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

Abstract

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.

Contents

0.1 Introduction . . 1

1 Preliminaries . . 3

1.1 Random measures and compensators . . 3

1.2 The weak property of predictable representation . . 8

1.3 Itô’s formula . . 11

2 Classical Results on BSDEs with jumps . . 13

2.1 Uniqueness . . 16

2.2 Existence for Lipschitz generators . . 18

2.3 Existence in general: Proof of Proposition 2.11 . . 21

3 BSDEs of a specific generator . . 29

3.1 The case of finite activity . . 31

3.2 Generators of difference type . . 35

3.3 A comparison theorem . . 40

3.4 The case of infinite activity . . 43

3.4.1 Uniqueness . . 44

3.4.2 Existence . . 45

4 Application to utility maximization . . 55

4.1 The financial market framework . . 55

4.2 Exponential utility maximization . . 56

5 Appendix A . . 63

5.1 Properties of weak convergence . . 63

5.2 Technical results . . 63

6 Theses . . 66

References . . 67

[The correct symbols are omitted from this preview]

0.1 Introduction

A solution of a Backward Stochastic Differential Equation (BSDE) is defined as a tuple (Y, Z) of processes in suitable spaces solving the equation

[Formula is omitted from this preview] (0.1)

for some fixed time horizon T , suitable terminal condition C and generator f under a Brownian filtration.

The case of Lipschitz generators (Pardoux and Peng, 1990) and generators with quadratic growth in z (Kobylanski, 2000) are well understood.

However, our focus is on BSDEs with jumps of the form

[Formula is omitted from this preview]

which involves another stochastic integral with respect to a compensated measure of an integer-valued random measure.

Its compensator is assumed to be absolutely continuous to the product measure A with a bounded density C where A is a o-finite measure satisfying a certain integrability condition.

Moreover, we assume that u and the Brownian Motion B have the weak property of predictable representation.

An existence and uniqueness result for Lipschitz generators has been proven by Barles, Buckdahn and Pardoux [1] in 1995.

Pardoux [13] extended these ideas and considered generators where the Lipschitz continuity in y is replaced by a continuity and monotonicity condition in y as well as a linear bound on f.

In 2006, Becherer [3] considered a specific class of generators under the assumption that the measure of A is finite.

The main contribution of this paper is the extension to the infinite case under some addi-tional assumptions on the generator.

The thesis is organized as follows:

In the preliminary part, we give an introduction on the topic of BSDEs with jumps, define random measures, compensators and the weak property of predictable representation and state important results in this setting such as ”Itô's formula”.

Secondly in Chapter 2, we review Pardoux’s work [13] and prove his results for inhomogenous compensators in detail by making reference to his later paper [14] where he considers the same problem for BSDE without jumps.

In Chapter 3, we recapitulate the existence and uniqueness result in case of finite activity (A is finite) presented by Becherer [3] under more general assumptions.

Next, we consider generators of difference type and show that the admit unique solutions such that the first component is controlled by the Lipschitz constant K_f of f and the supremum norm E of the terminal condition.

Using a linearization and measure change based argument as in Royer [16], we can state an comparison theorem which allows us to construct an increasing sequence of Y -components for an increasing sequence of generators.

These steps enables us to advance to the case of infinite activity.

After proving uniqueness of solutions by a comparison theorem we show their existence using the idea of monotone stability from Kobylanski [9] and Nutz [12] and techniques presented by Morlais [11].

Finally, in Chapter 4, we consider the Exponential Utility Maximization problem in a suitable financial market model.

We solve the problem by applying BSDE techniques and the general theory to our setting in the spirit of Becherer [3].

However, we do not rely on the finiteness of the measure A.

The Appendix contains properties of weak convergence and it gives an argument why the stochastic integrals in Ito’s formula vanish for our application when taking (conditional) expectations.

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