Sublinear Expectations¶

Introduction¶

Classical probability theory is built on a single probability measure $P$, which assigns precise probabilities to all events. In many settings --- particularly in finance under model uncertainty --- no single probability measure is known or trusted. Sublinear expectations, developed systematically by Shige Peng beginning in the mid-2000s, provide a rigorous mathematical framework for probability theory without a fixed probability measure.

The central idea is to replace the linear expectation operator $E_P[\cdot]$ with a sublinear operator $\hat{\mathbb{E}}[\cdot]$ that satisfies monotonicity, constant preservation, sub-additivity, and positive homogeneity. This framework naturally captures the worst-case evaluation over a family of probability measures, connecting directly to coherent risk measures, robust pricing, and ambiguity aversion.

Peng's crowning achievement is the construction of G-Brownian motion and the associated G-expectation, which provides a complete analog of classical stochastic calculus (Ito formula, martingale theory, stochastic differential equations) in a nonlinear setting where the volatility is uncertain.

Axioms of Sublinear Expectation¶

1. Definition and Basic Properties¶

Definition (Sublinear Expectation Space): A sublinear expectation space is a triple $(\Omega, \mathcal{H}, \hat{\mathbb{E}})$ where:

$\Omega$ is a sample space
$\mathcal{H}$ is a linear space of real-valued functions on $\Omega$ (interpreted as random variables), containing all constants, and closed under $\varphi(X_1, \ldots, X_n)$ for $X_1, \ldots, X_n \in \mathcal{H}$ and $\varphi \in C_{b,\text{Lip}}(\mathbb{R}^n)$
$\hat{\mathbb{E}}: \mathcal{H} \to \mathbb{R}$ is a functional satisfying:

(A1) Monotonicity: $X \geq Y$ implies $\hat{\mathbb{E}}[X] \geq \hat{\mathbb{E}}[Y]$

(A2) Constant preservation: $\hat{\mathbb{E}}[c] = c$ for all $c \in \mathbb{R}$

(A3) Sub-additivity: $\hat{\mathbb{E}}[X + Y] \leq \hat{\mathbb{E}}[X] + \hat{\mathbb{E}}[Y]$

(A4) Positive homogeneity: $\hat{\mathbb{E}}[\lambda X] = \lambda \hat{\mathbb{E}}[X]$ for all $\lambda \geq 0$

Remark: Axioms (A3) and (A4) together imply that $\hat{\mathbb{E}}$ is a sublinear functional in the sense of convex analysis. If (A3) is replaced by additivity, the functional becomes linear and reduces to a classical expectation.

2. Conjugate Expectation and Symmetry¶

Definition: The conjugate expectation is:

\[ \hat{\mathcal{E}}[X] = -\hat{\mathbb{E}}[-X] \]

Properties: The conjugate expectation is superlinear: $\hat{\mathcal{E}}[X + Y] \geq \hat{\mathcal{E}}[X] + \hat{\mathcal{E}}[Y]$.

For any $X \in \mathcal{H}$:

\[ \hat{\mathcal{E}}[X] \leq \hat{\mathbb{E}}[X] \]

with equality if and only if $\hat{\mathbb{E}}$ is linear on the subspace generated by $X$.

Financial Interpretation: $\hat{\mathbb{E}}[X]$ represents the upper price (worst-case expected payoff from the seller's perspective), while $\hat{\mathcal{E}}[X]$ represents the lower price (best-case from the seller's perspective, or worst-case from the buyer's). The gap $\hat{\mathbb{E}}[X] - \hat{\mathcal{E}}[X]$ measures the ambiguity spread.

3. Representation Theorem¶

Theorem (Representation via Linear Expectations): Let $\hat{\mathbb{E}}$ be a sublinear expectation on $\mathcal{H}$. Under regularity conditions (e.g., $\mathcal{H}$ is a lattice and $\hat{\mathbb{E}}$ is continuous from above), there exists a family $\mathcal{P}$ of probability measures on $\Omega$ such that:

\[ \hat{\mathbb{E}}[X] = \sup_{P \in \mathcal{P}} E_P[X] \]

for all $X \in \mathcal{H}$.

Proof sketch: This follows from the Hahn-Banach theorem and Riesz representation. The set $\mathcal{P}$ consists of all linear functionals dominated by $\hat{\mathbb{E}}$:

\[ \mathcal{P} = \{P : E_P[X] \leq \hat{\mathbb{E}}[X] \text{ for all } X \in \mathcal{H}, \; E_P[1] = 1, \; E_P[X] \geq 0 \text{ for } X \geq 0\} \]

The supremum is attained by a compactness argument when $\mathcal{H}$ is suitably topologized. $\square$

Significance: This theorem establishes the equivalence between sublinear expectations and worst-case evaluation over sets of probability measures, connecting Peng's axiomatic framework to the multiple-priors model of Gilboa and Schmeidler.

G-Normal Distribution¶

1. Definition¶

The G-normal distribution is the analog of the Gaussian distribution in the sublinear expectation framework.

Definition: A random variable $X$ on a sublinear expectation space $(\Omega, \mathcal{H}, \hat{\mathbb{E}})$ is G-normally distributed, written $X \sim \mathcal{N}(0, [\underline{\sigma}^2, \overline{\sigma}^2])$, if for any independent copy $Y$ of $X$ (in the sense of sublinear expectations):

\[ aX + bY \sim \sqrt{a^2 + b^2} \, X \]

for all $a, b \geq 0$, and the sublinear expectation of $\varphi(X)$ is characterized by the function:

\[ G(\alpha) = \frac{1}{2}\hat{\mathbb{E}}[\alpha X^2] = \frac{1}{2}\left(\overline{\sigma}^2 \alpha^+ - \underline{\sigma}^2 \alpha^-\right) \]

where $\alpha^+ = \max(\alpha, 0)$ and $\alpha^- = \max(-\alpha, 0)$.

2. Characterization via Nonlinear PDE¶

Theorem (Peng): Let $X \sim \mathcal{N}(0, [\underline{\sigma}^2, \overline{\sigma}^2])$. Then for $\varphi \in C_{b,\text{Lip}}(\mathbb{R})$:

\[ \hat{\mathbb{E}}[\varphi(X)] = u(1, 0) \]

where $u(t, x)$ solves the G-heat equation:

\[ \frac{\partial u}{\partial t} = G\left(\frac{\partial^2 u}{\partial x^2}\right) = \frac{1}{2}\left(\overline{\sigma}^2 \left(\frac{\partial^2 u}{\partial x^2}\right)^+ - \underline{\sigma}^2 \left(\frac{\partial^2 u}{\partial x^2}\right)^-\right) \]

with initial condition $u(0, x) = \varphi(x)$.

