∂²f/∂x²) dt + σ ∂f/∂x dW.
5Simplify. Where possible, rewrite the result in terms of f itself, for example dY = Y(…) dt + Y(…) dW.
6Check the result. The dW coefficient must be σ times ∂f/∂x. The extra ½σ² f″ term must appear in the drift.
7If asked for a distribution, integrate. Constant coefficients give a normal variable. State the mean and variance.

Quickest way: Shortcut: treat dX as a rule, not a limit

When to use it: Use this when f is a simple function such as ln X, X², eˣ or a product, and you want to avoid errors under time pressure.

Write df = f_t dt + f_x dX + ½ f_xx (dX)².
Replace dX with μ dt + σ dW in the middle term.
Replace (dX)² with σ² dt. This is the only second-order term that survives.
Collect dt terms and dW terms. Stop.
For GBM with f = ln S, remember the answer directly: drift μ − ½σ², volatility σ.

Common mistakes in Stochastic Differential Equations and Ito's Lemma

Leaving out the ½σ² f″ term.
Students apply the ordinary chain rule, where second-order terms vanish.
Fix: Always write the full second-order Taylor expansion first. Remind yourself that (dW)² = dt, so it is not negligible.
Using σ instead of σ² in the correction term.
Students forget that (dX)² = σ² dt, not σ dt.
Fix: Square the volatility. For GBM, the correction for ln S is −½σ², not −½σ.
Putting the correction term in the dW coefficient.
Students mix up which terms multiply dt and which multiply dW.
Fix: The ½σ² f″ term always goes with dt. The dW coefficient is only σ f′.
Confusing the drift μ of S with the drift of ln S.
Students assume ln S grows at the same rate as S.
Fix: The drift of ln S is μ − ½σ². Expected S(t) is S(0)e^(μt), but the median grows at e^((μ − ½σ²)t).
Treating dt · dW as non-zero or (dt)² as non-zero.
Students are unsure which products survive.
Fix: Memorise the three rules: (dW)² = dt, dt·dW = 0, (dt)² = 0. Only the first is non-zero.
Forgetting partial derivatives with respect to t when f depends on time explicitly.
Students focus on x and ignore the ∂f/∂t term.
Fix: Check if f has t in it, such as e^(−rt)S. If so, include ∂f/∂t in the drift.

Worked examples

Example 1

The share price S(t) follows dS = 0.08 S dt + 0.20 S dW. Find the SDE for Y = ln S, and state the distribution of ln S(t) given S(0) = 100.

Show the solution

Here μ = 0.08 and σ = 0.20 are constants. The function is f(S) = ln S.
Derivatives: f′ = 1/S and f″ = −1/S². There is no explicit t.
Drift of Y = μS × (1/S) + ½ σ²S² × (−1/S²) = μ − ½σ².
Compute: 0.08 − ½ × 0.04 = 0.08 − 0.02 = 0.06.
Volatility of Y = σS × (1/S) = σ = 0.20.
So dY = 0.06 dt + 0.20 dW.
Integrate: ln S(t) = ln 100 + 0.06t + 0.20 W(t). This is normal.
Mean = ln 100 + 0.06t. Variance = 0.04t.

Answer: dY = 0.06 dt + 0.20 dW. ln S(t) is normal with mean ln 100 + 0.06t and variance 0.04t.

Example 2

Let X follow dX = 0.05 dt + 0.3 dW. Find the SDE for Y = X².

Show the solution

Here μ = 0.05 and σ = 0.3 are constants. f(X) = X².
Derivatives: f′ = 2X and f″ = 2.
Drift = μ f′ + ½ σ² f″ = 0.05 × 2X + ½ × 0.09 × 2.
Compute: 0.10X + 0.09.
Volatility = σ f′ = 0.3 × 2X = 0.6X.
So dY = (0.10X + 0.09) dt + 0.6X dW.
Write in terms of Y if needed: X = √Y when X > 0, giving drift 0.10√Y + 0.09 and volatility 0.6√Y.

Answer: dY = (0.10X + 0.09) dt + 0.6X dW, where Y = X².

Exam tips

Write Ito's lemma in full at the start of every answer. Examiners award marks for the correct formula, even if arithmetic slips later.
Show each partial derivative on its own line. This protects method marks.
For GBM questions, state clearly that ln S is normal and S is lognormal. This is a frequent written-paper requirement.
Check the drift of ln S is μ − ½σ². A missing ½σ² is the most common lost mark.
In multiple-choice questions, look for options that omit the Ito correction. Those are usually the distractors.

Practice questions from Stochastic models for security prices

Stochastic Differential Equations and Ito's Lemma: frequently asked questions

What is the difference between an ordinary differential equation and an SDE?

An ordinary differential equation has a deterministic path. An SDE adds a random term driven by Brownian motion, so the path is random. Because of this, the usual chain rule must be replaced by Ito's lemma.

Why does (dW)² equal dt?

The increment dW has variance dt, so its square has expected value dt. Over many small steps, the sum of squared increments converges to the elapsed time. This is why we treat (dW)² as dt in calculations.

Why is the drift of ln S equal to μ − ½σ²?

Ito's lemma adds a correction ½σ² f″. For f = ln S, f″ = −1/S², and this gives −½σ². The correction reflects that the log function is concave, so random shocks reduce the growth rate of ln S.

Do I need to prove Ito's lemma for the exam?

You usually need to apply it, not prove it. Know the statement, the multiplication rules and how to use them. A short Taylor-series argument can help your understanding.