Martingale Theory and Itô Stochastic Analysis
In this chapter, we will briefly introduce the martingale theory and Itô’s stochastic analysis. First we introduce some basic concepts of continuous time stochastic processes and the definitions of four basic types of process: Markov process, martingale, Poisson process, and Brownian motion, as well as their basic properties, including Doob-Meyer’s decompositions of continuous local submartingales and quadratic variation processes of continuous local martingales and continuous semimartingales. Then we introduce the stochastic integrals of measurable adapted processes w.r.t. the Brownian motion. Finally, we introduce some useful tools for Itô calculus, such as Itô’s formula, Girsanov’s theorem, and the martingale representation theorem. Itô stochastic differential equations and Feynman-Kac formula are also presented. This chapter is self-contained. For a small number of results, we omit their proofs and refer to Karatzas and Shreve (1991) or Revuz and Yor (1999).
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