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Study Notes 2023 Draft

Probability & Stochastic Calculus

Measure-theoretic probability, Brownian motion, Itô's lemma, and the Feynman–Kac formula.

Stochastic Calculus Itô's Lemma Brownian Motion

Measure-Theoretic Foundations

A probability space (Ω, ℱ, ℙ) consists of a sample space, a σ-algebra of events, and a probability measure. Random variables are measurable functions X: Ω → ℝ. Expectation is integration with respect to ℙ.

Brownian Motion

A standard Brownian motion Wt satisfies: W₀ = 0, independent increments, Wt − Ws ~ N(0, t−s) for t > s, and continuous paths. It is nowhere differentiable, so classical calculus doesn't apply.

Itô's Lemma

For a smooth function f(t, Xt) where dXt = μ dt + σ dWt:

df = (∂f/∂t + μ ∂f/∂x + ½σ² ∂²f/∂x²) dt + σ ∂f/∂x dWt

The extra ½σ²∂²f/∂x² term (compared to classical chain rule) arises from the quadratic variation of Brownian motion: (dW)2 = dt.

Feynman–Kac Formula

The Feynman–Kac formula connects PDEs to stochastic processes. If u(t,x) solves a parabolic PDE with terminal condition, then u has a probabilistic representation as a conditional expectation under a suitable measure. This is the foundation for the risk-neutral pricing of derivatives.

Connection to Black–Scholes

Applying Itô's lemma to a portfolio of a derivative V(S,t) and Δ = ∂V/∂S shares of stock, and invoking no-arbitrage, yields the Black–Scholes PDE. The Feynman–Kac formula then gives the risk-neutral pricing formula V = e−r(T−t) Eℚ[payoff | St].

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