[Paper Review] G-expectations in infinite dimensional spaces and related PDEs

Anton Ibragimov|arXiv (Cornell University)|Jun 21, 2013

Stochastic processes and financial applications57 references3 citations

TL;DR

This paper extends G-expectation theory to infinite-dimensional Hilbert spaces, introducing G-Brownian motion, G-stochastic integrals, and a covariation set for G-normal random variables. It establishes a probabilistic representation of the unique viscosity solution to a fully nonlinear parabolic PDE with unbounded first-order term via the G-expectation of an infinite-dimensional Ornstein-Uhlenbeck process driven by G-Brownian motion.

ABSTRACT

In this paper, we extend the G-expectation theory to infinite dimensions. Such notions as a covariation set of G-normal distributed random variables, viscosity solution, a stochastic integral driven by G-Brownian motion are introduced and described in the given infinite dimensional case. We also give a probabilistic representation of the unique viscosity solution to the fully nonlinear parabolic PDE with unbounded first order term in Hilbert space in terms of G-expectation theory.

Motivation & Objective

• To extend the theory of G-expectations to infinite-dimensional Hilbert spaces, where standard finite-dimensional tools fail due to unbounded operators.
• To define a stochastic integral with respect to G-Brownian motion in Hilbert space, enabling the study of stochastic differential equations in infinite dimensions.
• To provide a probabilistic representation of the unique viscosity solution to a fully nonlinear parabolic PDE with unbounded first-order term in Hilbert space.
• To establish the existence and uniqueness of viscosity solutions using G-expectation and stochastic calculus in infinite dimensions.
• To generalize classical probabilistic representations (e.g., Feynman-Kac) to fully nonlinear, non-Markovian settings in infinite-dimensional spaces.

Proposed method

• Introduces a G-functional on the space of bounded, self-adjoint, compact operators in a Hilbert space, satisfying monotonicity, sublinearity, and continuity.
• Defines G-Brownian motion in Hilbert space as a stochastic process associated with a G-expectation, generalizing standard Brownian motion to sublinear expectations.
• Constructs a G-stochastic integral with respect to G-Brownian motion using a class of predictable, Hilbert-Schmidt operator-valued processes.
• Applies Itô’s isometry and BDG-type inequalities in the infinite-dimensional setting to ensure integrability and convergence of stochastic integrals.
• Derives a stochastic differential equation of Ornstein-Uhlenbeck type: $ dX_t = A X_t dt + dB_t $, with $ X_t $ driven by G-Brownian motion.
• Uses the solution $ X_T^{t,x} $ to define the candidate solution $ u(t,x) = \mathbb{E}[f(X_T^{t,x})] $, where $ \mathbb{E} $ is the G-expectation.

Experimental results

Research questions

• RQ1 How can G-expectation theory be consistently extended to infinite-dimensional Hilbert spaces?
• RQ2 What is the appropriate definition of a G-stochastic integral and G-Brownian motion in infinite dimensions?
• RQ3 Can the viscosity solution of a fully nonlinear parabolic PDE with unbounded first-order term be represented probabilistically in Hilbert space?
• RQ4 What conditions ensure the existence and uniqueness of such a solution using G-expectation methods?
• RQ5 How do standard stochastic calculus tools (Itô isometry, BDG inequality) generalize in this infinite-dimensional, sublinear setting?

Key findings

• A new class of G-functionals is defined on the space of bounded, self-adjoint, compact operators in Hilbert space, satisfying sublinearity, monotonicity, and continuity.
• The G-stochastic integral with respect to G-Brownian motion is constructed for predictable, Hilbert-Schmidt operator-valued integrands, ensuring convergence and integrability.
• The solution to the PDE $ \partial_t u + \langle Ax, D_x u \rangle + G(D_{xx}^2 u) = 0 $ is represented as $ u(t,x) = \mathbb{E}[f(X_T^{t,x})] $, where $ X_T^{t,x} $ solves the SDE driven by G-Brownian motion.
• The solution $ u $ is shown to be a viscosity solution via a perturbation argument and limit passage as $ \delta \to 0 $, yielding the correct generator condition.
• Continuity of $ u $ at $ (t,x) $ is established using moment bounds and convergence in $ L^2 $, relying on the BDG inequality and operator norm estimates.
• The solution inherits B-continuity and polynomial growth from the terminal condition $ f $, ensuring regularity under the G-expectation framework.