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Analysis and-linear-algebra-for-finance-part-i
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Patrick Roger
Analysis and Linear Algebra for Finance:
Part I
Patrick ROGER
LaRGE Research Center
EM Strasbourg Business School
University of Strasbourg
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3
Analysis and Linear Algebra for Finance: Part I
First edition
© 2013 Patrick Roger & bookboon.com (Ventus Publishing ApS)
ISBN 978-87-403-0412-1
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Analysis and Linear Algebra for Finance: Part I
4
Contents
Contents
	Introduction	 6
1	Preliminaries	 7
1.1	 Sets and subsets	 8
1.2	 Binary relations	 17
1.3	Mappings	 20
1.4	 Topology of
             
            
    
              
              
          
   
          
              
     
                  
                   
           
         

              
               
	25
2	 Functions of one variable	 43
2.1	 Definitions and notations	 44
2.2	 Limits and continuity	 50
2.3	Differentiation	 61
2.4	 Logarithms and exponential functions	 70
2.5	 Polynomial approximations and Taylor formula	 75
© Deloitte & Touche LLP and affiliated entities.
360°
thinking.
Discover the truth at www.deloitte.ca/careers
© Deloitte & Touche LLP and affiliated entities.
Dis
Deloitte & Touche LLP and affiliated entities.
360°
thinking.
Discover the truth at www.deloitte.ca/careers
© Deloitte & Touche LLP and affiliated entities.
360°
thinking.
Discover the truth at www.deloitte.ca/careers
Download free eBooks at bookboon.com
Pleaseclicktheadvert
Analysis and Linear Algebra for Finance: Part I
5
Contents
3	Integrals	 81
3.1	 Integral of a step function	 81
3.2	 General case	 86
3.3	Computations	 94
3.4	 Improper integrals	 98
4	Matrices	 101
4.1	Definitions	 101
4.2	 Elementary algebra on matrices	 104
4.3	 Linear equations	 115
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Analysis and Linear Algebra for Finance: Part I
6
Introduction
 

        
    
          
  
          
           
          
          
            
         
          
          
            
           
       
         
          
             
         

           
            
          
         
         
             

          
  

          
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Analysis and Linear Algebra for Finance: Part I
7
Preliminaries
 

         
          
            
        
          
           
             
          
 
        
          
            
            
     
    
            
             
              
            

          

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Analysis and Linear Algebra for Finance: Part I
8
Preliminaries
   
            
          
            
        
   
            
           
            
              
          
           
       
             
     
    
            
           
            
    
   
             
      
                   


            
   
               
      
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Analysis and Linear Algebra for Finance: Part I
9
Preliminaries
    
                    

            

               
  
                  
                
      
           

 
       
              
       
 

     
   


              
             
             
              
  
             
    
               
  
            
             
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Analysis and Linear Algebra for Finance: Part I
10
Preliminaries
   
             
              
     
                
 
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Analysis and Linear Algebra for Finance: Part I
11
Preliminaries
    
      
  
                
                 
         
               

               
           
   
               
        
             
    
               
                
             
    
          
                  
   
            
           
           
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Analysis and Linear Algebra for Finance: Part I
12
Preliminaries
   
          
            
            
              
           
  
    
        
           
           

             
               
 
                 
               
                
        
     

        
             
    
            
 
 
  
     
 
              
Download free eBooks at bookboon.com
Analysis and Linear Algebra for Finance: Part I
13
Preliminaries
Download free eBooks at bookboon.com
Analysis and Linear Algebra for Finance: Part I
14
Preliminaries
   
             
              
             
       
               
           
                   
                   
                   
          
               
            
    


                 
            
   
 

                 
              
            
          
    
 
          
            
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Analysis and Linear Algebra for Finance: Part I
15
Preliminaries
    
              
                


  
                
            
             
               
             
      
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Analysis and Linear Algebra for Finance: Part I
16
Preliminaries
   
      
          
             
             
         
    
             
             
          


             

    
            
     
          
                
  
            
             
      
       
  
       
   
              
            
             
     
        
               

       

              
                   
