42  Appendix A: Mathematical Foundations

This appendix collects the mathematical prerequisites and intermediate results needed for the course. It is organized by mathematical area, not by session. Use it as a reference; consult specific sections as needed.

Notation conventions: - \(\mathbb{R}\) denotes the real numbers; \(\mathbb{R}^n\) the \(n\)-dimensional Euclidean space - \(\mathbb{N} = \{0, 1, 2, ...\}\) denotes the non-negative integers - \(\mathbb{P}\) denotes a probability measure; \(E[\cdot]\) denotes expectation under \(\mathbb{P}\) - \(\mathcal{N}(\mu, \sigma^2)\) denotes the normal distribution with mean \(\mu\) and variance \(\sigma^2\) - \(W_t\) denotes a standard Brownian motion

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42.1 A.1 Probability Theory Essentials

42.1.1 A.1.1 Random Variables and Distributions

A random variable \(X\) is a measurable function from a probability space \((\Omega, \mathcal{F}, \mathbb{P})\) to \(\mathbb{R}\).

Expectation: \(E[X] = \int_\Omega X(\omega) d\mathbb{P}(\omega)\)

Variance: \(\text{Var}(X) = E[(X - E[X])^2] = E[X^2] - (E[X])^2\)

Conditional expectation: \(E[X | \mathcal{G}]\) for \(\sigma\)-algebra \(\mathcal{G} \subseteq \mathcal{F}\). The conditional expectation is itself a random variable, measurable with respect to \(\mathcal{G}\).

Key property (tower): \(E[E[X | \mathcal{G}]] = E[X]\)

42.1.2 A.1.2 The Normal Distribution

If \(X \sim \mathcal{N}(\mu, \sigma^2)\):

• Density: \(f_X(x) = \dfrac{1}{\sigma\sqrt{2\pi}} \exp\left(-\dfrac{(x-\mu)^2}{2\sigma^2}\right)\)
• Mean: \(\mu\); Variance: \(\sigma^2\)
• Moment generating function: \(M_X(t) = e^{\mu t + \sigma^2 t^2/2}\)
• Standardization: \(Z = (X - \mu)/\sigma \sim \mathcal{N}(0, 1)\)

Important moments:

Moment	Value
\(E[X]\)	\(\mu\)
\(\text{Var}(X)\)	\(\sigma^2\)
\(E[(X - \mu)^3]\)	0 (symmetry)
\(E[(X - \mu)^4]\)	\(3\sigma^4\)

42.1.3 A.1.3 Jensen’s Inequality (Foundational for GE-LAV)

Statement: For a convex function \(f: \mathbb{R} \to \mathbb{R}\) and any random variable \(X\) with finite expectation: \[f(E[X]) \leq E[f(X)]\] with strict inequality when \(f\) is strictly convex and \(X\) is non-degenerate.

For concave \(f\): the inequality reverses: \(f(E[X]) \geq E[f(X)]\).

Proof sketch (convex case): By the supporting hyperplane property, for any \(x_0\): \(f(x) \geq f(x_0) + f'(x_0)(x - x_0)\). Take \(x_0 = E[X]\) and expectations of both sides: \(E[f(X)] \geq f(E[X]) + f'(E[X]) \cdot E[X - E[X]] = f(E[X])\). ∎

Application to GE-LAV: The discount factor \(f(L) = e^{-r(L)T}\) is convex in \(L\) when \(r(L)\) is appropriately convex. This produces the systematic upward Jensen bias in LAV vs. DCF.

42.1.4 A.1.4 Conditional Expectation Properties

For random variables \(X, Y\):

• Linearity: \(E[aX + bY | \mathcal{G}] = aE[X | \mathcal{G}] + bE[Y | \mathcal{G}]\)
• Pull-out: If \(Z\) is \(\mathcal{G}\)-measurable: \(E[ZX | \mathcal{G}] = Z E[X | \mathcal{G}]\)
• Tower property: \(E[E[X | \mathcal{G}]] = E[X]\)
• Independence: If \(X\) independent of \(\mathcal{G}\): \(E[X | \mathcal{G}] = E[X]\)

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42.2 A.2 Stochastic Processes

42.2.1 A.2.1 Definition and Classification

A stochastic process \((X_t)_{t \in T}\) is a collection of random variables indexed by time. Common index sets: \(T = \mathbb{N}\) (discrete time) or \(T = [0, \infty)\) (continuous time).

