[Paper Review] A variational representation and large deviations for functionals of G-Brownian motion

Fuqing Gao|arXiv (Cornell University)|Apr 20, 2012

Stochastic processes and financial applications14 references3 citations

TL;DR

This paper establishes a variational representation for functionals of $G$-Brownian motion using a finite-dimensional approximation technique under $G$-expectation, overcoming limitations of classical methods like the Clark-Ocone formula. As an application, it proves a large deviation principle for stochastic flows driven by $G$-Brownian motion, providing a new proof for Boué-Dupuis variational representations and extending large deviations theory to nonlinear expectations.

ABSTRACT

A variational representation for functionals of G-Brownian motion is established by a finite-dimensional approximate technique. As an application of the variational representation, we obtain a large deviation principle for stochastic flows driven by G-Brownian motion.

Motivation & Objective

• To develop a variational representation for functionals of $G$-Brownian motion under the sublinear $G$-expectation framework.
• To overcome the inapplicability of classical tools like the Clark-Ocone formula and measurable selection in $G$-stochastic calculus.
• To establish a large deviation principle for stochastic flows driven by $G$-Brownian motion.
• To provide a new proof of the classical Boué-Dupuis variational representation using $G$-framework techniques.

Proposed method

• Develops a finite-dimensional approximation technique to handle the complexity of $G$-expectation, enabling approximation of functionals by sequences of $G$-stochastic differential equations.
• Uses the $G$-Girsanov transformation to prove the lower bound in the large deviation principle.
• Applies Gronwall's lemma to establish uniform convergence of approximating processes in the finite-dimensional approximation scheme.
• Employs a modified Kolmogorov criterion under $G$-expectation to prove tightness of families of stochastic processes.
• Utilizes bounded approximation and quasi-continuity to handle convergence in distribution under $\mathbb{E}^G$.
• Derives the variational representation via supremum over adapted controls $\eta \in (M^2(0,T))^d$, with a quadratic control term $H_T^G(\eta)$.

Experimental results

Research questions

• RQ1 Can a variational representation for functionals of $G$-Brownian motion be established in the absence of the Clark-Ocone formula and measurable selection?
• RQ2 How can large deviations be derived for stochastic flows driven by $G$-Brownian motion under model uncertainty?
• RQ3 Can the finite-dimensional approximation technique under $G$-expectation provide a new proof of the classical Boué-Dupuis variational representation?
• RQ4 What conditions ensure tightness and uniform convergence of $G$-stochastic processes under $\mathbb{E}^G$?

Key findings

• A variational representation is established: $\mathbb{E}^G(e^{\Phi(B)}) = \exp\left\{\sup_{\eta \in (M^2(0,T))^d} \mathbb{E}^G\left(\Phi(B^\eta) - H_T^G(\eta)\right)\right\}$, where $B^\eta_t = B_t + \int_0^t \eta_s d\langle B\rangle_s$.
• The finite-dimensional approximation technique successfully proves the upper bound in the large deviation principle by approximating $G$-SDEs.
• The lower bound is proven via the $G$-Girsanov transformation and bounded approximation, ensuring the validity of the large deviation principle.
• Tightness of the process family $\{Y_{\lambda,\epsilon}\}$ is established under $\mathbb{E}^G$ using a $G$-expectation version of the Kolmogorov criterion.
• Uniform convergence in distribution is achieved under $\mathbb{E}^G$, with $\lim_{\epsilon \to 0} \sup_{\lambda} \mathbb{E}^G(|\Phi(Y_{\lambda,\epsilon}) - \Phi(Y_\lambda)|) = 0$.
• The results provide a new proof of the classical Boué-Dupuis variational representation, extending its validity to the $G$-framework.