31  Session 22: Fokker-Planck · Master Equation (Concepts Only)

Unit 4 — Math Intuition Bridges
Book Chapters 14, 15 (concepts only — full derivation in Session 27 Track 2)
Track Common core (both tracks)

31.1 Learning Objectives

By the end of this session, students will be able to:

  1. State what the Fokker-Planck equation does — describe how a probability distribution evolves over time under SDE dynamics.
  2. Recognize the Fokker-Planck for the OU process and identify its three terms (drift transport, diffusion spreading, mean-reverting concentration).
  3. State what the master equation does — describe how the value function of an MFG evolves on the space of distributions.
  4. Connect Fokker-Planck to the well-known heat equation as a familiar analogue.
  5. Identify when computing distributions matters (regulatory stress testing, portfolio diagnostics) vs. when individual states suffice (single-asset valuation).

31.2 Pre-Class Assignment

  • Read: Book Chapters 14, 15 (concepts only — skim equations)

31.3 In-Class Outline (75 minutes)

Time Segment Format
0:00–0:05 Recap Sessions 19–21 · Today: tracking distributions Lecture
0:05–0:25 The Fokker-Planck equation Lecture + visual
0:25–0:45 Applying Fokker-Planck to the GE-LAV OU process Lecture
0:45–1:05 The master equation — value functions on distribution spaces Lecture
1:05–1:15 Why this matters for practitioners Discussion

31.4 Discussion Questions

  1. The Fokker-Planck equation is mathematically equivalent to a generalized heat equation. Does that connection give you confidence in the framework, or does it suggest the framework is just “thermodynamics applied to finance”? How are the two domains structurally similar / different?
  2. The master equation operates on the space of probability distributions (Wasserstein space). This is abstract. What’s the practical consequence of “values live on distribution spaces”?
  3. Computing the master equation requires substantial numerical infrastructure. Is this a barrier to GE-LAV adoption, or does the platform’s existence eliminate the barrier for practitioners?

31.5 Worked Example: Fokker-Planck Stress Test

Setup: A regulator wants to know: starting from current conditions (\(L_0 = -0.5\) for all LPs), what’s the probability that aggregate PE liquidity is worse than current (\(L_t < -0.5\)) at \(t = 2\) years?

Step 1: Solve OU Fokker-Planck

Using OU calibration (\(\kappa = 0.45\), \(\sigma_L = 0.32\), \(\bar{L} = 1.0\)):

Conditional density at \(t = 2\): - Mean: \(\bar{L} + (L_0 - \bar{L}) e^{-\kappa t} = 1.0 + (-1.5) \cdot e^{-0.9} = 1.0 + (-1.5)(0.407) = 0.39\) - Variance: \(\frac{\sigma_L^2}{2\kappa}(1 - e^{-2\kappa t}) = 0.1138 \cdot (1 - e^{-1.8}) = 0.1138 \cdot 0.835 = 0.0950\) - Standard deviation: 0.308

Step 2: Compute probability

\(P(L_2 < -0.5 | L_0 = -0.5) = \Phi\left(\dfrac{-0.5 - 0.39}{0.308}\right) = \Phi(-2.89) \approx 0.19\%\)

Interpretation: “Less than 0.2% probability that aggregate liquidity will be worse than current conditions 2 years from now.”

Note: This is a single-LP mean-reversion forecast. The MFG externality could amplify in extreme cases — but at \(L = -0.5\) we’re in the stable regime, so the single-LP Fokker-Planck is a good approximation.

31.6 What to Expect Next Session

Session 23 synthesizes everything from Sessions 19–22 into the LAV operator (book Ch. 16) and the GE equilibrium (book Ch. 17). We’ll see:

  • The Liquidity-Adjusted Valuation operator: how a single asset’s value depends on the entire \(L_t\) path
  • General equilibrium pricing: the fixed-point that determines \(\lambda^*\)
  • Market clearing in the GE-LAV framework

Reading: Book Chapters 16, 17 (concepts).

← Session 21 | Schedule | Next: Session 23 →