Applied Stochastic Control of Jump Diffusions

by Oksendal, Bernt; Sulem, Agnes; Ksendal, B. K.

• ISBN13: 9783540140238
• ISBN10: 3540140239
• Format: Paperback
• Copyright: 2005-01-30
• Publisher: Springer Verlag

Summary

The main purpose of the book is to give a rigorous, yet mostly nontechnical, introduction to the most important and useful solution methods of various types of stochastic control problems for jump diffusions and its applications. The types of control problems covered include classical stochastic control, optimal stopping, impulse control and singular control. Both the dynamic programming method and the maximum principle method are discussed, as well as the relation between them. Corresponding verification theorems involving the Hamilton-Jacobi Bellman equation and/or (quasi-)variational inequalities are formulated. There are also chapters on the viscosity solution formulation and numerical methods. The text emphasises applications, mostly to finance. All the main results are illustrated by examples and exercises appear at the end of each chapter with complete solutions. This will help the reader understand the theory and see how to apply it. The book assumes some basic knowledge of stochastic analysis, measure theory and partial differential equations.

Author Biography

Agnes Sulem is a researcher at INRIA, the French National Institute for Research in Computer Science and Control, in Rocquencourt near Paris, and teaches in the doctoral program at Paris-Dauphine.

Table of Contents

1 Stochastic Calculus with Jump diffusions	1	(26)
1.1 Basic definitions and results on Lévy Processes	1	(4)
1.2 The Itô formula and related results	5	(5)
1.3 Lévy stochastic differential equations	10	(2)
1.4 The Girsanov theorem and applications	12	(7)
1.5 Application to finance	19	(3)
1.6 Exercises	22	(5)
2 Optimal Stopping of Jump Diffusions	27	(12)
2.1 A general formulation and a verification theorem	27	(4)
2.2 Applications and examples	31	(5)
2.3 Exercises	36	(3)
3 Stochastic Control of Jump Diffusions	39	(20)
3.1 Dynamic programming	39	(7)
3.2 The maximum principle	46	(6)
3.3 Application to finance	52	(3)
3.4 Exercises	55	(4)
4 Combined Optimal Stopping and Stochastic Control of Jump Diffusions	59	(12)
4.1 Introduction	59	(1)
4.2 A general mathematical formulation	60	(5)
4.3 Applications	65	(4)
4.4 Exercises	69	(2)
5 Singular Control for Jump Diffusions	71	(10)
5.1 An illustrating example	71	(2)
5.2 A general formulation	73	(3)
5.3 Application to portfolio optimization with transaction costs	76	(2)
5.4 Exercises	78	(3)
6 Impulse Control of Jump Diffusions	81	(16)
6.1 A general formulation and a verification theorem	81	(4)
6.2 Examples	85	(9)
6.3 Exercices	94	(3)
7 Approximating Impulse Control of Diffusions by Iterated Optimal Stopping	97	(16)
7.1 Iterative scheme	97	(10)
7.2 Examples	107	(5)
7.3 Exercices	112	(1)
8 Combined Stochastic Control and Impulse Control of Jump Diffusions	113	(10)
8.1 A verification theorem	113	(3)
8.2 Examples	116	(4)
8.3 Iterative methods	120	(2)
8.4 Exercices	122	(1)
9 Viscosity Solutions	123	(26)
9.1 Viscosity solutions of variational inequalities	124	(3)
9.2 The value function is not always C¹	127	(3)
9.3 Viscosity solutions of HJBQVI	130	(10)
9.4 Numerical analysis of HJBQVI	140	(6)
9.5 Exercises	146	(3)
10 Solutions of Selected Exercises	149	(48)
10.1 Exercises of Chapter 1	149	(4)
10.2 Exercises of Chapter 2	153	(9)
10.3 Exercises of Chapter 3	162	(7)
10.4 Exercises of Chapter 4	169	(2)
10.5 Exercises of Chapter 5	171	(3)
10.6 Exercises of Chapter 6	174	(11)
10.7 Exercises of Chapter 7	185	(3)
10.8 Exercises of Chapter 8	188	(3)
10.9 Exercises of Chapter 9	191	(6)
References	197	(6)
Notation and Symbols	203	(4)
Index	207