Classification by state space: - Discrete state: e.g., Markov chains on \(\{0, 1, 2, ...\}\) - Continuous state: e.g., Brownian motion, OU process

Classification by index set: - Discrete time: \((X_n)_{n \in \mathbb{N}}\) - Continuous time: \((X_t)_{t \geq 0}\)

42.2.2 A.2.2 Markov Property

A process \((X_t)\) has the Markov property if, for all \(s < t\): \[\mathbb{P}(X_t \in A | \mathcal{F}_s) = \mathbb{P}(X_t \in A | X_s)\] where \(\mathcal{F}_s\) is the natural filtration.

In words: “future depends on past only through the present.” All the SDE-based processes in this course (Brownian motion, OU, GE-LAV state) are Markov.

42.2.3 A.2.3 Random Walks (Discrete-Time Building Block)

A simple random walk \((S_n)_{n=0,1,2,...}\): - \(S_0 = 0\) - \(S_n = X_1 + X_2 + ... + X_n\) - \(X_i\) i.i.d. with \(\mathbb{P}(X_i = +1) = \mathbb{P}(X_i = -1) = 1/2\)

Properties: - \(E[S_n] = 0\) - \(\text{Var}(S_n) = n\) (variance grows linearly with time) - Recurrent in 1D, 2D; transient in 3D+

Scaling limit: As \(n \to \infty\) with appropriate scaling, the random walk converges to Brownian motion (Donsker’s theorem).

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42.3 A.3 Brownian Motion

42.3.1 A.3.1 Definition

A stochastic process \(W_t\) is a standard Brownian motion if:

1. \(W_0 = 0\)
2. Paths \(t \mapsto W_t\) are continuous almost surely
3. Increments are independent: for \(s < t\), \(W_t - W_s\) is independent of \(\mathcal{F}_s\)
4. Increments are normally distributed: \(W_t - W_s \sim \mathcal{N}(0, t - s)\)

42.3.2 A.3.2 Key Properties

Martingale property: \(E[W_t | \mathcal{F}_s] = W_s\) for \(s \leq t\).

Quadratic variation: \([W, W]_t = t\) — Brownian motion accumulates quadratic variation at unit rate.

Self-similarity (scaling): For any \(c > 0\), the process \((c^{-1/2} W_{ct})\) has the same law as \((W_t)\).

Non-differentiability: Almost surely, \(W_t\) is nowhere differentiable.

Time-reversal: The process \((W_T - W_{T-t})_{0 \leq t \leq T}\) has the same law as \((W_t)_{0 \leq t \leq T}\) for any \(T > 0\).

42.3.3 A.3.3 Multidimensional Brownian Motion

\(d\)-dimensional standard Brownian motion: \(W_t = (W_t^1, W_t^2, ..., W_t^d)\) where each \(W_t^i\) is an independent 1D standard Brownian motion.

Cross-covariance: \(\text{Cov}(W_t^i, W_t^j) = t \cdot \delta_{ij}\) (Kronecker delta).

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42.4 A.4 Itô Calculus

42.4.1 A.4.1 Stochastic Integral

For an \(\mathcal{F}_t\)-adapted process \(f(t, \omega)\) satisfying \(\int_0^T E[f(t, \omega)^2] dt < \infty\), the Itô integral: \[\int_0^T f(t, \omega) dW_t\] is well-defined as the \(L^2\) limit of step-function approximations.

Key property (Itô isometry): \[E\left[\left(\int_0^T f dW_t\right)^2\right] = \int_0^T E[f^2] dt\]

This is the square of the standard \(L^2\) norm of \(f\) — the foundation for many calculations.

