Multidimensional quadratic and subquadratic BSDEs with special structure

We study multidimensional BSDEs of the form $$Y_t = \xi + \int_t^T f(s,Y_s,Z_s)ds - \int_t^T Z_s dW_s$$ with bounded terminal conditions $\xi$ and drivers $f$ that grow at most quadratically in $Z_s$. We consider three different cases. In the first one the BSDE is Markovian, and a solution can be obtained from a solution to a related FBSDE. In the second case, the BSDE becomes a one-dimensional quadratic BSDE when projected to a one-dimensional subspace, and a solution can be derived from a solution of the one-dimensional equation. In the third case, the growth of the driver $f$ in $Z_s$ is strictly subquadratic, and the existence and uniqueness of a solution can be shown by first solving the BSDE on a short time interval and then extending the solution recursively.Comment: 16 page

BSDEs with terminal conditions that have bounded Malliavin derivative

We show existence and uniqueness of solutions to BSDEs of the form $$Y_t = \xi + \int_t^T f(s,Y_s,Z_s)ds - \int_t^T Z_s dW_s$$ in the case where the terminal condition $\xi$ has bounded Malliavin derivative. The driver $f(s,y,z)$ is assumed to be Lipschitz continuous in $y$ but only locally Lipschitz continuous in $z$. In particular, it can grow arbitrarily fast in $z$. If in addition to having bounded Malliavin derivative, $\xi$ is bounded, the driver needs only be locally Lipschitz continuous in $y$. In the special case where the BSDE is Markovian, we obtain existence and uniqueness results for semilinear parabolic PDEs with non-Lipschitz nonlinearities. We discuss the case where there is no lateral boundary as well as lateral boundary conditions of Dirichlet and Neumann type

Robust Wasserstein Optimization and its Application in Mean-CVaR

We refer to recent inference methodology and formulate a framework for solving the distributionally robust optimization problem, where the true probability measure is inside a Wasserstein ball around the empirical measure and the radius of the Wasserstein ball is determined by the empirical data. We transform the robust optimization into a non-robust optimization with a penalty term and provide the selection of the Wasserstein ambiguity set's size. Moreover, we apply this framework to the robust mean-CVaR optimization problem and the numerical experiments of the US stock market show impressive results compared to other popular strategies

Data-driven Multiperiod Robust Mean-Variance Optimization

We study robust mean-variance optimization in multiperiod portfolio selection by allowing the true probability measure to be inside a Wasserstein ball centered at the empirical probability measure. Given the confidence level, the radius of the Wasserstein ball is determined by the empirical data. The numerical simulations of the US stock market provide a promising result compared to other popular strategies.Comment: 37 page

Backward Stochastic Differential Equations with Superlinear Drivers

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