Backward Stochastic Differential Equations with Superlinear Drivers

Abstract

This thesis focuses mainly on the well-posedness of backward stochastic differential equations:

[

Y_t=xi+int_t^Tf(s,Y_s,Z_s)ds-int_t^TZ_sdW_s

]

The most prevalent method for showing the well-posedness of BSDE is to use the Banach fixed point theorem on a space of stochastic processes. Another notable method is to use the comparison theorem and limiting argument. We present three other methods in this thesis:

1. Fixed point theorems on the space of random variables

2. BMO martingale theory and Girsanov transform

3. Malliavin calculus

Using these methods, we prove the existence and uniqueness of solution for multidimensional BSDEs with superlinear drivers which have not been studied in the previous literature. Examples include quadratic mean-field BSDEs with $L^2$ terminal conditions, quadratic Markovian BSDEs with bounded terminal conditions, subquadratic BSDEs with bounded terminal conditions, and superquadratic Markovian BSDEs with terminal conditions that have bounded Malliavin derivatives. Along the way, we also prove the well-posedness for backward stochastic equations, mean-field BSDEs with jumps, and BSDEs with functional drivers. In the last chapter, we explore the relationship between BSDEs with superquadratic driver and semilinear parabolic PDEs with superquadratic nonlinearities in the gradients of solutions. In particular, we study the cases where there is no boundary or there is a Dirichlet or Neumann lateral boundary condition.