Some contributions to stochastic differential equations

Thesis

This thesis elaborates topics on a type of McKean–Vlasov stochastic differential equations and forward–backward stochastic differential equations arising from physics models.

Mainly, this thesis is divided into three parts. Chapter 2 consists of preliminary definitions and results from the theory of parabolic equations, stochastic differential equations and backward stochastic differential equations. Some key results are summarised for subsequent developments.

Motivated by the random vortex method for the Navier–Stokes equation, Chapter 3 studies a special type of McKean–Vlasov stochastic differential equation (MVSDE), whose kernel function is assumed to be continuous except at few singularities in Rd. With some technical estimates, we establish the existence and uniqueness of both the weak and the strong solutions to the MVSDE theoretically. The numerical simulations are also conducted to approximate the MVSDE and the corresponding velocity field in the three dimensional space.

Chapter 4 is devoted to studying a system of forward-backward stochastic differential equations (FBSDEs) arising from the porous medium equation. The gererating function of the backward equation for this class of FBSDEs has quadratic growth in the variable Z. We construct the unique solution on a small time interval by using a fixed point argument, and further extend the study to a class of quadratic FBSDEs with more general growth conditions.

Authors: Yao, Y

• DOI: 10.5287/ora-e2eqq9g28
• Type of award: DPhil
• Level of award: Doctoral
• Awarding institution: University of Oxford
• Language: English
• Keywords: Cameron-Martin formula; forward-backward SDEs; McKean-Vlasov SDEs; vorticity equation; strong solution; diffusion processes; aronson estimates; quadratic BSDEs
• Subjects: Stochastic partial differential equations; Nonlinear partial differential operators