Proof: Define $u(t, x) = \hat{\mathbb{E}}[\varphi(x + \sqrt{t} X)]$. The self-similar structure of the G-normal distribution implies that $u$ satisfies the dynamic programming principle:

\[ u(t+s, x) = \hat{\mathbb{E}}[u(t, x + \sqrt{s} X)] \]

Taylor-expanding $u(t, x + \sqrt{s}X)$ to second order and using the definition of $G$ yields the G-heat equation as $s \to 0$. $\square$

Connection to Uncertain Volatility: The G-heat equation is precisely the Black-Scholes-Barenblatt equation from the uncertain volatility model: the worst-case evaluation of $\varphi(X)$ when $X$ has Gaussian distribution with unknown variance in $[\underline{\sigma}^2, \overline{\sigma}^2]$ selects $\overline{\sigma}^2$ when $\varphi$ is convex (positive second derivative) and $\underline{\sigma}^2$ when $\varphi$ is concave.

3. Special Cases¶

Case 1: $\underline{\sigma} = \overline{\sigma} = \sigma$: The G-normal distribution reduces to the classical normal $\mathcal{N}(0, \sigma^2)$, and the G-heat equation becomes the standard heat equation.

Case 2: $\underline{\sigma} = 0$, $\overline{\sigma} = \sigma$: This is the maximal distribution, corresponding to maximum ambiguity about whether volatility is present.

Case 3: General $[\underline{\sigma}, \overline{\sigma}]$: For a convex function $\varphi$:

\[ \hat{\mathbb{E}}[\varphi(X)] = E[\varphi(\overline{\sigma} Z)], \quad Z \sim \mathcal{N}(0,1) \]

and for a concave function $\varphi$:

\[ \hat{\mathbb{E}}[\varphi(X)] = E[\varphi(\underline{\sigma} Z)] \]

For general $\varphi$, the evaluation involves switching between volatilities according to the local convexity of $\varphi$.

G-Brownian Motion¶

1. Construction¶

Definition (G-Brownian Motion): A process $(B_t)_{t \geq 0}$ on a sublinear expectation space $(\Omega, \mathcal{H}, \hat{\mathbb{E}})$ is a G-Brownian motion if:

$B_0 = 0$
For each $t, s \geq 0$, the increment $B_{t+s} - B_t$ is independent of $(B_{t_1}, \ldots, B_{t_n})$ for $t_1 \leq \cdots \leq t_n \leq t$
$B_{t+s} - B_t \sim \sqrt{s} X$ where $X \sim \mathcal{N}(0, [\underline{\sigma}^2, \overline{\sigma}^2])$

Independence in the sublinear expectation sense means: $Y$ is independent of $X$ if for all $\varphi \in C_{b,\text{Lip}}(\mathbb{R}^2)$:

\[ \hat{\mathbb{E}}[\varphi(X, Y)] = \hat{\mathbb{E}}[\hat{\mathbb{E}}[\varphi(x, Y)]_{x=X}] \]

2. Properties of G-Brownian Motion¶

Theorem (Peng, 2007): G-Brownian motion has the following properties:

Continuous paths: $t \mapsto B_t(\omega)$ is continuous for quasi-surely all $\omega$
Stationary increments: $B_{t+s} - B_t$ has the same distribution as $B_s$
Quadratic variation: The quadratic variation process $\langle B \rangle_t$ exists and satisfies $\underline{\sigma}^2 t \leq \langle B \rangle_t \leq \overline{\sigma}^2 t$
Non-deterministic quadratic variation: Unlike classical Brownian motion, $\langle B \rangle_t$ is not deterministic; it is itself a stochastic process

Key Difference from Classical Brownian Motion: Property (4) is the essential distinction. For standard Brownian motion, $\langle B \rangle_t = t$ deterministically. For G-Brownian motion, $\langle B \rangle_t$ varies between $\underline{\sigma}^2 t$ and $\overline{\sigma}^2 t$, reflecting the volatility uncertainty. The process $\langle B \rangle_t - \underline{\sigma}^2 t$ is an increasing process, and $\overline{\sigma}^2 t - \langle B \rangle_t$ is also increasing, so $\langle B \rangle_t$ is constrained but not determined.

3. G-Ito Calculus¶

Theorem (G-Ito Formula): For $\Phi \in C^2(\mathbb{R})$ and G-Brownian motion $B_t$:

\[ \Phi(B_t) = \Phi(B_0) + \int_0^t \Phi'(B_s) \, dB_s + \frac{1}{2} \int_0^t \Phi''(B_s) \, d\langle B \rangle_s \]

where the stochastic integral $\int_0^t \Phi'(B_s) \, dB_s$ is defined through the G-expectation framework and the quadratic variation integral $\int_0^t \Phi''(B_s) \, d\langle B \rangle_s$ accounts for the uncertain volatility.

G-Martingale: A process $M_t$ is a G-martingale if:

\[ \hat{\mathbb{E}}[M_t \mid \mathcal{F}_s] = M_s \quad \text{for all } s \leq t \]

where the conditional G-expectation is defined through the G-BSDE theory.

Symmetric G-Martingale: $M_t$ is a symmetric G-martingale if both $M_t$ and $-M_t$ are G-martingales, meaning:

\[ \hat{\mathbb{E}}[M_t \mid \mathcal{F}_s] = \hat{\mathcal{E}}[M_t \mid \mathcal{F}_s] = M_s \]

The stochastic integral $\int_0^t \theta_s \, dB_s$ is a symmetric G-martingale, while $B_t$ itself is a symmetric G-martingale but $B_t^2 - \overline{\sigma}^2 t$ is a G-supermartingale (not a G-martingale unless $\underline{\sigma} = \overline{\sigma}$).

Capacity and Quasi-Sure Analysis¶

1. From Sublinear Expectation to Capacity¶

Since the sublinear expectation is not additive, it does not define a probability measure. Instead, it defines a capacity.

Definition (Capacity): The capacity associated with $\hat{\mathbb{E}}$ is:

\[ c(A) = \hat{\mathbb{E}}[\mathbb{1}_A] = \sup_{P \in \mathcal{P}} P(A) \]

for events $A \in \mathcal{F}$.

Properties:

$c(\emptyset) = 0$, $c(\Omega) = 1$
$c(A \cup B) \leq c(A) + c(B)$ (sub-additivity, but generally strict inequality)
$c$ is not additive: $c(A) + c(A^c) \geq 1$ in general

2. Quasi-Sure Properties¶

Definition: A property holds quasi-surely (q.s.) if the set where it fails has capacity zero:

\[ c(\{\omega : \text{property fails}\}) = 0 \]

Equivalently, the property holds $P$-almost surely for every $P \in \mathcal{P}$.

