math 396A: Introduction to Stochastic Differential Equations

The course gives an introduction to the topic of stochastic differential equations with focus on its application in financial mathematics. In chapter 6.3 I try to explain the famous Black &Scholes formula which won the Nobel price of Economy in 1973 and is the theoretical base of option pricing.
As preliminaries you should have passed a course similar to math141 (calculus). A complete course in probability theory or ordinary differential equations is not necessary, since we will briefly introduce the basic notions in the introducing chapters, but could be helpful as motivation as well as for easier understanding.

Textbook: B. K. Øksendal: Stochastic Differential Equations, An introduction with applications Springer, Berlin 2003
for complementary reading, I suggest: I. Karatzas, S. Shreve: Brownian motion and stochastic calculus Springer, New York 1991 (2nd edit)

Course outline (tentative)

The number in parentheses gives the approximate number of class periods.

1. Introduction
   1. Stochastic Analogs of Classical Differential Equations (1h)
   2. Examples: Optimal Stopping, Stochastic Control, Mathematical Finance (1h)
2. Mathematical Preliminaries I: Probability theory
   1. Probability Spaces, Random Variables (1h)
   2. Stochastic Processes (0.5h)
   3. An important example: Brownian Motion (1h)
3. Preliminaries II: Classical analog, Ordinary differential euqations
   1. First and n-th order ODE (1h)
   2. Vector valued ODE (0.5h)
   3. Existence and Uniqueness theorems (1h)
4. Itô Integrals, Itô formula and Martingale Representation Theorem
   1. Definition and elementary properties (1h)
   2. The 1-dim Itô-Fromula (1h)
   3. Multi-dim formula (0.5h)
   4. Martingale representation thm (1.5h)
5. Stochastic Differential Equations
   1. Examples with some Solution methods (2h)
   2. Existence and Uniqueness result (1h)
   3. Weak and Strong solutions (0.5h)
6. Application to Mathematical Finance
   1. Market, Portfolio and Arbitrage (1h)
   2. Attainability and Completeness (1h)
   3. Option pricing (1.5h)
7. Other topics as time permits, p.ex.
   • Diffusions (2h)
   • Applications to Boundary Value Problems (2h)
   • Application to Optimal stopping (2h)
   • Application to Stochastic Control (2.5h)

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