               
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Analysis and Linear Algebra for Finance: Part I
17
Preliminaries
   
      
  
 
   
 
  
             
                  
 
           
 
              
        
       
           
             
       
     
        
           
         
        
 
                
  
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Analysis and Linear Algebra for Finance: Part I
18
Preliminaries
   
            

              
            
            
                 
 
         
           
         
                
           
            
            
            
          
         
           
            


            
                  
            
             
              
              
                 
  
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Analysis and Linear Algebra for Finance: Part I
19
Preliminaries
   
        
  
          
            
            
          
             
            
     
  
            
  
             
             
  
      
           
             
                
               
            
            
      


          

           
           
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Analysis and Linear Algebra for Finance: Part I
20
Preliminaries
   
 
    
             
              
          
             
        
               
              
            
       
              
     
       
                
             
               
       
             
                
               
            
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Analysis and Linear Algebra for Finance: Part I
21
Preliminaries
  
             
               
      
       
 

  


 

  


            
               
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Analysis and Linear Algebra for Finance: Part I
22
Preliminaries
   
              
    
      

         

             
         
      
            
     
  

 


  




 


  




  
  
       

    
         
  
 
       
   

  
          
             
 
        
           
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Analysis and Linear Algebra for Finance: Part I
23
Preliminaries
  
           
              
                
   
              
       
        

      
          
   
 
  

  


  
        
    
           
 
             
 
-1 -0.5 0 0.5 1
0
0.2
0.4
0.6
0.8
1
x
f(x)
        
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Analysis and Linear Algebra for Finance: Part I
24
Preliminaries
   
  
                 
                 
         
        
      
            
            
       
              
       
     
              
   
      
   

          
       
       
               
                  

  
 
 
          
               
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Analysis and Linear Algebra for Finance: Part I
25
Preliminaries
    
               
      
   
           
          
                
          
           
             
             
          
          
          
          
          
           
            
          
            
           
  
     
       
    
          
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Analysis and Linear Algebra for Finance: Part I
26
Preliminaries
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Analysis and Linear Algebra for Finance: Part I
27
Preliminaries
    
                  
   
     
          
     
              
                    
              
             
              
             
   
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Analysis and Linear Algebra for Finance: Part I
28
Preliminaries
   
              
               
 
             
               
            
 
            
      
         
    

      



     
        

      
           


         
                
               
        
              

          
           
                 
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Analysis and Linear Algebra for Finance: Part I
29
Preliminaries
    
             
               
 
               
              
  
          
           
         
 
            
             
            
    
              
              
           
   
          
              
      
                  
                    
           
         

              
               
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Analysis and Linear Algebra for Finance: Part I
30
Preliminaries
   
                 
      

                 
                 
           
             
                
 
           
   
        
  
            
              
            
    
        
   
            

   
               
  
               
 
               

                
            
                
                
               

             
       
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Analysis and Linear Algebra for Finance: Part I
31
Preliminaries
    


            


  

 

           


                 
           
    
               
        
               
             
           
                
 
           
   
            
             
          
 
    
            
                
                
              
  
          
      
     
            

  
             

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Analysis and Linear Algebra for Finance: Part I
32
Preliminaries
   
              
            
         
            
         
 

 



 
      
            

 

 



             
            
            
       
  
         
     
              
               
   
         
     
 
    

      
   
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Analysis and Linear Algebra for Finance: Part I
33
Preliminaries
    
     



   



  
  

   


     
       
            
             
            
          
          
              
               
             
      
          
          
  
            
           
 

                
              
      

           
          

             
           
            
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Analysis and Linear Algebra for Finance: Part I
34
Preliminaries
   
                
        
           
             
               
           
           

               
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Analysis and-linear-algebra-for-finance-part-i