Example: G-Brownian motion has continuous paths quasi-surely, meaning that for every probability measure $P$ in the representing family $\mathcal{P}$, the paths are $P$-a.s. continuous.

3. Choquet Expectation¶

The capacity $c$ can be extended to a nonlinear integral via the Choquet integral:

\[ \int_\Omega X \, dc = \int_0^\infty c(\{X > t\}) \, dt + \int_{-\infty}^0 [c(\{X > t\}) - 1] \, dt \]

The Choquet integral provides an alternative nonlinear expectation that is comonotonic additive but generally different from $\hat{\mathbb{E}}$ (which is sublinear but not comonotonic additive).

Limit Theorems Under Sublinear Expectations¶

1. Central Limit Theorem¶

Theorem (Peng, 2008): Let $\{X_i\}_{i=1}^\infty$ be a sequence of i.i.d. random variables under a sublinear expectation $\hat{\mathbb{E}}$ with:

\[ \hat{\mathbb{E}}[X_1] = \hat{\mathcal{E}}[X_1] = 0, \quad \hat{\mathbb{E}}[X_1^2] = \overline{\sigma}^2, \quad \hat{\mathcal{E}}[X_1^2] = \underline{\sigma}^2 \]

Then:

\[ \frac{X_1 + X_2 + \cdots + X_n}{\sqrt{n}} \xrightarrow{d} \mathcal{N}(0, [\underline{\sigma}^2, \overline{\sigma}^2]) \]

in the sense that for all $\varphi \in C_{b,\text{Lip}}(\mathbb{R})$:

\[ \hat{\mathbb{E}}\left[\varphi\left(\frac{S_n}{\sqrt{n}}\right)\right] \to \hat{\mathbb{E}}[\varphi(X)] \]

where $X \sim \mathcal{N}(0, [\underline{\sigma}^2, \overline{\sigma}^2])$.

Significance: This is a profound generalization of the classical CLT. The limiting distribution is G-normal rather than Gaussian, reflecting the irreducible uncertainty in the variance. Even with infinitely many observations, if the variance is uncertain, the limit retains this uncertainty.

Proof sketch: Define $u(t,x)$ as the solution to the G-heat equation with initial condition $\varphi$. The key estimate is:

\[ \left|\hat{\mathbb{E}}\left[\varphi\left(\frac{S_n}{\sqrt{n}}\right)\right] - u(1, 0)\right| \leq \frac{C}{\sqrt{n}} \]

which follows from a nonlinear version of Lindeberg's method, replacing each $X_i/\sqrt{n}$ with a G-normal increment and bounding the error using the regularity of $u$. $\square$

2. Law of Large Numbers¶

Theorem (Peng, 2008): Under the same setup, but with $\hat{\mathbb{E}}[X_1] = \overline{\mu}$ and $\hat{\mathcal{E}}[X_1] = \underline{\mu}$ (not necessarily zero):

\[ \frac{X_1 + X_2 + \cdots + X_n}{n} \xrightarrow{d} \mathcal{M}([\underline{\mu}, \overline{\mu}]) \]

where $\mathcal{M}([\underline{\mu}, \overline{\mu}])$ is the maximal distribution supported on $[\underline{\mu}, \overline{\mu}]$:

\[ \hat{\mathbb{E}}[\varphi(X)] = \sup_{\mu \in [\underline{\mu}, \overline{\mu}]} \varphi(\mu) \]

for all $\varphi \in C_{b,\text{Lip}}(\mathbb{R})$.

Interpretation: The law of large numbers under sublinear expectations does not concentrate the average at a single point. Instead, the limit is a maximal distribution over the interval of possible means, reflecting the persistent uncertainty about the true drift.

Connection to Financial Mathematics¶

1. Uncertain Volatility and G-Expectation¶

The G-expectation framework provides a rigorous foundation for the uncertain volatility model of Avellaneda, Levy, and Paras (1995) and Lyons (1995).

Option Pricing: For a European option with payoff $\varphi(S_T)$ under uncertain volatility $\sigma_t \in [\underline{\sigma}, \overline{\sigma}]$:

\[ \text{Robust price} = \hat{\mathbb{E}}[\varphi(S_T)] = u(0, S_0) \]

where $u$ solves the fully nonlinear PDE:

\[ \frac{\partial u}{\partial t} + \frac{1}{2}\Sigma\left(\frac{\partial^2 u}{\partial S^2}\right)^2 S^2 \frac{\partial^2 u}{\partial S^2} + rS\frac{\partial u}{\partial S} - ru = 0 \]

with

\[ \Sigma(\Gamma) = \begin{cases} \overline{\sigma} & \text{if } \Gamma \geq 0 \\ \underline{\sigma} & \text{if } \Gamma < 0 \end{cases} \]

and terminal condition $u(T, S) = \varphi(S)$.

Example: For a European call $\varphi(S) = (S - K)^+$ (convex payoff, $\Gamma \geq 0$), the robust price equals the Black-Scholes price at the maximum volatility $\overline{\sigma}$. For a European put combined with a covered call (concave near the strike), the effective volatility switches.

2. Risk Measures and Coherence¶

The G-expectation defines a coherent risk measure:

\[ \rho(X) = \hat{\mathbb{E}}[-X] \]

Verification of Coherence Axioms:

Monotonicity: $X \leq Y \implies \rho(X) \geq \rho(Y)$ (from (A1))
Translation invariance: $\rho(X + c) = \rho(X) - c$ (from (A2))
Sub-additivity: $\rho(X + Y) \leq \rho(X) + \rho(Y)$ (from (A3))
Positive homogeneity: $\rho(\lambda X) = \lambda \rho(X)$ for $\lambda \geq 0$ (from (A4))

Dynamic Consistency: The conditional G-expectation $\hat{\mathbb{E}}_t[X] = \hat{\mathbb{E}}[X | \mathcal{F}_t]$ satisfies the tower property:

\[ \hat{\mathbb{E}}_s[\hat{\mathbb{E}}_t[X]] = \hat{\mathbb{E}}_s[X] \quad \text{for } s \leq t \]

This dynamic consistency is a key advantage over many ad hoc ambiguity models and ensures time-consistent decision making.

3. G-BSDEs and Nonlinear Pricing¶

Definition (G-BSDE): A G-backward stochastic differential equation is:

\[ Y_t = \xi + \int_t^T f(s, Y_s, Z_s) \, ds + \int_t^T g(s, Y_s, Z_s) \, d\langle B \rangle_s - \int_t^T Z_s \, dB_s - (K_T - K_t) \]

where $(Y_t, Z_t, K_t)$ is the solution triple and $K_t$ is a decreasing G-martingale accounting for the nonlinearity.

