Excerpt

## TABLE OF CONTENTS

**CERTIFICATION**

**DEDICATION**

**ACKNOWLEDGEMENT**

**CHAPTER ONE**

**1.1 INTRODUCTION**

**1.2 SCOPE OF WORK**

**1.3 OBJECTIVE**

**1.4 IMPORTANT DEFINITIONS AND NOTATION USED IN THIS WORK**

**1.5 Limitations**

**1.5 USEFULNESS OF STOCHASTIC DIFFERENTIAL EQUATIONS**

**1.6 Conclusion**

**CHAPTER TWO**

**2.1. BROWNIAN MOTION**

**2.2 Martingale Theorem**

**2.3 Quadratic Variation of Brownian Motion**

**2.4 ITO PRODUCT RULE**

**2.5 Properties of Ito Integral**

**CHAPTER THREE**

**3.1 STOCHASTIC DIFFERENTIAL EQUATIONS**

**3.2 STRONG AND WEAK SOLUTIONS**

**CHAPTER FOUR**

**APPLICATION TO MATHEMATICAL FINANCE**

**CHAPTER FIVE**

**5.1. SUMMARY**

**5.2. CONCLUSION**

**References**

## CERTIFICATION

This is to certify that this project work was carried out by ERHABOR OSAHON MOSES in the Department of Mathematics, Industrial Mathematics, University of Benin, Benin city

## DEDICATION

I dedicate this project to God Almighty for his grace, guidance and success over me.

This project is also dedicated to everyone in ERHABOR and OVIAWE family.

## ACKNOWLEDGEMENT

Firstly, I appreciate God Almighty who gave me strength, wisdom and grace during this compilation. I am grateful to my parents Pastor and Mrs. ERHABOR for their prayers, words of encouragement and financial support for the compilation of this project

I also extend my thanks to my uncle Mr. Oviawe Tony for his encouragement and financial support.

My thanks goes to my friends in the department who were instrumental in the compilation of this work, in the persons of Emmanuel Mitti, Alekhue Isaac Omokaro Uyi, and Ezekah Chikee

My profound gratitude goes to Dr. Chiedozie Ibe, my project supervisor for his patience, encouragement and guide throughout the compilation of this project.

## CHAPTER ONE

### 1.1 INTRODUCTION

A stochastic differential equation (SDE) is a differential equation in which one or more of the terms is a stochastic process resulting in a solution which is a stochastic process. Stochastic differential equation are used to model various phenomena such as stock prices. Typically, SDES contain a variable which represents random white noise calculated as the derivative of Brownian motion or the Wiener process

### 1.2 SCOPE OF WORK

This project covers the stochastic differentials, Brownian motion, Ito integral and Brownian motion and its application to finance

### 1.3 OBJECTIVE

The ultimate goal of this work is to introduce the solution to some stochastic differential equations

### 1.4 IMPORTANT DEFINITIONS AND NOTATION USED IN THIS WORK

The following are definitions and notations which will be useful in the cause of the study

1. Stochastic differential Equation: stochastic differential equation of the stochastic process t is defined as given the initial condition

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2. Brownian Motion: This is the random movement of particles suspended in a fluid

3. Strong Solution: A strong solution of the stochastic differential equation with initial condition is an adapted process with continuous paths such that for all

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4. Weak Solution: A weak solution of the stochastic differential equation with initial condition continuous stochastic process defined on some probability such that for some wiener process and some admissible filtration the process is adapted and satisfies the stochastic integral

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### 1.5 Limitations

At first, I encountered difficulty writing this topic because of its newness to me but I was able to come in terms with it from the knowledge of classical calculus

### 1.5 USEFULNESS OF STOCHASTIC DIFFERENTIAL EQUATIONS

Stochastic differential Equations is useful in the fields of Mathematics, Statistics, Sciences and Economics

### 1.6 Conclusion

Chapter one deals with the introduction, unique terms and notation and the usefulness in the project work.

Chapter two deals with Brownian motion, Ito integral and some simple problems were considered

Chapter three deals with stochastic differential equations and some simple problems were considered

Chapter four handles the application of Stochastic Differential Equations to finance

Chapter five concludes the project.

## CHAPTER TWO

### 2.1. BROWNIAN MOTION

The name Brownian motion was first mentioned by Robert Brown when he discovered that the motion of grain in water is zigzag and haphazard.

Definition of Brownian Motion: This is a random process in continuous time and continuous space

Definition 2.1.1: A random function X: [0 , ∞] R written as Xt is Brownian motion with variance starting at o if

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### 2.2 Martingale Theorem

Theorem 2.2.1: suppose that is Brownian motion. Thus

1.X_{t} is a continuous martingale

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### 2.3 Quadratic Variation of Brownian Motion

The quadratic variation for Brownian motion over the interval [0,T], denoted by

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Clearly is a function of the sample points w . Hence the quadratic variation calculated for Brownian motion for each partition itself is a random variable. Note that, the limits is taken over all partition of [0, T] with an n

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Since for each partition is a random variable, how to find limiting distribution of for large n? The question would be what is the proper mode for convergence in these random variables. We shall use convergence in mean square (convergence in L2). The definition of convergence in L2 is as follows

Definition:

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In the case of Brownian motion, we say that

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When the above results hold good, we say that the quadratic variation accumulated by the Brownian motion over the interval [0 ,T] is T almost surely and is denoted as [W, W](T) = T. The above result is proved in the following theorems

Theorem

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Since for fixed is normal distribution with mean 0 and variance

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Using the fourth order moment of normal distribution with mean 0 and variance ( is 3( 2, we get

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### 2.4. ITO INTEGRAL

Suppose Brownian motion = Wt

Quadratic variation (dWt)= dt

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Considering two variable (t, x), we want to evaluate F(t, x) =F(t, wt)

Now using Taylor’s series expansion

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Note: The Ito lemma above is also referred to as Ito chain rule

**[...]**

- Quote paper
- Erhabor Moses (Author), 2019, Stochastic Differential Equations and Their Application in Finance. An Overview, Munich, GRIN Verlag, https://www.grin.com/document/513307

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