Content
Chapter 1 BSDEs Driven by Lévy Processes 1
1.1 Preliminaries: notations and theorems 2
1.2 BSDEs for Lévy processes 4
1.2.1 Comparison theorem 4
1.2.2 An existence and uniqueness theorem 8
1.3 BSDEs with reflecting barriers 13
1.3.1 Introduction and preliminaries 13
1.3.2 BSDEs with one reflecting barrier: comparison 16
1.3.3 BSDEs with two reflecting barriers 18
1.3.4 Comparison theorem 23
1.4 RBSDEs with time delayed generators 26
1.4.1 Introduction 26
1.4.2 Preliminaries and notations 26
1.4.3 Priori estimates 29
1.4.4 Existence and uniqueness of the solution 39
1.5 Lp-solutions for RBSDEs with time delayed generators 42
1.5.1 Preliminaries and notations 42
1.5.2 Priori estimates 45
1.5.3 Existence and uniqueness of the solution 60
1.6 BSPDES for Lévy processes 65
1.6.1 Introduction 65
1.6.2 Preliminaries: notations and lemmas 68
1.6.3 BSPDEs driven by Lévy processes 69
1.6.4 Concluding remarks 81
Chapter 2 Financial Markets Driven by Lévy Processes 82
2.1 The power utility maximization problem 82
2.1.1 Introduction 82
2.1.2 The formulation of the problem 83
2.1.3 Solution in terms of triplets 84
2.1.4 A particular case 88
2.1.5 Appendix 89
2.2 Optimal investment for an insurer: the martingale approach 91
2.2.1 Introduction 91
2.2.2 Problem formulation 92
2.2.3 CARA Utility 96
2.3 Cooperative hedging in two explicit model 100
2.3.1 Introduction 100
2.3.2 Preliminary and notation 101
2.3.3 Optimal cooperative hedging of the complete case 104
2.3.4 Optimal cooperative hedging of a volatility jump model 108
2.4 Cooperative hedging with a higher interest rate for borrowing 110
2.4.1 Introduction 110
2.4.2 The model 111
2.4.3 The optimal cooperative hedging strategy 113
2.4.4 Two lemmas about BSDE 114
2.5 Two-agent Pareto optimal cooperative investment 116
2.5.1 Introduction 116
2.5.2 The model 117
2.5.3 Motivation 120
2.5.4 Main results 123
2.5.5 Calculating u(x, T0) explicitly 127
2.5.6 Concluding remarks 130
2.6 Cooperative hedging under g-expectation constraint 130
2.6.1 Introduction 130
2.6.2 The preliminaries about Neyman-Pearson lemma 132
2.6.3 The problem formulation 135
2.6.4 Optimal cooperative hedging of the complete case 136
Chapter 3 Optimal Control via Malliavin Calculus 140
3.1 Mean-field stochastic maximum principle 140
3.1.1 Introduction and preliminaries 140
3.1.2 A brief review of Malliavin calculus for Lévy processes 143
3.1.3 The stochastic maximum principle 147
3.2 Partial information maximum principle via Malliavin calculus 157
3.2.1 Introduction 157
3.2.2 The stochastic maximum principle 161
3.2.3 An application 174
3.3 Stochastic maximum principle for jump-diffusion mean-field FBSDEs 176
Chapter 4 Pricing Vulnerable Options 186
4.1 Variable default boundary under jump-diffusion model 188
4.1.1 The model 188
4.1.2 Valuation of European vulnerable options 190
4.1.3 Three specific examples 195
4.1.4 Appendix 198
4.2 Random corporate liabilities 207
4.2.1 The model 208
4.2.2 Valuation of European vulnerable options 210
4.2.3 Specific cases of the pricing formula 215
4.2.4 Conclusion 220
4.2.5 Appendix 221
Bibliography 225