by J Yeh (University of California, Irvine)

This book is a thorough and self-contained treatise of martingales as a tool in stochastic analysis, stochastic integrals and stochastic differential equations. The book is clearly written and details of proofs are worked out.

• Stochastic Processes:
  • Generated σ-Algebras
  • Stochastic Processes
  • Stopping Times
  • Convergence in L_p and Uniform Integrability
• Martingales:
  • Martingale, Submartingale and Supermartingale
  • Fundamental Submartingale Inequalities
  • Convergence of Submartingales
  • Uniformly Integrable Submartingales
  • Regularity of Sample Functions of Submartingales
  • Increasing Processes
• Stochastic Integrals:
  • L₂-Martingales and Quadratic Variation Processes
  • Stochastic Integrals with Respect to Martingales
  • Adapted Brownian Motions
  • Extensions of the Stochastic Integral
  • Itô's Formula
  • Itô's Stochastic Calculus
• Stochastic Differential Equations:
  • The Space of Continuous Functions
  • Definition and Function Space Representation of Solution
  • Existence and Uniqueness of Solutions
  • Strong Solutions