Financial Interpretation: The solution $Y_0$ of a G-BSDE with terminal condition $\xi = \varphi(S_T)$ gives the robust price of the claim $\varphi$ under volatility uncertainty, with $Z_t$ providing the hedging strategy and $K_t$ representing the "ambiguity premium" from model uncertainty.

1. Sublinear Expectations vs g-Expectations¶

Feature	g-Expectation	G-Expectation
Probability space	Fixed $(\Omega, \mathcal{F}, P)$	No reference measure
Generator	$g(t, y, z)$	$G(\alpha) = \frac{1}{2}(\overline{\sigma}^2 \alpha^+ - \underline{\sigma}^2 \alpha^-)$
Nonlinearity source	Drift uncertainty	Volatility uncertainty
PDE connection	Quasilinear	Fully nonlinear
Stochastic integral	Classical Ito	G-Ito

2. Sublinear Expectations vs Multiple Priors¶

The representation theorem establishes that every sublinear expectation corresponds to a worst-case evaluation over a set of priors $\mathcal{P}$. However, the sublinear expectation framework offers advantages:

Constructive: G-Brownian motion and G-Ito calculus provide explicit computational tools
Limit theorems: The G-CLT and G-LLN have no direct analog in the multiple-priors framework
Dynamic consistency: Built into the framework via the tower property

3. Sublinear Expectations vs Choquet Expectations¶

Choquet expectations (based on non-additive capacities) are comonotonic additive but generally not sublinear. Conversely, sublinear expectations are generally not comonotonic additive. The two frameworks coincide only in special cases (e.g., when the capacity is 2-alternating).

Numerical Example¶

Setup: Consider a G-Brownian motion with $\underline{\sigma} = 0.15$, $\overline{\sigma} = 0.25$, and a European call option with $S_0 = 100$, $K = 100$, $T = 1$, $r = 0$.

G-expectation price: Since the call payoff is convex:

\[ \hat{\mathbb{E}}[(S_T - K)^+] = C_{\text{BS}}(S_0, K, \overline{\sigma}, T) \approx \$9.95 \]

Conjugate expectation price: Since the call payoff is convex, the lower bound uses $\underline{\sigma}$:

\[ \hat{\mathcal{E}}[(S_T - K)^+] = C_{\text{BS}}(S_0, K, \underline{\sigma}, T) \approx \$5.97 \]

Ambiguity spread: $\$9.95 - \$5.97 = \$3.98$, representing the price range due to volatility uncertainty.

Butterfly spread (concave in the middle): For $\varphi(S) = (S-95)^+ - 2(S-100)^+ + (S-105)^+$, the effective volatility switches between $\overline{\sigma}$ and $\underline{\sigma}$ depending on the moneyness region, requiring numerical solution of the G-heat equation.

Summary and Key Takeaways¶

Axiomatic foundation: Sublinear expectations generalize classical probability through four axioms (monotonicity, constant preservation, sub-additivity, positive homogeneity), providing a rigorous framework for model uncertainty
Representation theorem: Every sublinear expectation equals the worst-case expectation over a family of probability measures, connecting to the multiple-priors and coherent risk measure frameworks
G-normal distribution: The analog of the Gaussian distribution under volatility uncertainty, characterized by the G-heat equation (a fully nonlinear PDE equivalent to the Black-Scholes-Barenblatt equation)
G-Brownian motion: A process with uncertain quadratic variation, providing the foundation for stochastic calculus under volatility uncertainty with a complete G-Ito theory
Nonlinear limit theorems: The G-CLT converges to a G-normal distribution (not a point), and the G-LLN converges to a maximal distribution (not a constant), reflecting persistent uncertainty
Financial applications: G-expectation provides the rigorous mathematical underpinning for uncertain volatility pricing, coherent dynamic risk measures, and robust derivative valuation
Dynamic consistency: The conditional G-expectation satisfies the tower property, ensuring time-consistent evaluation --- a crucial property for dynamic hedging and risk management

Exercises¶

Exercise 1. Verify the four axioms of a sublinear expectation for $\hat{\mathbb{E}}[X] = \sup_{P \in \mathcal{P}} \mathbb{E}_P[X]$ where $\mathcal{P}$ is a convex set of probability measures. Specifically, check: (a) monotonicity, (b) constant preservation $\hat{\mathbb{E}}[c] = c$, (c) sub-additivity $\hat{\mathbb{E}}[X + Y] \leq \hat{\mathbb{E}}[X] + \hat{\mathbb{E}}[Y]$, and (d) positive homogeneity $\hat{\mathbb{E}}[\lambda X] = \lambda \hat{\mathbb{E}}[X]$ for $\lambda > 0$.

Solution to Exercise 1

Goal. Verify the four axioms of a sublinear expectation for $\hat{\mathbb{E}}[X] = \sup_{P \in \mathcal{P}} E_P[X]$ where $\mathcal{P}$ is a convex set of probability measures.

(a) Monotonicity: Let $X \geq Y$ (pointwise, i.e., $X(\omega) \geq Y(\omega)$ for all $\omega$). For every $P \in \mathcal{P}$, the linear expectation $E_P$ is monotone, so $E_P[X] \geq E_P[Y]$. Taking the supremum over $P \in \mathcal{P}$:

\[ \hat{\mathbb{E}}[X] = \sup_{P \in \mathcal{P}} E_P[X] \geq \sup_{P \in \mathcal{P}} E_P[Y] = \hat{\mathbb{E}}[Y] \]

since each term in the first supremum dominates the corresponding term in the second. $\checkmark$

(b) Constant preservation: Let $c \in \mathbb{R}$ be a constant. For every $P \in \mathcal{P}$, $E_P[c] = c$ (since $c$ is deterministic). Therefore:

\[ \hat{\mathbb{E}}[c] = \sup_{P \in \mathcal{P}} E_P[c] = \sup_{P \in \mathcal{P}} c = c \]

since the supremum of a constant over any non-empty set is that constant. $\checkmark$

(c) Sub-additivity: For any $P \in \mathcal{P}$, by linearity of $E_P$:

\[ E_P[X + Y] = E_P[X] + E_P[Y] \leq \sup_{P' \in \mathcal{P}} E_{P'}[X] + \sup_{P'' \in \mathcal{P}} E_{P''}[Y] = \hat{\mathbb{E}}[X] + \hat{\mathbb{E}}[Y] \]

