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.
Table of Contents
0.1 Introduction
1 Preliminaries
1.1 Random measures and compensators
1.2 The weak property of predictable representation
1.3 Itô’s formula
2 Classical Results on BSDEs with jumps
2.1 Uniqueness
2.2 Existence for Lipschitz generators
2.3 Existence in general: Proof of Proposition 2.11
3 BSDEs of a specific generator
3.1 The case of finite activity
3.2 Generators of difference type
3.3 A comparison theorem
3.4 The case of infinite activity
3.4.1 Uniqueness
3.4.2 Existence
4 Application to utility maximization
4.1 The financial market framework
4.2 Exponential utility maximization
5 Appendix A
5.1 Properties of weak convergence
5.2 Technical results
6 Theses
Objectives and Research Scope
This thesis investigates backward stochastic differential equations (BSDEs) driven by Brownian motion and random measures, specifically focusing on the existence and uniqueness of solutions in the case of infinite activity, extending beyond classical Lipschitz conditions by employing monotonicity properties of the generator and comparison theorems.
- Theoretical foundation of BSDEs with jumps in filtered probability spaces.
- Extension of existence and uniqueness results for generators under monotonicity conditions.
- Development and application of comparison theorems for specific classes of generators.
- Analysis of the utility maximization problem within financial market models using BSDE techniques.
Excerpt from the Book
1.3 Itô’s formula
Let us introduce the generalized Itô’s formula for a solution (Yt, Zt, Ut)0≤t≤T ∈ S2 × L2(B) × L2(˜μ) of the above BSDE (1.1) with data (ξ,f):
Theorem 1.17. (Itô’s formula) For any C2-function F on Rd it holds:
F(Yt) = F(ξ) − ∫_t^T ⟨F'(Ys−), Zs dBs⟩ + ∫_t^T ⟨F'(Ys−), fs(Ys−, Zs, Us)⟩ ds − 1/2 ∫_t^T F''(Ys−)|Zs|2ds − ∫_t^T ∫_E (F(Ys− + Us(e)) − F(Ys−)) ˜μ(ds, de) − ∫_t^T ∫_E (F(Ys− + Us(e)) − F(Ys−) − ⟨F'(Ys−), Us(e)⟩) ν(ds, de)
Proof. By the Itô formula for semimartingales ([4], Thm. 9.35.), it follows ... [Note: The OCR math formatting is complex; the actual book text follows this structure].
Summary of Chapters
Preliminaries: Provides fundamental definitions, random measures, compensators, and the "Itô's formula" necessary for understanding BSDEs.
Classical Results on BSDEs with jumps: Reviews existing work on BSDEs, specifically focusing on uniqueness and existence proofs for Lipschitz and monotone generators.
BSDEs of a specific generator: Recapitulates results for finite activity and extends them to the infinite activity case using comparison theorems and monotone stability.
Application to utility maximization: Applies the developed BSDE theory to solve an exponential utility maximization problem in a financial market.
Appendix A: Contains auxiliary technical results including properties of weak convergence and essential lemmas for the main proofs.
Theses: Summarizes the core mathematical contributions and theoretical findings of the work.
Key Keywords
Backward Stochastic Differential Equations, BSDEs, Jumps, Brownian Motion, Random Measures, Compensators, Itô’s Formula, Lipschitz generators, Monotonicity, Infinite activity, Utility maximization, BMO-martingales, Comparison theorem, Financial markets, Stochastic integration.
Frequently Asked Questions
What is the primary subject of this thesis?
The work primarily deals with Backward Stochastic Differential Equations (BSDEs) that include jump components, specifically those driven by Brownian motion and a random measure.
What is the main objective?
The core objective is to establish existence and uniqueness results for solutions to these BSDEs under conditions where the generator satisfies monotonicity properties rather than the standard global Lipschitz conditions.
What mathematical methods are employed?
The thesis utilizes stochastic analysis, including the theory of random measures, Itô’s formula for jump processes, Girsanov’s theorem, and comparison theorems for BSDEs.
How is the "infinite activity" case approached?
The infinite activity case is addressed by starting with finite activity results, establishing a comparison theorem, and using monotone stability techniques to extend the findings.
What application is presented?
The thesis applies these BSDE techniques to solve the Exponential Utility Maximization problem within a defined financial market model.
What are the key keywords characterizing the work?
Key terms include BSDEs, Jumps, Infinite Activity, Monotonicity, and Utility Maximization.
Does the thesis rely on the finiteness of the measure λ?
No, one of the main contributions is the extension of existing results to the infinite case, where the measure λ does not need to be finite.
What is the role of the BMO-martingales in this work?
BMO-martingales are crucial in providing the necessary estimates for the existence proofs and ensuring the stability of the solution processes.
- 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
