G-Brownian Motion: Theory and Applications

• G-Brownian motion is defined under a sublinear expectation that captures volatility uncertainty via a family of martingale measures.
• It generalizes classical stochastic analysis by extending tools such as Girsanov’s theorem, variational representations, and nonlinear PDE connections.
• Its robust framework supports practical applications in areas like financial risk management, optimal control, and rough path analysis.

G-Brownian motion generalizes classical Brownian motion by being defined under a sublinear expectation (“G-expectation”), which models volatility uncertainty and leads to a nonlinear, non-dominated law for the process. Within this framework the “law” is defined as a supremum over expectations corresponding to a family of martingale measures, and the behavior of both the process and its quadratic variation are governed by fully nonlinear partial differential equations. G-Brownian motion provides a flexible and robust mathematical foundation for modeling stochastic systems with uncertain volatility, under which classical results in stochastic analysis, such as Girsanov’s formula, variational representations, large deviations, rough path theory, and backward stochastic differential equations, can be generalized and developed.

1. Foundations and Definition

G-Brownian motion (B) is a continuous process defined on a canonical space of paths and equipped with a sublinear expectation called the G-expectation, denoted as $\mathbb{E}$. Its increments are independent and stationary in the sense of G-expectation, and each increment is G-normally distributed, i.e., its law is characterized by a fully nonlinear PDE of the G-heat type: $$\partial_t u(t,x) - G(\partial_{xx}^2 u(t,x)) = 0,\quad u(0,x) = \varphi(x),$$ where $G$ is a continuous, convex, sublinear function reflecting the volatility uncertainty. For a d-dimensional random vector $X$, the G-normal distribution is defined by the property that for any $a, b \geq 0$,

$aX + b\tilde{X} \equiv \sqrt{a^2 + b^2} X,$

where $\tilde{X}$ is an independent copy of $X$, and $G(A) = \frac{1}{2}\mathbb{E}\left[(AX,X)\right]$ for $A \in S_d$. The canonical G-Brownian motion results in a sublinear expectation that is represented as the supremum of linear expectations over a class of martingale measures, as established by Denis, Hu, and Peng.

G-Brownian motion possesses a non-deterministic quadratic variation process $\langle B\rangle_t$, with independent and stationary increments under G-expectation, capturing the uncertainty in volatility. In the multidimensional setting, the components are generally correlated due to uncertainty, making the covariance structure non-trivial.

2. Nonlinear Girsanov Theorem and Martingale Measure Enlargement

Classically, Girsanov’s theorem describes a change of measure that transforms Brownian motion with drift into a standard Brownian motion. In the G-framework, the law is not tied to a single measure but to a family of them. The key innovation is that if $B$ is a G-Brownian motion and $h$ is a suitable integrable process, then the process

$$D_t = \exp\left\{\int_0^t h_s \cdot dB_s - \frac{1}{2} \int_0^t h_s \cdot d\langle B\rangle_s h_s\right\}$$

acts as a “density” in the sublinear expectation sense.

Defining a new process: $$\tilde{B}_t = B_t - \int_0^t d\langle B\rangle_s h_s$$ and the transformed expectation $\hat{\mathbb{E}}[X] = \mathbb{E}[X D_T]$, $\tilde{B}$ is a G-Brownian motion under $\hat{\mathbb{E}}$. The proof uses enlargement of the class of martingale measures to rigorously justify the change of sublinear expectation. This enlargement is critical in handling multidimensional cases, overcoming limitations of approaches relying solely on martingale characterizations suitable only for the one-dimensional setting (Osuka, 2011).

3. Variational Representation and Large Deviations Principles

Under G-expectation, exponential functionals of G-Brownian motion admit a variational (control-theoretic) representation analogous to the Boué-Dupuis formula in the classical case. For a bounded functional $\varphi$ on path space, the representation is: $$\mathbb{E}_G\left[\exp\left\{\varphi(B)\right\}\right] = \exp\left\{\sup_n \mathbb{E}_G\left[\varphi(B^n) - H_T(n)\right]\right\}$$ with $B^n_t = B_t + \int_0^t n_s \, d\langle B \rangle_s$ and $$H_T(n) = \frac{1}{2} \int_0^t n_s^{\top} d\langle B \rangle_s n_s$$ (Gao, 2012).

This formulation underpins the large deviation principle (LDP) for stochastic flows driven by G-Brownian motion. For small-perturbation SDEs driven by G-Brownian motion, the LDP is governed by a rate function arising naturally from the variational representation. The associated “good rate function” for flows $\phi$ is

$$I(\phi) = \inf\left\{ J(f,g) : \phi = Y(f,g)\right\}$$

with $J(f,g)$ and $Y(f,g)$ defined via corresponding controlled deterministic equations.