42.4.2 A.4.2 Itô SDEs

A stochastic differential equation in Itô form: \[dX_t = \mu(X_t, t) dt + \sigma(X_t, t) dW_t\] with initial condition \(X_0\), has a unique strong solution under standard conditions: - \(\mu, \sigma\) Lipschitz continuous - \(\mu, \sigma\) have linear growth bound

42.4.3 A.4.3 Itô’s Lemma (Chain Rule)

For a smooth function \(f: \mathbb{R} \to \mathbb{R}\) and an Itô process \(X_t\): \[df(X_t) = f'(X_t) dX_t + \frac{1}{2} f''(X_t) \sigma^2(X_t, t) dt\]

The Itô correction \(\frac{1}{2} f''(X_t) \sigma^2 dt\) is what distinguishes stochastic calculus from ordinary calculus. It is the second-order Taylor term that becomes first-order because \((dW_t)^2 \to dt\) at order 1.

Multivariate version: For \(f: \mathbb{R}^d \to \mathbb{R}\): \[df(X_t) = \sum_i \partial_i f(X_t) dX_t^i + \frac{1}{2} \sum_{i,j} \partial_{ij} f(X_t) [dX^i, dX^j]_t\] where \([dX^i, dX^j]_t = (\sigma\sigma^T)_{ij}(X_t, t) dt\).

42.4.4 A.4.4 Connection to Jensen’s Inequality

Apply Itô’s lemma to \(f(X) = e^{-rX}\) with \(r > 0\) (convex function): - \(f'(X) = -r e^{-rX}\) - \(f''(X) = r^2 e^{-rX}\)

If \(X\) is constant (zero drift, zero diffusion): no correction. If \(X\) is stochastic: \(E[f(X_T)] > f(E[X_T])\) by Jensen, with magnitude \(\frac{r^2}{2} \text{Var}(X_T)\) to leading order.

This is exactly the Jensen bias mechanism that makes \(V^{LAV} > V^{DCF}\) when \(r(L)\) is convex in \(L\) and \(L\) is stochastic.

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42.5 A.5 Ornstein-Uhlenbeck Process

42.5.1 A.5.1 Definition

The OU process is the Itô SDE: \[dL_t = \kappa(\bar{L} - L_t) dt + \sigma dW_t\] with parameters \(\kappa > 0\) (mean reversion speed), \(\bar{L}\) (long-run mean), \(\sigma > 0\) (volatility).

42.5.2 A.5.2 Closed-Form Solution

Using the integrating factor \(e^{\kappa t}\): \[L_t = e^{-\kappa t} L_0 + \bar{L}(1 - e^{-\kappa t}) + \sigma \int_0^t e^{-\kappa(t-s)} dW_s\]

The integral is Gaussian (linear in \(dW\)), so \(L_t\) given \(L_0\) is Gaussian.

Conditional distribution: \[L_t | L_0 \sim \mathcal{N}\left(\bar{L} + (L_0 - \bar{L})e^{-\kappa t}, \frac{\sigma^2}{2\kappa}(1 - e^{-2\kappa t})\right)\]

42.5.3 A.5.3 Stationary Distribution

As \(t \to \infty\), the OU distribution converges: \[L_\infty \sim \mathcal{N}\left(\bar{L}, \frac{\sigma^2}{2\kappa}\right)\]

This is the invariant distribution of the OU process — independent of starting point \(L_0\).

42.5.4 A.5.4 Calibrated Parameters (GE-LAV)

Parameter	Value	Interpretation
\(\kappa\)	0.45/yr	Mean reversion speed
\(\sigma\)	0.32	Volatility coefficient
\(\bar{L}\)	1.0 (or 0.0 depending on normalization)	Long-run mean
\(\text{Var}(L_\infty)\)	0.1138	Stationary variance
Std\((L_\infty)\)	0.337	Stationary standard deviation
Half-life	1.54 years	\(\ln(2)/\kappa\)

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42.6 A.6 Variational Inequalities and Free Boundaries

42.6.1 A.6.1 The American Option Analogue

The GE-LAV exit problem is an optimal stopping problem. The value function satisfies: \[\min\left\{ -\mathcal{L}V + rV - C, \quad V - G \right\} = 0\]

where: - \(\mathcal{L}\) is the HJB operator (drift + diffusion + time) - \(G(L)\) is the exit payoff - The first argument is the continuation condition - The second argument is the exercise condition

This is mathematically identical to American option pricing.

42.6.2 A.6.2 Smooth Pasting

At the free boundary \(L^*(t)\): 1. Value match: \(V(L^*, t) = G(L^*)\) 2. Smooth pasting: \(\partial_L V(L^*, t) = G'(L^*)\)

These two conditions uniquely determine \(L^*(t)\).

42.6.3 A.6.3 Numerical Methods

For the GE-LAV exit problem: - Discretize the state space: \(L \in [-3, 3]\) on a 200-point grid - Discretize time: Backward induction from \(t = T\) to \(t = 0\) - At each step: Solve the HJB PDE assuming continuation, then take maximum with exit payoff - Output: The exit boundary \(L^*(t)\) emerges from the switching surface

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42.7 A.7 Convex Analysis Essentials

42.7.1 A.7.1 Convex Functions

A function \(f: \mathbb{R} \to \mathbb{R}\) is convex if: \[f(\lambda x + (1-\lambda) y) \leq \lambda f(x) + (1-\lambda) f(y)\] for all \(x, y \in \mathbb{R}\) and \(\lambda \in [0, 1]\).

Strict convexity: Inequality is strict whenever \(x \neq y\).

Equivalent for twice-differentiable \(f\): \(f''(x) \geq 0\) for all \(x\) (strictly convex if \(f''(x) > 0\)).

42.7.2 A.7.2 Convexity in GE-LAV

The premium function \(\pi(L) = \pi_0 - \pi_1 L + \pi_2 L^2\) with \(\pi_2 > 0\) is strictly convex.

The discount factor \(e^{-r(L)T}\) as a function of \(L\) is convex (since composition of convex with affine of \(L\) is convex; with \(r(L)\) convex, even more curvature).

This convexity is the source of the Jensen bias in private market valuation.

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42.8 A.8 Game Theory: Nash Equilibrium and Mean-Field

42.8.1 A.8.1 Nash Equilibrium

A Nash equilibrium for a game with \(N\) players is a profile \((s_1^*, ..., s_N^*)\) such that: \[U_i(s_i^*, s_{-i}^*) \geq U_i(s_i, s_{-i}^*) \quad \forall s_i, \forall i\]

Each player’s strategy is a best response given others’ strategies; mutually consistent.

Existence (Nash 1950): Every finite game has at least one Nash equilibrium in mixed strategies. For games with continuous strategies, additional regularity needed.

42.8.2 A.8.2 Mean-Field Approximation

When \(N\) is large, tracking individual strategies is intractable. Replace with the distribution of strategies across the population: \[\mu_t = \text{distribution of } (s_t^i) \text{ across } i\]

Each player reacts to \(\mu_t\) rather than to individual others. As \(N \to \infty\), this becomes exact (propagation of chaos).

42.8.3 A.8.3 Fixed-Point Structure

A mean-field equilibrium is a distribution \(\mu^*\) such that: 1. Each player’s optimal strategy, given \(\mu^*\), is \(s^*(\mu^*)\) 2. The aggregate distribution of \(s^*(\mu^*)\) across the population equals \(\mu^*\)

This is a fixed point: \(\mu^* = \Phi(\mu^*)\) where \(\Phi\) is the mapping from population state to individual best response back to aggregate.

Existence: Schauder fixed-point theorem (for compact convex sets in topological vector spaces).

Uniqueness: Banach fixed-point theorem under contraction (stability condition).

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42.9 A.9 Optimization and Lagrange Methods

42.9.1 A.9.1 Unconstrained Optimization

For smooth \(f: \mathbb{R}^n \to \mathbb{R}\), an interior critical point satisfies \(\nabla f = 0\).

Second-order condition: Hessian \(\nabla^2 f\) positive (negative) definite for local minimum (maximum).

42.9.2 A.9.2 Constrained Optimization

For \(\max f(x)\) subject to \(g(x) = 0\): Lagrangian: \(L(x, \lambda) = f(x) - \lambda g(x)\)

First-order conditions: \(\nabla_x L = 0\), \(\nabla_\lambda L = 0\) (i.e., \(g(x) = 0\)).

