Introduction to Stochastic Integration

INTRODUCTION TO STOCHASTIC INTEGRATION

1.  Introduction

i.  Integrals

ii.  Random walks

2.  Brownian Motion

i.  Definition of Brownian motion

ii.  Simple properties of Brownian motion

iii.  Wiener integral

iv.  Conditional expectation

v.  Martingales

vi.  Series expansion of Wiener integrals

3.  Constructions of Brownian motion

i.  Wiener space

ii.  Borel-Cantelli lemma

iii.  Kolgomorov’s extension and continuity theorems

iv.  Levy’s interpolation method

4.  Stochastic Integrals

i.  Stochastic integrals

ii.  Simple examples of stochastic integrals

iii.  Doob submartingale inequality

iv.  Stochastic processes defined by Ito integrals

v.  Riemann sums and stochastic integrals

5.  An extension of stochastic integrals

i.  A larger class of integrands

ii.  The key lemma

iii.  General stochastic integrals

iv.  Stopping times

v.  Associated stochastic processes

6.  Stochastic integrals for martingales

i.  Poisson processes

ii.  Predictable stochastic processes

iii.  Doob-Mayer decomposition theorem

iv.  Martingales as integrators

v.  Extension for integrands

7.  The Ito Formula

i.  Ito’s formula in the simplest form

ii.  Proof of Ito’s formula

iii.  Ito’s formula slightly generalized

iv.  Ito’s formula in the general form

v.  Multi-dimensional Ito’s formula

vi.  Ito’s formula for martingales

8.  Applications of the Ito Formula

i.  Evaluation of stochastic integrals

ii.  Decomposition and compensators

iii.  Stratonovich integral

iv.  Levy’s characterization theorem

v.  Multidimensional Brownian motions

vi.  Tanaka’s formula and local time

vii. Exponential processes

viii.  Transformation of probability measures

ix.  Girsanov theorem

9.  Multiple Wiener-Ito integrals

i.  Double Wiener-Ito integrals

ii.  Hermite polynomials

iii.  Homogeneous chaos

iv.  Orthonormal basis for homogeneous chaos

v.  Multiple Wiener-Ito integrals

vi.  Wiener-Ito theorem

vii. Representation of martingales

10.  Stochastic differential equations

i.  Definition and some examples

ii.  Existence and uniqueness theorem

iii.  Properties of the solution

iv.  Semigroups and diffusion processes

v.  Approximation of the solution