Backward stochastic differential equations with unbounded coefficients and their applications

Li, Jiajie Backward stochastic differential equations with unbounded coefficients and their applications. PhD thesis, University of Liverpool.

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Abstract

In this thesis, we focus on problems on the theory of Backward Stochastic Differential Equations (BSDEs). In particular, BSDEs with an unbounded generator are considered, under various conditions (on the generator). Using more general (or weaker) conditions, the classical results on BSDEs are improved and some associated problems on mathematical finance are resolved. Chapter 1 introduces some of the literature, general setting and ideas in this field and emphasises the motivations which has led to the study of these equations. In addition, some mathematical preliminaries we used throughout this thesis are included in Chapter 2. In Chapter 3, we consider nonlinear BSDEs with an unbounded generator. Under a Lipschitz-type condition, we show sufficient conditions for the existence and uniqueness of solutions to nonlinear BSDEs, which are weaker than the existing ones. We also give a comparison theorem as a generalisation of Peng's result. Chapter 4 studies a class of backward stochastic differential equations whose generator satisfies linear growth and continuity conditions, which can also be unbounded. We prove the existence of the solution pair for this class of equations which is more general than the existing ones. In Chapter 5, we consider the problem of solvability for linear backward stochastic differential equations with unbounded coefficients. New and weaker sufficient conditions for the existence of a unique solution pair are given. It is shown that certain exponential processes have stronger integrability in this case. As applications, we solve the problems of completeness in a market with a possibly unbounded coefficients and optimal investment with power utility in a market with unbounded coefficients. Chapter 6 studies the classical Stochastic Differential Equations where the drift and diffusion coefficients satisfy Lipschitz-type and linear growth conditions, which can also be unbounded. We give sufficient conditions for the existence of a unique solution to unbounded SDEs. The method of proof is that of Picard iterations and the resulting conditions are new. We also prove a comparison theorem. Chapter 7 summaries the results in this thesis and outlines possible directions for future works based on current results.

Item Type:	Thesis (PhD)
Additional Information:	Date: 2014-10 (completed)
Depositing User:	Symplectic Admin
Date Deposited:	01 Sep 2015 13:55
Last Modified:	17 Dec 2022 01:18
DOI:	10.17638/02010039
URI:	https://livrepository.liverpool.ac.uk/id/eprint/2010039