28 Session 19: Brownian Motion · Itô · OU (Concepts Only)
| Unit | 4 — Math Intuition Bridges |
| Book Chapters | 10, 12 (concepts only — full derivations in Session 25 Track 2) |
| Track | Common core (both tracks) |
This unit teaches the conceptual content of Part 2 of the book — what the math means, what it does, and why it works — without proofs or detailed derivations. The goal is for all students to leave Unit 4 with a working intuition about the GE-LAV machinery.
Track 1 students are assessed only on conceptual understanding. PS3 includes intuition-level questions for Track 1.
Track 2 students will revisit each topic in Sessions 25–29 with full mathematical rigor — proofs, derivations, numerical implementations. PS3 for Track 2 includes mathematical work.
If you find Unit 4 too easy, you’ll love Sessions 25–29. If you find it too hard, the framework’s applications (Unit 3 and Session 25+ Track 1) don’t require this level of detail.
28.1 Learning Objectives
By the end of this session, students will be able to:
- Explain Brownian motion in plain language — what it is, why it’s the canonical model of randomness in continuous time.
- State the four key properties of Brownian motion (continuity, independent increments, normal distribution of increments, scale).
- State Itô’s lemma conceptually — the “chain rule” of stochastic calculus — and why it has an extra term compared to ordinary calculus.
- Re-derive (informally) the OU process as a special case of an Itô SDE with mean-reverting drift.
- Recognize when Itô calculus is the right tool (continuous random processes) and when it isn’t (jumps, discrete states).
28.2 Pre-Class Assignment
- Read: Book Chapter 10 sections relevant to Brownian motion and Itô calculus — read for concepts, not formulas. Skim the technical derivations.
- Watch: Pre-class video (10 min) explaining Brownian motion intuitively (link on course site)
28.3 In-Class Outline (75 minutes)
| Time | Segment | Format |
|---|---|---|
| 0:00–0:10 | Why we need this · Math intuition bridges begin | Lecture |
| 0:10–0:30 | Brownian motion: the canonical continuous random process | Lecture + simulation |
| 0:30–0:50 | Itô’s lemma: chain rule with a twist | Lecture + visual |
| 0:50–1:05 | OU as an Itô SDE | Lecture |
| 1:05–1:15 | What Itô calculus can and can’t do | Discussion |
28.4 Discussion Questions
- The Itô correction term \(\frac{1}{2} f''(X_t) \sigma^2 dt\) has the same mathematical form as the Jensen convexity correction \(\frac{1}{2} f''(X) \text{Var}(X)\). Why are these two expressions the same? What’s the deep connection?
- The OU process has \(\kappa = 0.45\)/yr, implying half-life of liquidity shocks ~18 months. Many practitioners would estimate half-life closer to 6-12 months (faster mean reversion). What evidence would resolve this?
- Real liquidity has jumps (March 2020). The GE-LAV core uses pure Itô (no jumps). For the purposes of valuation, is this approximation acceptable? When does it break?
28.5 Worked Example: Applying Itô’s Lemma
Setup: Let \(L_t\) follow OU with \(\kappa = 0.45\), \(\sigma_L = 0.32\), \(\bar{L} = 1.0\). Compute the expected value and variance of \(f(L_t) = L_t^2\) at \(t = 5\), given \(L_0 = 0\).
Step 1: Apply Itô’s lemma to \(f(L) = L^2\)
\(f'(L) = 2L\), \(f''(L) = 2\)
\(df(L_t) = 2L_t dL_t + \frac{1}{2}(2)\sigma_L^2 dt = 2L_t [\kappa(\bar{L} - L_t)dt + \sigma_L dW_t] + \sigma_L^2 dt\)
Take expectations (Brownian term vanishes in expectation): \(\frac{d}{dt} E[L_t^2] = 2\kappa \bar{L} E[L_t] - 2\kappa E[L_t^2] + \sigma_L^2\)
Step 2: We need \(E[L_t]\) first
\(E[L_t | L_0 = 0] = \bar{L}(1 - e^{-\kappa t}) = 1.0 \cdot (1 - e^{-2.25}) = 1.0 \cdot 0.894 = 0.894\)
So at \(t = 5\), \(E[L_5] = 0.894\) (close to long-run mean).
Step 3: Variance
\(\text{Var}[L_t | L_0] = \frac{\sigma_L^2}{2\kappa}(1 - e^{-2\kappa t}) = \frac{0.1024}{0.9}(1 - e^{-4.5}) = 0.1138 \cdot 0.989 = 0.1126\)
Step 4: \(E[L_t^2]\)
\(E[L_t^2] = \text{Var}[L_t] + (E[L_t])^2 = 0.1126 + 0.894^2 = 0.1126 + 0.799 = 0.912\)
Interpretation: Starting at \(L_0 = 0\) (stressed but not crisis), 5 years later we expect \(E[L] \approx 0.89\) (close to long-run mean) with variance \(\approx 0.11\) (close to stationary). The \(E[L^2]\) value \(\approx 0.91\) matters for the quadratic premium function \(\pi(L)\).
28.6 What to Expect Next Session
Session 20 introduces stochastic control and the HJB equation. We’ll cover:
- The optimal control problem: choose actions \(u_t\) to maximize expected utility
- The dynamic programming principle (Bellman’s logic)
- The HJB equation: a PDE that the value function must satisfy
- Application: the LP’s exit timing problem (Session 11–13 revisited mathematically)
Reading: Book Chapter 11 (concepts only — skim the proofs).
For Track 2: Session 25 derives the HJB for the GE-LAV exit problem with full smooth pasting and verification theorems. Today is the conceptual setup.