The inequality holds because $E_P[X] \leq \sup_{P'} E_{P'}[X]$ and $E_P[Y] \leq \sup_{P''} E_{P''}[Y]$ for each fixed $P$. Taking the supremum over $P$ on the left side:

\[ \hat{\mathbb{E}}[X + Y] = \sup_{P \in \mathcal{P}} E_P[X + Y] = \sup_{P \in \mathcal{P}} \{E_P[X] + E_P[Y]\} \leq \hat{\mathbb{E}}[X] + \hat{\mathbb{E}}[Y] \]

Note: convexity of $\mathcal{P}$ is not needed for sub-additivity; this holds for any set $\mathcal{P}$. The key is that the supremum of a sum is at most the sum of the suprema, since the optimizer for $X + Y$ jointly may not be the best individual optimizer for $X$ or $Y$ separately. $\checkmark$

(d) Positive homogeneity: Let $\lambda > 0$. For every $P \in \mathcal{P}$, $E_P[\lambda X] = \lambda E_P[X]$ by linearity. Therefore:

\[ \hat{\mathbb{E}}[\lambda X] = \sup_{P \in \mathcal{P}} E_P[\lambda X] = \sup_{P \in \mathcal{P}} \lambda E_P[X] = \lambda \sup_{P \in \mathcal{P}} E_P[X] = \lambda \hat{\mathbb{E}}[X] \]

The third equality uses the fact that for $\lambda > 0$, $\sup_P \lambda f(P) = \lambda \sup_P f(P)$ (scaling commutes with supremum for positive scalars). $\checkmark$

All four axioms are verified, confirming that $\hat{\mathbb{E}}[X] = \sup_{P \in \mathcal{P}} E_P[X]$ defines a sublinear expectation. $\square$

Exercise 2. The G-normal distribution $X \sim \mathcal{N}(0, [\underline{\sigma}^2, \overline{\sigma}^2])$ satisfies $\hat{\mathbb{E}}[\varphi(X)] = u(1, 0)$ where $u$ solves the G-heat equation $\partial_t u = G(\partial_{xx} u)$ with $G(a) = \frac{1}{2}(\overline{\sigma}^2 a^+ - \underline{\sigma}^2 a^-)$. For $\varphi(x) = x^2$, compute $\hat{\mathbb{E}}[X^2]$ and $\hat{\mathbb{E}}[-X^2]$. Verify that $\hat{\mathbb{E}}[X^2] = \overline{\sigma}^2$ and $\hat{\mathbb{E}}[-X^2] = -\underline{\sigma}^2$.

Solution to Exercise 2

Goal. Compute $\hat{\mathbb{E}}[X^2]$ and $\hat{\mathbb{E}}[-X^2]$ for $X \sim \mathcal{N}(0, [\underline{\sigma}^2, \overline{\sigma}^2])$.

Setup. The G-normal distribution is characterized by $\hat{\mathbb{E}}[\varphi(X)] = u(1, 0)$ where $u$ solves the G-heat equation:

\[ \frac{\partial u}{\partial t} = G\left(\frac{\partial^2 u}{\partial x^2}\right), \quad u(0, x) = \varphi(x) \]

with $G(a) = \frac{1}{2}(\overline{\sigma}^2 a^+ - \underline{\sigma}^2 a^-)$.

Computation of $\hat{\mathbb{E}}[X^2]$: Set $\varphi(x) = x^2$. We look for a solution of the form $u(t, x) = x^2 + c \cdot t$ for some constant $c$. Then:

\[ \frac{\partial u}{\partial t} = c, \quad \frac{\partial^2 u}{\partial x^2} = 2 \]

The G-heat equation requires $c = G(2)$. Since $2 > 0$:

\[ G(2) = \frac{1}{2}\overline{\sigma}^2 \cdot 2 = \overline{\sigma}^2 \]

So $u(t, x) = x^2 + \overline{\sigma}^2 t$, and:

\[ \hat{\mathbb{E}}[X^2] = u(1, 0) = 0 + \overline{\sigma}^2 = \overline{\sigma}^2 \]

Computation of $\hat{\mathbb{E}}[-X^2]$: Set $\varphi(x) = -x^2$. Look for $u(t, x) = -x^2 + c \cdot t$. Then:

\[ \frac{\partial u}{\partial t} = c, \quad \frac{\partial^2 u}{\partial x^2} = -2 \]

The G-heat equation requires $c = G(-2)$. Since $-2 < 0$:

\[ G(-2) = -\frac{1}{2}\underline{\sigma}^2 \cdot 2 = -\underline{\sigma}^2 \]

So $u(t, x) = -x^2 - \underline{\sigma}^2 t$, and:

\[ \hat{\mathbb{E}}[-X^2] = u(1, 0) = 0 - \underline{\sigma}^2 = -\underline{\sigma}^2 \]

Verification. Using the representation $\hat{\mathbb{E}}[X] = \sup_{P \in \mathcal{P}} E_P[X]$ where under each $P \in \mathcal{P}$, $X$ has distribution $N(0, \sigma_P^2)$ with $\sigma_P \in [\underline{\sigma}, \overline{\sigma}]$:

$\hat{\mathbb{E}}[X^2] = \sup_{\sigma \in [\underline{\sigma}, \overline{\sigma}]} E[(\sigma Z)^2] = \sup_{\sigma \in [\underline{\sigma}, \overline{\sigma}]} \sigma^2 = \overline{\sigma}^2$ $\checkmark$
$\hat{\mathbb{E}}[-X^2] = \sup_{\sigma \in [\underline{\sigma}, \overline{\sigma}]} E[-(\sigma Z)^2] = \sup_{\sigma \in [\underline{\sigma}, \overline{\sigma}]} (-\sigma^2) = -\underline{\sigma}^2$ $\checkmark$

Note the key asymmetry: $\hat{\mathbb{E}}[X^2] = \overline{\sigma}^2$ uses the maximum volatility (worst case for the "long variance" position), while $\hat{\mathbb{E}}[-X^2] = -\underline{\sigma}^2$ uses the minimum volatility (worst case for the "short variance" position). Furthermore, $\hat{\mathbb{E}}[X^2] + \hat{\mathbb{E}}[-X^2] = \overline{\sigma}^2 - \underline{\sigma}^2 > 0$ when $\underline{\sigma} < \overline{\sigma}$, showing that $\hat{\mathbb{E}}$ is genuinely nonlinear (a linear expectation would give $\hat{\mathbb{E}}[X^2] + \hat{\mathbb{E}}[-X^2] = 0$). $\square$

Exercise 3. Prove the representation theorem for sublinear expectations: if $\hat{\mathbb{E}}$ is a sublinear expectation on a finite state space $\Omega = \{\omega_1, \ldots, \omega_n\}$, then there exists a compact convex set $\mathcal{P}$ of probability measures such that $\hat{\mathbb{E}}[X] = \max_{P \in \mathcal{P}} \mathbb{E}_P[X]$. Hint: use the supporting hyperplane theorem.

