Contributions to the numerics of quadratic backward SDEs and probability on turbulence

Thesis

In this thesis, we first study the numerical scheme of path-dependent quadratic backward stochastic differential equations (quadratic BSDE or QBSDE for short). Under the assumption that the terminal condition is of the form Φ(X), where Φ is a Lipschitz function on D(R^m) (the space of càdlàg paths) and X is the solution to a Lipschitz SDE satisfying linear growth conditions, we derive the path-regularity result which characterises the solution (Y,Z). Motivated by the modified implicit BTZ scheme, we propose the modified (inner) explicit BTZ scheme and conduct a perturbation analysis on Z-part of the scheme, followed by concluding the error of the scheme is O(N^−1/2)) with N as the number of timesteps instead of O(N^−1/2+ε)) for some ε > 0 in the modified implicit BTZ scheme. Numerical experiments support our conclusion on the convergence rate.

The second purpose of this thesis is to derive a partial differential equation (PDE) for the single-time single-point probability density function (PDF) of the velocity field of a turbulent flow. The PDF PDE is a highly nonlinear parabolic-transport equation, which depends on two conditional statistical characteristics of important physical meaning. The PDF PDE is a general form of the classical Reynolds mean flow equation, and is a precise formulation of the PDF transport equation. The PDF PDE provides us with a new method for modelling turbulence. An explicit example is constructed, though the example is seemingly artificial, it demonstrates the PDF method based on the new PDF PDE.

Lastly, vorticity random fields of turbulent flows are singled out as the main dynamical variables for the description of turbulence, and the evolution equation of the PDF of the vorticity field has been obtained. This PDF evolution equation is a mixed type PDE of the second order which depends only on the conditional mean (which is a first order statistic) of the underlying turbulent flow. This is in contrast with Reynolds’ mean flow equation which relies on a quadratic statistic. The PDF equation is interesting on its own and is worthy of study as a PDE of second order.

Authors: Zhou, M

• DOI: 10.5287/ora-mj0ovmawx
• Type of award: DPhil
• Level of award: Doctoral
• Awarding institution: University of Oxford
• Language: English
• Keywords: BSDE; Turbulence; Numerical method
• Subjects: Mathematics; Turbulence; Stochastic analysis