This diploma thesis is concerned with backward stochastic differential equations (BSDEs) with jumps which are driven by a Brownian Motion and a random measure.
We derive existence and uniqueness results for bounded solutions to such BSDEs when the generator posses a certain monotonicity property instead of the usual global Lipschitz condition.
Starting with results in the case of finite activity, considering generators of difference type and showing a comparison theorem, allows us to advance to the case 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.
Frequently Asked Questions
What are Backward Stochastic Differential Equations (BSDEs)?
BSDEs are a class of stochastic equations where the solution is specified by a terminal condition at a future time. They are widely used in financial mathematics for option pricing and risk management.
What is the difference between finite and infinite activity in BSDEs?
Finite activity assumes a limited number of jumps in any time interval. Infinite activity deals with jump processes (like Lévy processes) that can have an infinite number of small jumps in a finite time period.
Why is the monotonicity property important for the generator?
Standard results often require a global Lipschitz condition. The monotonicity property allows for a broader class of generators, enabling existence and uniqueness results for more complex financial models.
How are BSDEs applied to utility maximization?
In financial markets, BSDEs help find the optimal investment strategy that maximizes an investor's expected exponential utility, accounting for market jumps and risks.
What is the "property of predictable representation"?
It is a technical condition ensuring that any square-integrable martingale can be represented as a stochastic integral, which is a fundamental requirement for solving BSDEs.
- Quote paper
- Martin Büttner (Author), 2011, On Backward Stochastic Differential Equations (BSDEs) with jumps of infinite activity, Munich, GRIN Verlag, https://www.grin.com/document/323586