Solution to Exercise 3

Goal. Prove the representation theorem on a finite state space $\Omega = \{\omega_1, \ldots, \omega_n\}$.

Step 1: Identify random variables with $\mathbb{R}^n$. On $\Omega = \{\omega_1, \ldots, \omega_n\}$, a random variable $X$ is a vector $\mathbf{x} = (x_1, \ldots, x_n) \in \mathbb{R}^n$ where $x_i = X(\omega_i)$. The sublinear expectation $\hat{\mathbb{E}}: \mathbb{R}^n \to \mathbb{R}$ is a functional satisfying:

(A1) $\mathbf{x} \geq \mathbf{y}$ componentwise $\implies$ $\hat{\mathbb{E}}[\mathbf{x}] \geq \hat{\mathbb{E}}[\mathbf{y}]$
(A2) $\hat{\mathbb{E}}[c \mathbf{1}] = c$ for $c \in \mathbb{R}$
(A3) $\hat{\mathbb{E}}[\mathbf{x} + \mathbf{y}] \leq \hat{\mathbb{E}}[\mathbf{x}] + \hat{\mathbb{E}}[\mathbf{y}]$
(A4) $\hat{\mathbb{E}}[\lambda \mathbf{x}] = \lambda \hat{\mathbb{E}}[\mathbf{x}]$ for $\lambda > 0$

Step 2: $\hat{\mathbb{E}}$ is a sublinear functional on $\mathbb{R}^n$. Properties (A3) and (A4) mean that $\hat{\mathbb{E}}$ is sublinear (positively homogeneous and subadditive) on $\mathbb{R}^n$. This is equivalent to saying $\hat{\mathbb{E}}$ is the support function of some convex set.

Step 3: Apply the supporting hyperplane theorem. By the Hahn-Banach theorem (or its finite-dimensional version, the supporting hyperplane theorem), for any sublinear functional $p: \mathbb{R}^n \to \mathbb{R}$, we have:

\[ p(\mathbf{x}) = \max_{\ell \in \mathcal{L}} \ell(\mathbf{x}) \]

where $\mathcal{L} = \{\ell : \mathbb{R}^n \to \mathbb{R} \text{ linear} : \ell(\mathbf{x}) \leq p(\mathbf{x}) \text{ for all } \mathbf{x}\}$ is the subdifferential of $p$ at the origin.

Step 4: Identify linear functionals with probability measures. Each linear functional $\ell$ on $\mathbb{R}^n$ has the form $\ell(\mathbf{x}) = \sum_{i=1}^n p_i x_i$ for some $(p_1, \ldots, p_n) \in \mathbb{R}^n$. We need to show each $\ell \in \mathcal{L}$ corresponds to a probability measure, i.e., $p_i \geq 0$ and $\sum_i p_i = 1$.

Non-negativity: Take $\mathbf{x} = -\mathbf{e}_i$ (negative of the $i$-th basis vector, so $x_j = -\delta_{ij}$). Then $\mathbf{x} \leq \mathbf{0}$, so by monotonicity $\hat{\mathbb{E}}[\mathbf{x}] \leq \hat{\mathbb{E}}[\mathbf{0}] = 0$. For $\ell \in \mathcal{L}$: $\ell(-\mathbf{e}_i) = -p_i \leq \hat{\mathbb{E}}[-\mathbf{e}_i] \leq 0$, so $p_i \geq 0$.
Normalization: Take $\mathbf{x} = \mathbf{1}$ (all ones). By constant preservation, $\hat{\mathbb{E}}[\mathbf{1}] = 1$. Also $\ell(\mathbf{1}) = \sum_i p_i \leq \hat{\mathbb{E}}[\mathbf{1}] = 1$. Take $\mathbf{x} = -\mathbf{1}$: $\hat{\mathbb{E}}[-\mathbf{1}] = -1$ and $\ell(-\mathbf{1}) = -\sum_i p_i \leq -1$, giving $\sum_i p_i \geq 1$. Combining: $\sum_i p_i = 1$.

Step 5: Form the set $\mathcal{P}$. Define:

\[ \mathcal{P} = \left\{P = (p_1, \ldots, p_n) : p_i \geq 0, \sum_i p_i = 1, \sum_i p_i x_i \leq \hat{\mathbb{E}}[\mathbf{x}] \text{ for all } \mathbf{x} \in \mathbb{R}^n\right\} \]

This is a compact (closed and bounded subset of the simplex) convex set.

Step 6: Conclude. By the supporting hyperplane theorem:

\[ \hat{\mathbb{E}}[\mathbf{x}] = \max_{P \in \mathcal{P}} E_P[\mathbf{x}] = \max_{P \in \mathcal{P}} \sum_{i=1}^n p_i x_i \]

The maximum (rather than supremum) is attained because $\mathcal{P}$ is compact and $P \mapsto E_P[\mathbf{x}]$ is continuous. $\square$

Exercise 4. State the G-Central Limit Theorem: if $X_1, X_2, \ldots$ are i.i.d. under a sublinear expectation with $\hat{\mathbb{E}}[X_i] = \overline{\mu}$, $\hat{\mathbb{E}}[-X_i] = -\underline{\mu}$, then $\frac{1}{\sqrt{n}}\sum_{i=1}^n (X_i - \mu_n)$ converges to a G-normal distribution. Explain how this differs from the classical CLT and why the limit is an interval rather than a point.

Solution to Exercise 4

Statement of the G-Central Limit Theorem.

Let $\{X_i\}_{i=1}^\infty$ be a sequence of independent, identically distributed random variables on a sublinear expectation space $(\Omega, \mathcal{H}, \hat{\mathbb{E}})$ satisfying:

\[ \hat{\mathbb{E}}[X_i] = \hat{\mathcal{E}}[X_i] = 0 \quad (\text{symmetric mean}) \]

\[ \hat{\mathbb{E}}[X_i^2] = \overline{\sigma}^2, \quad \hat{\mathcal{E}}[X_i^2] = \underline{\sigma}^2 \]

where $0 < \underline{\sigma} \leq \overline{\sigma}$. Then the normalized sum:

\[ S_n = \frac{X_1 + X_2 + \cdots + X_n}{\sqrt{n}} \]

converges in distribution (under $\hat{\mathbb{E}}$) to the G-normal distribution $\mathcal{N}(0, [\underline{\sigma}^2, \overline{\sigma}^2])$:

\[ \lim_{n \to \infty} \hat{\mathbb{E}}[\varphi(S_n)] = \hat{\mathbb{E}}[\varphi(X)] \]

for all $\varphi \in C_{b, \text{Lip}}(\mathbb{R})$, where $X \sim \mathcal{N}(0, [\underline{\sigma}^2, \overline{\sigma}^2])$.

