Intro; Introduction; Contents; Part I Courses; Scaling Limits of Markov-Branching Trees and Applications; 1 Introduction; 2 Discrete Trees, Examples and Motivations; 2.1 Discrete Trees; 2.2 First Examples; 2.3 The Markov-Branching Property; 3 The Example of Galton-Watson Trees and Topological Framework; 3.1 Real Trees and the Gromov-Hausdorff Topology; 3.2 Scaling Limits of Conditioned Galton-Watson Trees; 4 Scaling Limits of Markov-Branching Trees; 4.1 A Markov Chain in the Markov-Branching Sequence of Trees; 4.2 Scaling Limits of Non-increasing Markov Chains

4.3 Self-Similar Fragmentation Trees4.3.1 Self-Similar Fragmentation Processes; 4.3.2 Self-Similar Fragmentation Trees; 4.4 Scaling Limits of Markov-Branching Trees; 5 Applications; 5.1 Galton-Watson Trees; 5.1.1 Galton-Watson Trees with n Vertices; 5.1.2 Galton-Watson Trees with Arbitrary Degree Constraints; 5.2 Pólya Trees; 5.3 Dynamical Models of Tree Growth; 5.3.1 Ford's Alpha Model; 5.3.2 k-Ary Growing Trees; 5.3.3 Marginals of Stable Trees; 5.4 Cut-Trees; 6 Further Perspectives; 6.1 Multi-Type Markov-Branching Trees and Applications; 6.2 Local Limits

6.3 Related Random Geometric StructuresReferences; Optimality of Two-Parameter Strategies in Stochastic Control; 1 Introduction; 1.1 One-Parameter Strategies; 1.2 Two-Parameter Strategies; 1.2.1 Two-Sided Singular Control; 1.2.2 Impulse Control; 1.2.3 Zero-Sum Games Between Two Players; 1.3 Fluctuation Theory of Spectrally One-Sided Lévy Processes; 1.4 Solution Procedures; 1.4.1 Selection of the Two Parameters; 1.4.2 Verification of Optimality; 1.5 Comparison with Other Approaches; 1.6 Computation; 2 Spectrally Negative Lévy Processes and Scale Functions; 2.1 Path Variations and Regularity

2.2 Scale Functions2.3 Smoothness of Scale Functions; 2.4 Fluctuation Identities for Spectrally Negative Lévy Processes; 2.4.1 Two-Sided Exit; 2.4.2 Resolvent Measures; 2.5 Fluctuation Identities for the Infimum and Reflected Processes; 2.5.1 Fluctuation Identities for the Infimum Process; 2.5.2 Fluctuation Identities for tb; 2.5.3 Fluctuation Identities for Yta; 2.6 Fluctuation Identities for Doubly Reflected Lévy Processes; 2.7 Other Properties of the Scale Function; 2.7.1 Asymptotics as x →∞; 2.7.2 Log-Concavity; 2.7.3 Martingale Properties; 2.8 Some Further Notations

3 Two-Sided Singular Control3.1 The Double Reflection Strategy; 3.2 Smoothness of the Value Function; 3.3 Existence of (a*, b*); 3.3.1 The Case of Example 3.1; 3.3.2 The Case of Example 3.2; 3.3.3 The Case of Example 3.3; 3.4 Variational Inequalities and Verification; 4 Impulse Control; 4.1 The (s,S)-Strategy; 4.2 Smoothness of the Value Function; 4.2.1 The Case of Example 4.3; 4.2.2 Brief Remarks on the Cases of Examples 4.1 and 4.2; 4.3 Quasi-Variational Inequalities and Verification; 4.3.1 The Case of Example 4.3; 4.3.2 Brief Remarks on the Cases of Examples 4.1 and 4.2