On Backward Stochastic Differential Equations (BSDEs) with jumps of infinite activity

Inhaltsverzeichnis (Table of Contents)

• Introduction
• Preliminaries
  • Random measures and compensators
  • The weak property of predictable representation
  • Itô's formula
• Classical Results on BSDEs with jumps
  • Uniqueness
  • Existence for Lipschitz generators
  • Existence in general: Proof of Proposition 2.11
• BSDEs of a specific generator
  • The case of finite activity
  • Generators of difference type
  • A comparison theorem
  • The case of infinite activity
    • Uniqueness
    • Existence
• Application to utility maximization
  • The financial market framework
  • Exponential utility maximization
• Appendix A
  • Properties of weak convergence
  • Technical results

Zielsetzung und Themenschwerpunkte (Objectives and Key Themes)

This diploma thesis explores backward stochastic differential equations (BSDEs) with jumps driven by a Brownian Motion and a random measure. The main objective is to derive existence and uniqueness results for bounded solutions to such BSDEs, particularly when the generator possesses a monotonicity property instead of the usual global Lipschitz condition. The thesis investigates the extension of existing results from finite activity to the infinite activity case.

• Existence and uniqueness of solutions to BSDEs with jumps
• Monotonicity properties of generators in BSDEs
• Extension of results from finite activity to infinite activity
• Application of BSDE techniques to utility maximization problems
• The weak property of predictable representation in the context of BSDEs

Zusammenfassung der Kapitel (Chapter Summaries)

The thesis commences with an introduction to BSDEs with jumps, defining key concepts such as random measures, compensators, and the weak property of predictable representation. Chapter 2 revisits Pardoux's work on BSDEs with jumps, proving his results for inhomogeneous compensators. Chapter 3 delves into a specific class of generators, exploring the case of finite activity and generators of difference type. It then introduces a comparison theorem and extends the analysis to the case of infinite activity, demonstrating both uniqueness and existence of solutions. Finally, Chapter 4 applies the developed theory to the Exponential Utility Maximization problem within a suitable financial market model.

Schlüsselwörter (Keywords)

The primary focus of this thesis lies in backward stochastic differential equations (BSDEs) with jumps, specifically those driven by a Brownian Motion and a random measure. Key themes include the existence and uniqueness of solutions to such BSDEs, particularly when the generator exhibits monotonicity properties instead of standard Lipschitz conditions. The thesis explores the extension of existing results from finite activity to infinite activity cases. Additionally, the work investigates the application of BSDE techniques to utility maximization problems in financial markets.