# Math 438: Modeling with Dynamics and Control 2

## Contents

## Catalog Information

### Title

Differential and Integral Equations 2

### (Credit Hours:Lecture Hours:Lab Hours)

(3:3:0)

### Offered

W

### Prerequisite

Math 436, Math 402; concurrent with Math 439, Math 404

### Description

An introduction to the theory of integral equations, of the calculus of variations, of stochastic differential equations and of optimal stochastic control. An introduction to the algorithms that are commonly used to study these systems

## Desired Learning Outcomes

### Prerequisites

Math 436, Math 402; concurrent with Math 439, Math 404

### Minimal learning outcomes

Students will have a solid understanding of the concepts listed below. They will be able to prove theorems that are central to this material, including theorems that they have not seen before. They will understand the model specifications for the algorithms, and be able to recognize whether they apply to a given application or not. They will be able to perform the relevant computations on small, simple problems. They will have a basic knowledge of the capabilities of commercial software that available for these problems.

- Integral Equations
- Classification and Origins
- Relationship to Differential Equations
- Fredholm Equations
- Symmetric Kernels
- Volterra Equations
- General Integral Equations

- Calculus of Variations
- Variational Problems
- Euler-Lagrange Condition
- Second Variation
- Generalizations of the Variational Problem
- Hamiltonian Theory

- Optimal Control
- Problem Formulation
- Hamilton-Jacobi-Bellman Equation
- The Adjoint Equation
- Sufficient Conditions
- Linear Quadratic Regulator (LQR)

- Stochastic Differential Equations
- Brownian Motion and Diffusion
- Weiner Processes
- Itô Processes and Itô's Lemma
- Black-Scholes Equation

- Stochastic Optimal Control
- Problem Formulation
- Hamilton-Jacobi-Bellman Equation
- Optimal Stopping Times
- Linear Quadratic Gaussian (LQG)
- Investment-Consumption Problems

### Textbooks

Possible textbooks for this course include (but are not limited to):