Quasi-continuity and tightness of G-stochastic flows are established using Kolmogorov’s criterion and the capacity theory associated with G-expectation, allowing the LDP to be established both in Laplace and strong classical forms.

4. Stochastic Calculus, BSDEs, and Nonlinear PDE Connections

G-stochastic calculus adapts Itô’s formalism to the sublinear expectation context. The decomposition of a G-martingale includes both a symmetric G-martingale and an additional decreasing process $K$, leading to the canonical BSDE: $$Y_t = \xi + \int_t^T f(s, Y_s, Z_s) ds + \int_t^T g(s, Y_s, Z_s) d\langle B\rangle_s - \int_t^T Z_s dB_s - (K_T - K_t)$$ The existence and uniqueness of the triple $(Y, Z, K)$ are established under Lipschitz conditions using novel Galerkin-type approximations and sublinear martingale analysis (Hu et al., 2012).

The theory directly connects to fully nonlinear PDEs: via the nonlinear Feynman–Kac formula, the value function $u$ associated with a G-BSDE is a viscosity solution to

$\partial_t u - G(D^2 u) = 0$

with appropriate boundary conditions.

For stochastic optimal control, the dynamic programming principle (DPP) under G-expectation requires an extension involving a backward semigroup, and the value function is shown to solve a fully nonlinear second-order PDE in the viscosity sense. Control problems are specified by G-driven SDEs with drift, volatility, and quadratic covariation terms all possibly controlled, and the backward semigroup represents the nonlinear flow of costs under optimal choices (Zheng et al., 2013).

5. Rough Path Structure and Pathwise Analysis

G-Brownian motion quasi-surely admits Hölder continuous paths of any order $\alpha < 1/2$, and can be canonically enhanced to a geometric rough path of roughness $2 < p < 3$, enabling pathwise solutions to rough differential equations (RDEs) driven by G-Brownian motion. For such enhancement, the iterated integrals used to define the second rough path level are constructed as the $L^2$-limits of dyadic piecewise linear approximations.

The equivalence of SDE and RDE formulations, modulo Itô–Stratonovich correction terms involving the quadratic variation, is established quasi-surely. For a differentiable manifold $M$, SDEs driven by G-Brownian motion can be formulated intrinsically, and “G-Brownian motion on manifolds” is constructed by rolling the Euclidean G-Brownian motion using horizontal lifts and projecting to $M$. When the G-function’s invariant group is $O(d)$, the resulting process leads to intrinsic nonlinear PDEs on $M$ for the law’s characterization (Geng et al., 2013).

Quantitative estimates of roughness and pathwise Norris lemmas within the G-framework further enable pathwise uniqueness and stability analysis for RDEs driven by G-Brownian motion (Peng et al., 2015).

6. Path Properties, Capacity, and Function Spaces

Key sample path features of G-Brownian motion include:

• The zero set at any fixed level $a$ is closed and has zero Lebesgue measure quasi-surely: $\int_0^T 1_{\{B_s = a\}} ds = 0$ q.s.
• The set of local maxima is dense; G-Brownian motion is quasi-surely nowhere monotone and almost nowhere differentiable.
• Indicator functions of regular events, such as $1_{\{B_t \in O\}}$, belong to the integrable space $L_G^1(\Omega)$, provided $O$ is a regular Borel set. This provides essential closure properties needed for further development of G-martingale and stopping time analyses (Wang et al., 2014).

7. Applications: Control, Finance, and Large Deviations

G-Brownian motion forms the foundation for robust mathematical finance under volatility uncertainty, supporting bid-ask pricing of European claims via sublinear expectations and extending classical Clark-Ocone and Girsanov formulas to this context (Chen, 2013). Stochastic optimal control and recursive utility problems under model ambiguity are naturally formulated with G-Brownian motion as the noise driver, and their value functions connect to fully nonlinear PDEs.

The variational approach developed for functionals of G-Brownian motion enables the weak convergence and large deviations analysis necessary for rare event estimation, robust risk management, and uncertain volatility option pricing (Gao, 2012). The self-normalized G-Brownian motion and its generalization of classical Donsker invariance principles further emphasize the breadth of asymptotic and ergodic results available in the G-framework (Zhang, 2015).

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The theory of G-Brownian motion thus forms a cohesive and multifaceted framework supporting stochastic analysis, control theory, financial mathematics, rough path theory, and asymptotic probability under uncertainty. It unifies nonlinear sublinear expectation with complex sample path properties, extending classical probabilistic paradigms to fundamentally uncertain environments.