stochastic calculus

Simplified from Ji-Ha Kim’s blog post.

Brownian motion

The Wiener process describes the position of a particle undergoing Brownian motion:

An increment over is , and in the infinitesimal limit:

The sample path is continuous but nowhere differentiable. The rate of change over a small interval :

which diverges. This rules out standard calculus.

Itô calculus

The stochastic differential is

with and .

and , negligible as . So in the mean-square sense. Unlike ordinary calculus where vanishes, is on the same scale as .

Itô’s lemma

For , ordinary calculus gives . Brownian motion’s roughness requires a second-order correction. Taylor-expand:

As : and vanish, but stays. This gives Itô’s lemma (the stochastic chain rule):

The extra captures the curvature from Brownian motion.

Itô’s lemma for a general SDE

For and , since keep terms up to :

Stochastic differential equations

SDEs blend deterministic behaviour with stochastic noise:

is the drift (average direction), is the diffusion (strength of random jitter). If , this is a standard ODE; if , pure scaled Brownian motion.

Constant drift and diffusion

Setting and (constants, independent of and ):

with :

A process drifting linearly with noise spreading over time.

Geometric Brownian motion

For systems where changes scale with size (e.g. stock prices):

The percentage change has a trend and randomness. To solve, apply Itô’s lemma with , so , , :

Integrate from to :

The drift is adjusted by from the Itô correction. This underlies the Black-Scholes model.

Itô vs Stratonovich

The Itô integral evaluates the integrand at the left endpoint of each interval — non-anticipating (only uses information up to the current time). Natural in finance.

Stratonovich calculus evaluates at the midpoint instead. This preserves the ordinary chain rule with no second-derivative correction, because the midpoint evaluation absorbs the correction into the integral definition.

Conversion between Itô drift and Stratonovich drift :

where . The diffusion term is the same in both.

Stratonovich suits physical systems where noise has slight smoothness or continuity — the Wong-Zakai theorem shows that smooth noise approximations converge to Stratonovich SDEs in the limit. Itô dominates in finance for its non-anticipating, martingale-friendly properties.