Partial differential equation for Stochastic calculus finance

[kdmfe]

Dear experts,


I want to learn/study Partial Differential equations relevant from understanding Stochastic calculus for finance.

I have written my notes for Partial differential equation at graduate level.

Very recently I have written my notes for Stochastic calculus based on Steven Shreve’s book 2, continuous time models.

Kindly suggest a book for a beginner level course in PDE for Stochastic calculus and a book which demonstrates high level of understanding required for Stochastic calculus in finance.

Thank you

 

[Daniel Duffy]

PDE for Stochastic calculus
What do you mean by this?

SDE ?

Have you seen this?

Numerical Methods in Computational Finance: A Partial Differential Equation (PDE/FDM) Approach

<p>This book is a detailed and step-by-step introduction to the mathematical foundations of ordinary and partial differential equations, their approximation by the finite difference method and applications to computational finance. The book is structured so that it can be read by beginners...

www.wiley.com

BTW I am probably the only author in this space with (primary) background in PDE/FDM. SDEs are the starting-point in many books and SDE != PDE. The devil is in the details. PDE is seldom done at business school.

 

[Daniel Duffy]

Also online ODE/PDE course

Online Courses :: Datasim

www.datasim.nl

 

[SamDQuant]

Have you seen this?

Numerical Methods in Computational Finance: A Partial Differential Equation (PDE/FDM) Approach

<p>This book is a detailed and step-by-step introduction to the mathematical foundations of ordinary and partial differential equations, their approximation by the finite difference method and applications to computational finance. The book is structured so that it can be read by beginners...

www.wiley.com

BTW I am probably the only author in this space with (primary) background in PDE/FDM. SDEs are the starting-point in many books and SDE != PDE. The devil is in the details. PDE is seldom done at business school.

The excerpt from your new book gives one an exciting overview of numerical methods in computational finance. The fascinating aspect is that you touched on different areas of mathematics, as listed below.

- Real Analysis
- Differential Equations (ODEs, PDEs and SDEs)
- Linear Algebra
- Derivatives
- Interest rate models
- Of course, Numerical Methods with a focus on FDMs and their application.

I think it would make for a good read. Thanks for sharing your knowledge, Dr. Duffy.

 

[kdmfe]

PDE for Stochastic calculus
What do you mean by this?

SDE ?

Have you seen this?

Numerical Methods in Computational Finance: A Partial Differential Equation (PDE/FDM) Approach

<p>This book is a detailed and step-by-step introduction to the mathematical foundations of ordinary and partial differential equations, their approximation by the finite difference method and applications to computational finance. The book is structured so that it can be read by beginners...

www.wiley.com

BTW I am probably the only author in this space with (primary) background in PDE/FDM. SDEs are the starting-point in many books and SDE != PDE. The devil is in the details. PDE is seldom done at business school.

Partial Differential Equation

 

[kdmfe]

All,
I have been reading/trying to understand topic covered in Steven Shreve's book 2- Continuous time models.

My post here for a reference book separately dedicated for Partial Differential Equation is see if there is more aspects to understand/know topics in Chapter 6 - Connections with Partial Differential Equations.

Of course, I could PArtial Differetial Equations covered in Chapter 10 - Term Structure models, i.e, closed form solutions satisfying the initial condition and the Partial Differential equation.

Moreover, I liked this topic, rather than Brownian motion, martingale.

I find those topics far too complicated, being from non-math background, than Partial Differential equation, This is just my feeling.

Hence this post.

Kindly advice.

 

[Daniel Duffy]

Chapters 6 and 10 are a bridge too far.

I can give a short answer to your woes: See my book above where I do all these in great detail. (esp chapters 13, 25,26)

And most PDEs in real life don't have closed solutions (a kind of myth) ... numerical methods are needed.

A modest proposal is to acquaint yourself with my PDE work. I have been directly addressing these concerns in academia and industry for more than 20 years.

 

[Daniel Duffy]

The excerpt from your new book gives one an exciting overview of numerical methods in computational finance. The fascinating aspect is that you touched on different areas of mathematics, as listed below.

- Real Analysis
- Differential Equations (ODEs, PDEs and SDEs)
- Linear Algebra
- Derivatives
- Interest rate models
- Of course, Numerical Methods with a focus on FDMs and their application.

I think it would make for a good read. Thanks for sharing your knowledge, Dr. Duffy.

Well spotted!
Many (most?) people in finance do not have hard-core training in the joys of ODE/PDE/FDM.
I have also included several chapters on Linear Analysis and Numerical Linear Algebra to make the book more self[-contained.

And detailed PDE for interest rate problems.

 

[kdmfe]

Chapters 6 and 10 are a bridge too far.

I can give a short answer to your woes: See my book above where I do all these in great detail. (esp chapters 13, 25,26)

And most PDEs in real life don't have closed solutions (a kind of myth) ... numerical methods are needed.

A modest proposal is to acquaint yourself with my PDE work. I have been directly addressing these concerns in academia and industry for more than 20 years.

Dear Daniel,

I was not even aware until you mentioned this '
And most PDEs in real life don't have closed solutions (a kind of myth) ... numerical methods are needed.'

While taking notes from my PhD sir, I found this of great interest. Hence I have requested for guidance in this forum.

Also, in Chapter 6, at the very beginning, Steven Shreve's book mentioned that to solve we can use either a PDE or Monte-Carlo technique (6.1 Intrduction).