More generally, if $\hat{\mathbb{E}}[X_i] = \overline{\mu}$ and $\hat{\mathcal{E}}[X_i] = \underline{\mu}$ (not necessarily zero), the CLT has an additional drift term, and one considers the centered sum $\frac{1}{\sqrt{n}}\sum_{i=1}^n(X_i - \mu_n)$ for an appropriate centering sequence.

How the G-CLT differs from the classical CLT:

Limit is a distribution, not a point variance. In the classical CLT, $S_n \xrightarrow{d} N(0, \sigma^2)$ for a single, known variance $\sigma^2$. In the G-CLT, the limit is $\mathcal{N}(0, [\underline{\sigma}^2, \overline{\sigma}^2])$, a G-normal distribution characterized by a range of variances. This is not a single Gaussian but a family of Gaussians parameterized by $\sigma \in [\underline{\sigma}, \overline{\sigma}]$.
Persistent uncertainty. In the classical setting, the variance of the limit is determined by the single true variance $\sigma^2 = E[X_i^2]$. In the G-CLT, even with infinitely many observations, the variance remains uncertain: the "true" variance could be any value in $[\underline{\sigma}^2, \overline{\sigma}^2]$. This reflects the fact that under model uncertainty (multiple priors), aggregating data does not resolve the ambiguity about which prior is correct.
The limit is an interval, not a point. For the G-expectation of a convex function $\varphi$:

\[ \hat{\mathbb{E}}[\varphi(X)] = E[\varphi(\overline{\sigma} Z)], \quad Z \sim N(0,1) \]

while for concave $\varphi$:

\[ \hat{\mathbb{E}}[\varphi(X)] = E[\varphi(\underline{\sigma} Z)] \]

So the G-expectation of any nonlinear function of the limit depends on the convexity structure, with different volatilities being selected for different test functions. The classical CLT, by contrast, gives a single answer for each $\varphi$.

Nonlinear PDE characterization. The G-normal distribution is characterized by the G-heat equation (a fully nonlinear PDE), whereas the classical normal is characterized by the linear heat equation. The G-heat equation selects the worst-case volatility at each point in space-time, making the limit inherently nonlinear.
Financial interpretation. Consider a portfolio of $n$ i.i.d. returns $X_i$ with uncertain variance. The classical CLT says the portfolio's normalized return converges to a Gaussian with known variance. The G-CLT says the portfolio's normalized return converges to a G-normal distribution --- the variance remains uncertain. This means that even extreme diversification cannot eliminate volatility uncertainty, which has profound implications for risk management: model risk is irreducible through diversification alone. $\square$

Exercise 5. For the G-Brownian motion $(B_t)_{t \geq 0}$ with $\hat{\mathbb{E}}[B_t^2] = \overline{\sigma}^2 t$ and $\hat{\mathbb{E}}[-B_t^2] = -\underline{\sigma}^2 t$, compute the G-expectation of the payoff $\varphi(B_T) = (B_T - K)^+$ for $K = 0$ and $T = 1$. Show that $\hat{\mathbb{E}}[(B_1)^+] = \overline{\sigma}/\sqrt{2\pi}$ and interpret this as the worst-case call price under uncertain volatility.

Solution to Exercise 5

Goal. Compute $\hat{\mathbb{E}}[(B_1)^+]$ where $B_1$ is a G-Brownian motion at time $T = 1$ with $\hat{\mathbb{E}}[B_1^2] = \overline{\sigma}^2$.

Step 1: Use the G-normal characterization. Since $B_1 \sim \mathcal{N}(0, [\underline{\sigma}^2, \overline{\sigma}^2])$, we have for any $\varphi \in C_{b,\text{Lip}}(\mathbb{R})$:

\[ \hat{\mathbb{E}}[\varphi(B_1)] = \sup_{\sigma \in [\underline{\sigma}, \overline{\sigma}]} E[\varphi(\sigma Z)] \]

where $Z \sim N(0,1)$ under the classical probability.

Step 2: Analyze convexity of the payoff. The function $\varphi(x) = x^+ = \max(x, 0)$ is convex (it is the maximum of two linear functions: $x$ and $0$). For convex $\varphi$, the G-expectation is achieved at the maximum volatility:

\[ \hat{\mathbb{E}}[(B_1)^+] = E[(\overline{\sigma} Z)^+] = \overline{\sigma} E[Z^+] \]

To verify this, note that for convex $\varphi$, the map $\sigma \mapsto E[\varphi(\sigma Z)]$ is increasing (by Jensen's inequality applied to the scaling). Specifically, for $\varphi(x) = x^+$:

\[ E[(\sigma Z)^+] = \sigma E[Z \cdot \mathbb{1}_{\{Z > 0\}}] = \sigma \int_0^\infty z \frac{1}{\sqrt{2\pi}} e^{-z^2/2} \, dz = \sigma \cdot \frac{1}{\sqrt{2\pi}} \]

This is increasing in $\sigma$, confirming the supremum is at $\overline{\sigma}$.

Step 3: Compute the integral. We need:

\[ E[Z^+] = E[Z \cdot \mathbb{1}_{\{Z > 0\}}] = \int_0^\infty z \frac{1}{\sqrt{2\pi}} e^{-z^2/2} \, dz \]

Substituting $u = z^2/2$, $du = z \, dz$:

\[ E[Z^+] = \frac{1}{\sqrt{2\pi}} \int_0^\infty e^{-u} \, du = \frac{1}{\sqrt{2\pi}} \]

Step 4: Final result.

\[ \hat{\mathbb{E}}[(B_1)^+] = \overline{\sigma} \cdot \frac{1}{\sqrt{2\pi}} = \frac{\overline{\sigma}}{\sqrt{2\pi}} \]

Interpretation as worst-case call price. Consider an asset with zero drift and uncertain volatility $\sigma \in [\underline{\sigma}, \overline{\sigma}]$, with dynamics $dS_t = \sigma_t S_t \, dB_t$ (in a simplified setting where $S_t = S_0 + B_t$ for a normalized asset, or more precisely, $B_T$ represents the normalized log-return).

The payoff $(B_1)^+$ is analogous to a call option payoff (the positive part of the terminal value). The G-expectation $\hat{\mathbb{E}}[(B_1)^+] = \overline{\sigma}/\sqrt{2\pi}$ gives the worst-case (super-replication) price under volatility uncertainty.

