Stochastic Calculus for Quantitative Finance

In 1994 and 1998 F. Delbaen and W. Schachermayer published two breakthrough papers where they proved continuous-time versions of the Fundamental Theorem of Asset Pricing. This is one of the most remarkable achievements in modern Mathematical Finance which led to intensive investigations in many applications of the arbitrage theory on a mathematically rigorous basis of stochastic calculus.

Mathematical Basis for Finance: Stochastic Calculus for Finance provides detailed knowledge of all necessary attributes in stochastic calculus that are required for applications of the theory of stochastic integration in Mathematical Finance, in particular, the arbitrage theory. The exposition follows the traditions of the Strasbourg school.

This book covers the general theory of stochastic processes, local martingales and processes of bounded variation, the theory of stochastic integration, definition and properties of the stochastic exponential; a part of the theory of Lévy processes. Finally, the reader gets acquainted with some facts concerning stochastic differential equations.

• Contains the most popular applications of the theory of stochastic integration
• Details necessary facts from probability and analysis which are not included in many standard university courses such as theorems on monotone classes and uniform integrability
• Written by experts in the field of modern mathematical finance

Graduate students and professors worldwide working in all subdisciplines of economics and finance

• Basic Notation
• Preface
• List of Statements
  • Definitions
  • Examples
  • Exercises
  • Remarks
  • Propositions
  • Theorem
  • Lemmas
  • Corollaries
• 1. General Theory of Stochastic Processes
  • Abstract
  • 1.1 Stochastic basis and stochastic processes
  • 1.2 Stopping times
  • 1.3 Measurable, progressively measurable, optional and predictable σ-algebras
  • 1.4 Predictable stopping times
  • 1.5 Totally inaccessible stopping times
  • 1.6 Optional and predictable projections
• 2. Martingales and Processes with Finite Variation
  • Abstract
  • 2.1 Elements of the theory of martingales
  • 2.2 Local martingales
  • 2.3 Increasing processes and processes with finite variation
  • 2.4 Integrable increasing processes and processes with integrable variation. Doléans measure
  • 2.5 Locally integrable increasing processes and processes with locally integrable variation
  • 2.6 Doob–Meyer decomposition
  • 2.7 Square-integrable martingales
  • 2.8 Purely discontinuous local martingales
• 3. Stochastic Integrals
  • Abstract
  • 3.1 Stochastic integrals with respect to local martingales
  • 3.2 Semimartingales. Stochastic integrals with respect to semimartingales: locally bounded integrands. Itô’s formula
  • 3.3 Stochastic exponential
  • 3.4 Stochastic integrals with respect to semimartingales: the general case
  • 3.5 σ-martingales
• Appendix
  • A.1 Theorems on monotone classes
  • A.2 Uniform integrability
  • A.3 Conditional expectation
  • A.4 Functions of bounded variation
• Bibliographical Notes
  • Chapter 1
  • Chapter 2
  • Chapter 3
  • Appendix
• Bibliography
• Index