Unkowingly, with little or no knowlege, I thought the above two areas will be of great value and to invest my time.

Thank you once again for your guidance.

 

[Daniel Duffy]

Glad to be of help.

 

[Daniel Duffy]

to solve we can use either a PDE or Monte-Carlo technique (6.1 Introduction).

Author NOT exactly saying that..

(2) numerical methods
(1) risk-neutral expected payoff

 

[Daniel Duffy]

Here is an article that discusses PDE for Black Scholes and its numerical approximation. Hopefully to clear up things a bit.
BTW the QN C++ course has modules on Monte Carlo and PDE _and_ how to implement them in C++

 

[kdmfe]

Hello Daniel,

Thank your advice.

I have purchased the book on Kindle. I am writing my notes for the chapters. I have done it so far for Chpt 1, 4,5 and 2. Most likely will start with the next chapter, Chapter 3 in a couple of days.

I have the following questions:
1. Certain aspects in (ODE and its solution) Chapter 2 are not clear to me, i.e., how those are derived. Also, the text in the chapter refers to other books. I am not sure if I need to purchase those books. Can you guide me how the other students are handling such situations? Given that your book is a highly specialised knowledge book.

2. I am happy with the new learnings I am getting from writing my notes on this book. I have a general question, just curious, will this knowledge enhance my chances of getting a job.


Thank you in advance for your guidance.

 

[Daniel Duffy]

Hi
1. Which aspects?
My coached students who do ODE/PDE course send solutions to me etc.

2. Depends on the employer.

 

[Daniel Duffy]

Here's a list of books.
S. Goldberg Introduction to Difference Equations Dover 1986

Useful for solutions of difference equations, divided differences and applications.



Kreider, Kuller, Ostberg and Perkins, An Introduction to Linear Analysis Addison-Wesley

Al-round book on linear analysis and applications, aka KKOP



M. R. Spiegel, Vector Analysis, Schaum Outline Series.

2d and 3d vector analysis; useful for later.



P.A. Schmidt, F. Ayres College Mathematics, Schaum Outline Series.

High-school refresher, just in case you have forgotten some maths.

5A. “Vector Spaces of Finite Dimension”, G.C. Shephard, Oliver and Boyd 1966. (Good front-end to 5B)

== 5B “Linear Functional Analysis”, B.P. Rynne and M.A. Youngson Springer 2000

Good readable intro.

• “Elements of the Theory of Functions and Functional Analysis”, A.N. Kolmogorov and S.V. Fomin Dover 1961
• "Real Analysis" N.B. Haaser and J.A. Sullivan Dover 1991
• “Linear Algebra”, G.E. Shilov Dover 1977

“

// Numerical Analysis

”Numerical Methods”, G. Dahlquist an A. Bjorck Dover 1974

Great, practical book. ODEs as well.

“Qualitative Theory of Ordinary Differential Equations”, F. Brauer, J.A. Nohel Dover 1969.

Good for theory of ODE.

Partial Differential Equations, P. Duchateau, D.W. Zachman, Schaum Series.



“Operations Research”, R. Bronson, G. Naadimuthu, Schaum Series 1997

Very good for optimization. Lots of hands-on examples.



“Matrix Operations”, R. Bronson 1989

Very good for matrices. Lots of hands-on examples.

• Linear Algebra, G.E. Shilov Dover 1977
• Financial Instrument Pricing using C++ Second Edition, Daniel J. Duffy 2018.
• Theory and Problems of Differential Equations, F. Ayres Schaum Series 1972 (LOTS of worked examples of ODEs).
• Mathematical Techniques for Biology and Medicine, W. Simon Dover 1972.

• College Mathematics P.A. Schmidt et al Schaum Series.
• 
  
• Theory and Problems of Modern Algebra, F. Ayres Schaum Series.
• 
  
• Analysis of Numerical Methods, E. Isaacson and H.B. Keller Dover.
• 
  
• 
  
• Calculus of Finite Differences and Difference Equation, Schaum’s Outlines.
• (very practical book on learning and applying finite differences; useful later on for sure).

Spiegel, M. (1999) Complex Variables Schaum’s Outlines.



**book on how to solve maths problems



How to Solve it

How to Solve It - Wikipedia



T. Apostol, Calculus Volume I John Wiley & Sons New York 1967 (esp. chapters 3, 4, 6, 9, 10,11,14)

 

[kdmfe]

On my pt 1. I will send you one particular example where I struggled to understand.

Can you please tell me if your online course (Online Courses :: Datasim) helps the student in understanding the methods used in your book?

Numerical Methods in Computational Finance: A Partial Differential Equation (PDE/FDM) Approach)​

In otherwords, is the book and online course related to each other?

Thank you

 

[Daniel Duffy]

Yes, This book is an update of my fdm book 2006 PLUS mathematical foundations of ODE/PDE.

 

[kdmfe]

Ok I think I did not post my question properly.

Is the online course and the content in your books complement each other.

Thank you

 

[Daniel Duffy]

Ok I think I did not post my question properly.

Is the online course and the content in your books complement each other.

Thank you

Yes. The student exercises are all tested and solved. And they were used to help me write the chapters.

 

[Daniel Duffy]

It is so that readers of the book are comfortable with calculus, real analysis etc. before embarking on the ODE/PDE course.