The worst case selects $\sigma = \overline{\sigma}$ because $(B_1)^+$ is convex: a long call position has positive gamma, meaning higher volatility increases the expected payoff. The seller of the call faces the worst scenario when volatility is maximal. This is consistent with the Black-Scholes-Barenblatt equation, which prescribes $\overline{\sigma}$ for positive-gamma positions.

Conversely, the conjugate expectation gives the lower bound:

\[ \hat{\mathcal{E}}[(B_1)^+] = -\hat{\mathbb{E}}[-(B_1)^+] = \frac{\underline{\sigma}}{\sqrt{2\pi}} \]

The interval $[\underline{\sigma}/\sqrt{2\pi}, \overline{\sigma}/\sqrt{2\pi}]$ is the bid-ask spread for this call option under volatility uncertainty. $\square$

Exercise 6. The G-Ito formula states that for $f \in C^2(\mathbb{R})$ and G-Brownian motion $B_t$: $f(B_t) = f(0) + \int_0^t f'(B_s) \, dB_s + \frac{1}{2}\int_0^t f''(B_s) \, d\langle B \rangle_s$. Apply this to $f(x) = e^x$ and explain why the quadratic variation process $\langle B \rangle_t$ takes values in $[\underline{\sigma}^2 t, \overline{\sigma}^2 t]$ rather than being deterministic. What does this imply for hedging under volatility uncertainty?

Solution to Exercise 6

Application of the G-Ito formula to $f(x) = e^x$.

Step 1: Apply the G-Ito formula. With $f(x) = e^x$, we have $f'(x) = e^x$ and $f''(x) = e^x$. The G-Ito formula gives:

\[ e^{B_t} = e^{B_0} + \int_0^t e^{B_s} \, dB_s + \frac{1}{2}\int_0^t e^{B_s} \, d\langle B \rangle_s \]

Since $B_0 = 0$, this becomes:

\[ e^{B_t} = 1 + \int_0^t e^{B_s} \, dB_s + \frac{1}{2}\int_0^t e^{B_s} \, d\langle B \rangle_s \]

Step 2: Why $\langle B \rangle_t$ is not deterministic. In the G-Brownian motion framework, the quadratic variation $\langle B \rangle_t$ is defined as the limit:

\[ \langle B \rangle_t = \lim_{|\pi| \to 0} \sum_{i=0}^{n-1} (B_{t_{i+1}} - B_{t_i})^2 \]

where the limit is taken in the quasi-sure sense. The key difference from classical Brownian motion is that under each measure $P \in \mathcal{P}$, the quadratic variation may differ:

Under measure $P_\sigma$ (corresponding to volatility $\sigma$), $\langle B \rangle_t = \sigma^2 t$
Different measures in $\mathcal{P}$ assign different values to $\langle B \rangle_t$

Since no single measure is privileged, $\langle B \rangle_t$ is not a deterministic function. It satisfies:

\[ \underline{\sigma}^2 t \leq \langle B \rangle_t \leq \overline{\sigma}^2 t \quad \text{quasi-surely} \]

but the exact value depends on which measure $P \in \mathcal{P}$ governs the path. The process $\langle B \rangle_t$ is increasing and continuous, but it is a genuine stochastic process (not deterministic) because the "true" volatility at each instant is unknown.

More formally, the G-Brownian motion increment $B_{t+s} - B_t$ is G-normally distributed with $\hat{\mathbb{E}}[(B_{t+s} - B_t)^2] = \overline{\sigma}^2 s$ and $\hat{\mathcal{E}}[(B_{t+s} - B_t)^2] = \underline{\sigma}^2 s$. The gap between these means the realized quadratic variation is not predetermined. The quadratic variation process $\langle B \rangle_t$ is itself a source of randomness, distinct from the randomness in $B_t$.

Step 3: Implications for hedging under volatility uncertainty. The G-Ito formula for $e^{B_t}$ reveals the fundamental challenge of hedging under volatility uncertainty:

\[ e^{B_t} - 1 = \underbrace{\int_0^t e^{B_s} \, dB_s}_{\text{hedgeable}} + \underbrace{\frac{1}{2}\int_0^t e^{B_s} \, d\langle B \rangle_s}_{\text{not fully hedgeable}} \]

The first integral $\int_0^t e^{B_s} \, dB_s$ is a symmetric G-martingale and represents the hedgeable component --- the P&L from delta-hedging. The hedge ratio is $\Delta_s = e^{B_s}$ (or $f'(B_s)$ in general).

The second integral $\frac{1}{2}\int_0^t e^{B_s} \, d\langle B \rangle_s$ depends on the realized quadratic variation $\langle B \rangle_s$, which is uncertain. Since $d\langle B \rangle_s$ takes values between $\underline{\sigma}^2 \, ds$ and $\overline{\sigma}^2 \, ds$, this term has an uncertain contribution:

\[ \frac{1}{2}\underline{\sigma}^2 \int_0^t e^{B_s} \, ds \leq \frac{1}{2}\int_0^t e^{B_s} \, d\langle B \rangle_s \leq \frac{1}{2}\overline{\sigma}^2 \int_0^t e^{B_s} \, ds \]

This uncertainty is the gamma P&L risk: since $f''(x) = e^x > 0$ (positive gamma), higher realized volatility benefits a long position in $e^{B_t}$. The hedger cannot eliminate this risk through delta-hedging alone.

Practical implications:

Delta-hedging is insufficient: Unlike classical Black-Scholes, where delta-hedging perfectly replicates the payoff, under G-Brownian motion the quadratic variation integral introduces an unhedgeable residual.
Gamma exposure determines risk: The term $f''(B_s) \, d\langle B \rangle_s$ shows that the exposure to volatility uncertainty is proportional to gamma ($f'' = e^{B_s}$ in this case). Positive gamma means the hedger benefits from high volatility; a super-replicating seller must assume the worst case ($\overline{\sigma}$).
Need for vega hedging or robust strategies: To reduce exposure to the uncertain $\langle B \rangle$, one must either (a) hedge gamma using other options (vega hedging), effectively neutralizing $f''$, or (b) accept the uncertainty and price it via the BSB equation, which automatically selects the worst-case volatility at each point. $\square$

Feature	g-Expectation	G-Expectation
Probability space	Fixed \((\Omega, \mathcal{F}, P)\)	No reference measure
Generator	\(g(t, y, z)\)	\(G(\alpha) = \frac{1}{2}(\overline{\sigma}^2 \alpha^+ - \underline{\sigma}^2 \alpha^-)\)
Nonlinearity source	Drift uncertainty	Volatility uncertainty
PDE connection	Quasilinear	Fully nonlinear
Stochastic integral	Classical Ito	G-Ito