  • 2. Download free eBooks at bookboon.com 2 Patrick Roger Analysis and Linear Algebra for Finance: Part I Patrick ROGER LaRGE Research Center EM Strasbourg Business School University of Strasbourg
  • 3. Download free eBooks at bookboon.com 3 Analysis and Linear Algebra for Finance: Part I First edition © 2013 Patrick Roger & bookboon.com (Ventus Publishing ApS) ISBN 978-87-403-0412-1
  • 4. Download free eBooks at bookboon.com Pleaseclicktheadvert Analysis and Linear Algebra for Finance: Part I 4 Contents Contents Introduction 6 1 Preliminaries 7 1.1 Sets and subsets 8 1.2 Binary relations 17 1.3 Mappings 20 1.4 Topology of                                                                                                                                                                                                           25 2 Functions of one variable 43 2.1 Definitions and notations 44 2.2 Limits and continuity 50 2.3 Differentiation 61 2.4 Logarithms and exponential functions 70 2.5 Polynomial approximations and Taylor formula 75 © Deloitte & Touche LLP and affiliated entities. 360° thinking. Discover the truth at www.deloitte.ca/careers © Deloitte & Touche LLP and affiliated entities. Dis Deloitte & Touche LLP and affiliated entities. 360° thinking. Discover the truth at www.deloitte.ca/careers © Deloitte & Touche LLP and affiliated entities. 360° thinking. Discover the truth at www.deloitte.ca/careers
  • 5. Download free eBooks at bookboon.com Pleaseclicktheadvert Analysis and Linear Algebra for Finance: Part I 5 Contents 3 Integrals 81 3.1 Integral of a step function 81 3.2 General case 86 3.3 Computations 94 3.4 Improper integrals 98 4 Matrices 101 4.1 Definitions 101 4.2 Elementary algebra on matrices 104 4.3 Linear equations 115 Increase your impact with MSM Executive Education For more information, visit www.msm.nl or contact us at +31 43 38 70 808 or via admissions@msm.nl the globally networked management school For more information, visit www.msm.nl or contact us at +31 43 38 70 808 or via admissions@msm.nl For almost 60 years Maastricht School of Management has been enhancing the management capacity of professionals and organizations around the world through state-of-the-art management education. Our broad range of Open Enrollment Executive Programs offers you a unique interactive, stimulating and multicultural learning experience. Be prepared for tomorrow’s management challenges and apply today. Executive Education-170x115-B2.indd 1 18-08-11 15:13
  • 6. Download free eBooks at bookboon.com Analysis and Linear Algebra for Finance: Part I 6 Introduction                                                                                                                                                                                                                                                                                                         
  • 7. Download free eBooks at bookboon.com Analysis and Linear Algebra for Finance: Part I 7 Preliminaries                                                                                                                                                                                                                             
  • 8. Download free eBooks at bookboon.com Analysis and Linear Algebra for Finance: Part I 8 Preliminaries                                                                                                                                                                                                                                                                                                     
  • 9. Download free eBooks at bookboon.com Analysis and Linear Algebra for Finance: Part I 9 Preliminaries                                                                                                                                                                                                                                                                                                  
  • 10. Download free eBooks at bookboon.com Pleaseclicktheadvert Analysis and Linear Algebra for Finance: Part I 10 Preliminaries                                                           Get “Bookboon’s Free Media Advice” Email kbm@bookboon.com See the light! The sooner you realize we are right, the sooner your life will get better! A bit over the top? Yes we know! We are just that sure that we can make your media activities more effective.
  • 11. Download free eBooks at bookboon.com Analysis and Linear Algebra for Finance: Part I 11 Preliminaries                                                                                                                                                                                                                                                                                    
  • 12. Download free eBooks at bookboon.com Analysis and Linear Algebra for Finance: Part I 12 Preliminaries                                                                                                                                                                                                                                                                                        
  • 13. Download free eBooks at bookboon.com Analysis and Linear Algebra for Finance: Part I 13 Preliminaries
  • 14. Download free eBooks at bookboon.com Analysis and Linear Algebra for Finance: Part I 14 Preliminaries                                                                                                                                                                                                                                                                                                                            
  • 15. Download free eBooks at bookboon.com Pleaseclicktheadvert Analysis and Linear Algebra for Finance: Part I 15 Preliminaries                                                                                                                            GOT-THE-ENERGY-TO-LEAD.COM We believe that energy suppliers should be renewable, too. We are therefore looking for enthusiastic new colleagues with plenty of ideas who want to join RWE in changing the world. Visit us online to find out what we are offering and how we are working together to ensure the energy of the future.
  • 16. Download free eBooks at bookboon.com Analysis and Linear Algebra for Finance: Part I 16 Preliminaries                                                                                                                                                                                                                                                                                                                                                                               