42.9.3 A.9.3 Variational Methods

For functionals \(J[u] = \int F(u, u', t) dt\), the Euler-Lagrange equation: \[\frac{\partial F}{\partial u} - \frac{d}{dt}\frac{\partial F}{\partial u'} = 0\]

The HJB equation arises from applying variational methods to the value function in optimal control.

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42.10 A.10 Quick Reference Formulas

42.10.1 A.10.1 OU Calibration

At canonical values \(\kappa = 0.45\), \(\sigma = 0.32\): - Stationary variance: \(0.1138\) - Stationary std: \(0.337\) - Half-life: \(1.54\) yr - \(1 - e^{-\kappa t}\) at \(t = 5\): \(0.895\) - \(1 - e^{-2\kappa t}\) at \(t = 5\): \(0.989\)

42.10.2 A.10.2 Jensen Bias

• Affine approximation: \(B(T) = A \cdot T + C\) with \(A \approx 0.16-0.18\%\)/yr by asset class
• Closed-form: \(B(T) = (\pi_2/2) \Pi_{liq}(T)\) where \(\Pi_{liq}(T) = \sigma^2[T - (1-e^{-2\kappa T})/(2\kappa)]\)
• At calibrated \(\pi_2 = 0.021\): \(B(5) \approx 0.4\%\), \(B(15) \approx 1.5\%\), \(B(20) \approx 2.0\%\)

42.10.3 A.10.3 Effective Rate

• DCF rate: typically constant at \(r_f + \beta \cdot ERP + \pi_0 \approx 7.5\%\)
• LAV rate: path-dependent at \(r(L_t) = r_f + \pi(L_t)\)
• GE-LAV rate: \(r_{GE}(L, \mu) = r(L) + \text{equilibrium uplift}\)
• At GFC depth (\(L = -1.5\)): \(r_{GE} \approx 32\%\) vs. DCF \(7.5\%\) → 4.31× amplification

42.10.4 A.10.4 Welfare

• Welfare gap: \(\Delta W \approx 2.3\%\)/yr
• Aggregate: \(\$13T \times 2.3\% = \$300B/\)yr
• Pigouvian tax at GFC depth: \(\tau^* \approx 7\%\) on secondary transactions

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42.11 A.11 Mathematical Maturity Required for Each Track

42.11.1 Track 1 (Practitioner)

Required: - Probability theory through expectations and variances (A.1) - Comfortable with the normal distribution (A.1.2) - Aware of Jensen’s inequality and its direction (A.1.3, A.7) - Basic familiarity with stochastic processes as concepts (A.2)

Recommended but not required: - Recognition of SDE notation - Awareness of HJB equation as a concept (not derivation) - Comfort reading mathematical statements in lectures

42.11.2 Track 2 (Researcher)

Required (in addition to Track 1): - Itô calculus, including Itô’s lemma derivation (A.4) - OU process derivation and properties (A.5) - Variational inequalities and free boundary problems (A.6) - Convex analysis (A.7) - Game theory and mean-field methods (A.8)

Helpful: - Functional analysis (Hilbert spaces, weak topologies) - Numerical analysis (finite differences, iterative methods) - Measure theory (Lebesgue integration, Radon-Nikodym)

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42.12 A.12 Suggested Self-Diagnostic

Before the course (or by Session 4 at the latest), confirm understanding of:

For Track 1: - [ ] Can you state Jensen’s inequality and identify whether \(E[X^2] \geq (E[X])^2\) is consistent with it? - [ ] If \(X \sim \mathcal{N}(0, 1)\), what is \(\mathbb{P}(X > 1.96)\)? Approximately? - [ ] If a random variable has mean 0 and variance 9, what is its standard deviation? - [ ] What is a Markov chain, informally?

For Track 2: - [ ] Can you derive Itô’s lemma from a Taylor expansion? - [ ] Can you compute \(E\left[\int_0^t e^{-\kappa(t-s)} dW_s\right]\) using the Itô isometry? - [ ] Can you state the HJB equation for a generic control problem? - [ ] What is a fixed-point theorem? Name one.

If you cannot answer the Track 1 questions, the practitioner track will be challenging — additional preparation recommended. If you cannot answer the Track 2 questions, the researcher track will be very challenging — strongly recommend remediation first.

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