Nonlinear Expectations and Nonlinear Stochastic Calculus
Classical stochastic calculus is built on linear probability theory, where uncertainty is modelled using a fixed probability measure. However, in many real-world situations—particularly in finance—there is ambiguity or uncertainty about the underlying probabilistic model itself. This has led to the development of nonlinear expectation theory, which provides a framework for modelling uncertainty beyond classical probability.
One of the most influential developments in this direction is the theory of G-expectation, introduced by Shige Peng. This framework generalises classical expectations and leads to new stochastic processes, such as G-Brownian motion, as well as a corresponding stochastic calculus. This project explores the foundations and implications of this theory.
In this project, students will begin by reviewing the fundamental concepts of stochastic calculus, including Brownian motion, Itô’s formula, and stochastic differential equations. Building on this background, the project will introduce the notion of G-expectation, a nonlinear expectation defined via a nonlinear heat equation.
Students will study the associated concepts of G-normal distributions and G-Brownian motion, which generalise classical Gaussian distributions and Brownian motion to this nonlinear setting. The project will then develop the corresponding stochastic calculus, including Itô-type stochastic integrals with respect to G-Brownian motion.
A key aspect of the project will be the study of stochastic differential equations under G-expectation, including results on existence and uniqueness of solutions. Throughout, emphasis will be placed on understanding how this framework differs from classical probability theory. In particular, unlike the earlier theory of g-expectations, the G-expectation framework is intrinsic and does not rely on a pre-specified linear probability space.
Depending on interest, the project may take a more theoretical direction, focusing on the structure and properties of nonlinear expectations, or explore connections to applications such as uncertainty modelling and robust finance.
Possible directions for further exploration include:
Students may also propose alternative topics in consultation with the supervisor.
The project will be based on independent study supported by regular meetings with the supervisor. Emphasis will be placed on developing a deep conceptual understanding of the theory and its mathematical structure.
Students will engage with advanced literature, work through key results, and explore examples where appropriate. Evidence of learning will include:
Indicative resources include: