Room 201, Astronomy-Mathematics Building, NTU

**Speaker:**

Tzuu-Shuh Chiang (Academia Sinica)

Shuenn-Jyi Sheu (National Central University)

Wei-Da Chen (National Central University)

**Organizers:**

Lung-Chi Chen (National Chengchi University)

Abstract

The purpose of this short course is to give an introduction about the relation of probability theory and partial differential equation. The probability distribution of a Markov process is governed by its Kolmogorov’s equation. The heat equation is the Kolmogorov’s equation for Brownian motion. This brings the connection of solution of heat equation and the distribution of the Brownian motion. When we study the heat equation in a region with boundary conditions, we consider the hitting time of the Brownain motion to the boundary. The Markov property of the Brownian motion is used to calculate the distribution of hitting time. We further consider the Kolmogorov’s equation for the Brownian motion with drift which is given by a generalized heat equation, a second order parabolic equation. The stochastic integration and Ito’s differential rule are tools to study such partial differential equation by probability theory. We also illustrate some applications, we discuss the Black-Scholes option pricing formular and its hedging strategy, the minimization of ruin probability with risk income.

Main Text:

S.R.S. Varadhan (1989), Lectures on Diffusion Problems and Partial Differential Equations, Tata Institute of Fundamental Research, Bombay

Additional Reading:

I. Karatzas and S.E. Shreve (1998), Brownian Motion and Stochastic Calculus, 2nd Edition, Springer I. Karatzas and S.E. Shreve (1998), Method of Mathematical Finance, Springer D. Stroock (2008), Partial Differential Equations for Probabilists, Cambridge University Press

Brief Content

1. Heat Equation and Brownian motion

1.1: Solution of heat equation and distribution of Brownian motion.

1.2: Wiener measure and Kolmogorov’s Theorem.

1.3: Independent increment and Markov property.

2. Heat equation with Boundary Value in 1-dimension.

2.1: 1-d heat equation with Dirichlet boundary value.

2.2: Reflection principle and the hitting time and hitting distribution.

2.3: 1-d heat equation with Neumann boundary value.

3. Dirichlet Problem and Brownian Motion.

4. Stochastic Integration and Ito’s Differential Rule

4.1: Stochastic integration

4.2: Ito’s formula.

4.3: Poisson equation.

5. Brownian Motion with Drift

5.1: Integral equation and stochastic differential equation.

5.2: Carmeron-Martin Girsanov Theorem.

5.3: Application Carmeron-Martin Girsanov Theorem

6. Applications to Mathematical Finance

6.1: Black-Scholes option pricing formula

6.2: Minimizing ruin probability

**Contact:**
Risa, 02-3366-8811, risalu@ncts.ntu.edu.tw

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