  • 17. Download free eBooks at bookboon.com Analysis and Linear Algebra for Finance: Part I 17 Preliminaries                                                                                                                                                                                                                
  • 18. Download free eBooks at bookboon.com Analysis and Linear Algebra for Finance: Part I 18 Preliminaries                                                                                                                                                                                                                                                                                                                                                 
  • 19. Download free eBooks at bookboon.com Analysis and Linear Algebra for Finance: Part I 19 Preliminaries                                                                                                                                                                                                                                                                                            
  • 20. Download free eBooks at bookboon.com Analysis and Linear Algebra for Finance: Part I 20 Preliminaries                                                                                                                                                                                                                                                                              
  • 21. Download free eBooks at bookboon.com Pleaseclicktheadvert Analysis and Linear Algebra for Finance: Part I 21 Preliminaries                                                                                                ... BrowserTexting SMS from your computer ...Sync'd with your Android phone & number SMS from your computer ...Sync'd with your Android phone & number SMS from your computer ...Sync'd with your Android phone & number SMS from your computer ...Sync'd with your Android phone & number Go to BrowserTexting.com and start texting from your computer! FREE 30 days trial!
  • 22. Download free eBooks at bookboon.com Analysis and Linear Algebra for Finance: Part I 22 Preliminaries                                                                                                                                                                                                                          
  • 23. Download free eBooks at bookboon.com Analysis and Linear Algebra for Finance: Part I 23 Preliminaries                                                                                                                                                                     -1 -0.5 0 0.5 1 0 0.2 0.4 0.6 0.8 1 x f(x)         
  • 24. Download free eBooks at bookboon.com Analysis and Linear Algebra for Finance: Part I 24 Preliminaries                                                                                                                                                                                                                                                                    
  • 25. Download free eBooks at bookboon.com Analysis and Linear Algebra for Finance: Part I 25 Preliminaries                                                                                                                                                                                                                                                                                
  • 26. Download free eBooks at bookboon.com Analysis and Linear Algebra for Finance: Part I 26 Preliminaries
  • 27. Download free eBooks at bookboon.com Pleaseclicktheadvert Analysis and Linear Algebra for Finance: Part I 27 Preliminaries                                                                                                                                                      Contact us to hear more kbm@bookboon.com Who is your target group? And how can we reach them? At Bookboon, you can segment the exact right audience for your advertising campaign. Our eBooks offer in-book advertising spot to reach the right candidate.
  • 28. Download free eBooks at bookboon.com Analysis and Linear Algebra for Finance: Part I 28 Preliminaries                                                                                                                                                                                                                                                                                  
  • 29. Download free eBooks at bookboon.com Analysis and Linear Algebra for Finance: Part I 29 Preliminaries                                                                                                                                                                                                                                                                                                                                    
  • 30. Download free eBooks at bookboon.com Analysis and Linear Algebra for Finance: Part I 30 Preliminaries                                                                                                                                                                                                                                                                                                                                                                                     
  • 31. Download free eBooks at bookboon.com Analysis and Linear Algebra for Finance: Part I 31 Preliminaries                                                                                                                                                                                                                                                                                                                                                          
  • 32. Download free eBooks at bookboon.com Analysis and Linear Algebra for Finance: Part I 32 Preliminaries                                                                                                                                                                                                                                                 
  • 33. Download free eBooks at bookboon.com Analysis and Linear Algebra for Finance: Part I 33 Preliminaries                                                                                                                                                                                                                                                                                                                              
  • 34. Download free eBooks at bookboon.com Pleaseclicktheadvert Analysis and Linear Algebra for Finance: Part I 34 Preliminaries                                                                                                                  THE BEST MASTER IN THE NETHERLANDS