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## Über dieses Buch

This book introduces readers to the basic concepts of and latest findings in the area of differential equations with uncertain factors. It covers the analytic method and numerical method for solving uncertain differential equations, as well as their applications in the field of finance. Furthermore, the book provides a number of new potential research directions for uncertain differential equation. It will be of interest to researchers, engineers and students in the fields of mathematics, information science, operations research, industrial engineering, computer science, artificial intelligence, automation, economics, and management science.

## Inhaltsverzeichnis

### Chapter 1. Introduction

Abstract
Uncertain differential equation is a type of differential equations involving uncertain processes. So far, uncertain differential equation driven by the Liu process, uncertain differential equation with jumps, multi-dimensional uncertain differential equation, and high-order uncertain differential equation have been proposed, which are all covered in this book.
Kai Yao

### Chapter 2. Uncertain Variable

Abstract
Uncertainty theory was founded by Liu [36] in 2007 and perfected by Liu [38] in 2009 to deal with human’s belief degree based on four axioms, and uncertain variable is the main tool to model a quantity with human uncertainty in uncertainty theory. The emphases of this chapter are on uncertain measure, uncertain variable, uncertainty distribution, inverse uncertainty distribution, operational law, expected value, and variance.
Kai Yao

### Chapter 3. Uncertain Process

Abstract
Uncertain process is a sequence of uncertain variables indexed by the time. The emphases of this chapter are on the concepts about an uncertain process including its uncertainty distribution, extreme value, and time integral.
Kai Yao

### Chapter 4. Contour Process

Abstract
Contour process is a type of uncertain processes with some special structures so that the set of contour processes is closed under the extreme value operator, time integral operator, and monotone function operator. This chapter introduces contour processes, including their inverse uncertainty distributions, extreme values, time integrals, and monotone functions.
Kai Yao

### Chapter 5. Uncertain Calculus

Abstract
Uncertain calculus deals with the integration and differentiation of uncertain processes. This chapter introduces uncertain calculus with respect to Liu process. The emphases of this chapter are on Liu integral and Liu process as well as the fundamental theorem and integration by parts.
Kai Yao

### Chapter 6. Uncertain Differential Equation

Abstract
Uncertain differential equation is a type of differential equations involving uncertain processes. This chapter introduces uncertain differential equations driven by Liu processes, including analytic methods, Yao–Chen formula, numerical methods, existence and uniqueness theorem, and stability theorems as well as their applications in stock markets.
Kai Yao

### Chapter 7. Uncertain Calculus with Renewal Process

Abstract
Uncertain renewal process is a nonnegative integer valued uncertain process, which counts the number of renewals that an uncertain system occurs. Uncertain calculus with renewal process deals with the integration and differentiation of uncertain processes with respect to renewal processes. The emphases of this chapter are on Yao integral and Yao process as well as the fundamental theorem and integration by parts.
Kai Yao

### Chapter 8. Uncertain Differential Equation with Jumps

Abstract
Uncertain differential equation with jumps is a type of differential equations driven by both canonical Liu processes and uncertain renewal processes. This chapter introduces uncertain differential equation with jumps, including existence and uniqueness theorem, and stability theorems as well as its application in stock markets.
Kai Yao

### Chapter 9. Multi-Dimensional Uncertain Differential Equation

Abstract
Multi-dimensional uncertain differential equation is a system of uncertain differential equations. The emphases of this chapter are on multi-dimensional Liu process, multi-dimensional uncertain calculus, and multi-dimensional uncertain differential equation. For simplicity, we employ the infinite norm in this chapter and write
$$|{\varvec{x}}|=\bigvee _{i=1}^n|x_i|,\quad |A|=\bigvee _{i=1}^m\sum _{j=1}^n |a_{ij}|$$
for an n-dimensional vector $${\varvec{x}}=(x_1,x_2,\ldots ,x_n)^T$$ and an $$m\times n$$ matrix $$A=[a_{ij}],$$ respectively.
Kai Yao

### Chapter 10. High-Order Uncertain Differential Equation

Abstract
High-order uncertain differential equation is a type of uncertain differential equations involving the high-order derivatives of uncertain processes. This chapter introduces high-order uncertain differential equation including its equivalent transformation to a multi-dimensional uncertain differential equation, Yao formula, numerical method, and existence and uniqueness theorem.
Kai Yao

### Backmatter

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