Stochastic Calculus

Define a sigma algebra as a collection of subsets of omega closed under complement and countable unions, illustrating information, generated sigma algebras, and filtration concepts in probability.

Explore measurable functions and sigma algebras, define random variables as measurable mappings from omega to Y, and use the smallest sigma algebra sigma(X) to describe X.

Define the expectation of a measurable random variable as the integral with respect to P, and show how the indicator function of A yields the probability via its expectation.

Explore the independence of random variables, including product density and expectation factorization, plus covariance, variance, correlation, and independence of sigma-algebras and variables.

Explore conditional expectations as random variables on a probability space, with G as a sub-sigma algebra of F, and view E[X|G] as a projection onto G.

Explore the conditional expectation properties, including linearity, taking out G-measurable X, the tower rule, independence implications, and the total expectation rule, as practical tools.

Explore stochastic processes as time-evolving random variables and filtrations of sigma-algebras that capture information up to time t, as shown in a discrete X0, X1, X2 example.

Explore martingales, a central category in stochastic calculus, where integrable, adapted processes satisfy E[Xt|Fs] = Xs, implying zero expected increments and constant expectation, as seen with Brownian motion.

Explore the Brownian motion as a central martingale in stochastic calculus. Starts at zero and becomes almost surely continuous, while showing independent, normally distributed increments with variance t-s.

Verify a martingale by checking integrability, adaptation, and Xs = E[Xt|Fs], then illustrate with Xt = Bt^2 - t using increments and independence.

Define the stochastic (Ito) integral by approximating x_t with elementary constant blocks and taking the limit of their integrals, contrasting with Riemann and Stratonovic approaches.

Learn Ito's lemma for Brownian motion, derive the stochastic integral via a stochastic Taylor expansion, and connect quadratic variation to the diamond notation and dt.

Apply Ito's lemma to Ito processes, extending from Brownian motion to drift and diffusion. Derive the formula for functions of time and Ito processes, including quadratic variation and cross variation.

Apply Ito's lemma to Brownian motion and an Ito process to compute a stochastic integral, yielding the closed form W_t^2/2 - t/2 and illustrating its practical power.

Define quadratic variation through mesh limits and cross-variation, noting Brownian motion has t and deterministic functions have zero; for Ito processes, it equals the integral of sigma squared dt.

Demonstrate Ito's isometry by equating E[(∫ x_t dW_t)^2] with E[∫ x_t^2 dt] for x in L2, via an elementary-process sketch of Brownian increments.

Geometric brownian motion with drift mu and volatility sigma is derived using Ito's formula to yield xt = x0 exp((mu - sigma^2/2) t + sigma w_t).

Demonstrates that the stochastic integral ∫_0^t x_s dW_s is a martingale for any x in L2. Proves this by computing E[y_t|F_s] and using independence of Brownian increments.

explores how to change measure between probabilities on a sample space, introduces absolute continuity and the Radon-Nikodym theorem, and defines the Radon-Nikodym derivative to transition from P to Q.

Apply the Radon-Nikodym and Girsanov theorems to change measure from a drifted to a driftless Brownian motion, via a transition function; constant drift yields geometric Brownian motion.

This lecture defines stopping times as random variables mapping outcomes to times within a filtration, and presents Doob's optional stopping theorem for martingales under certain conditions, showing E[X_tau] = E[X_0].

Apply the continuous Doop's optional stopping theorem to a stopped martingale and compute the probability that Brownian motion first hits the upper barrier, yielding p_up = d/(u+d).