Partial differential equation for Stochastic calculus finance

[kdmfe]

Hello Daniel,

I have written notes on Sequence, Infinite series, Limits and Continuity, Differentiation and Reimann Integral. Also I have written my notes on Measure theory (focused on Outer measure).

So to be rewording the sentence.

So if I take the online course on ODE/PDE, I will be able to use the notes from the online course to complete writing my notes on your book. Am I correct in my understanding?

My Phd sir told me the online course covers many more topics than covered in your book.

Thank you

 

[Daniel Duffy]

My Phd sir told me the online course covers many more topics than covered in your book.

True. So?
For students, modules A,B,F,G are the topics for finance students. This is what makes it unique.The other modules are there for completeness.
There are numerous books on the other topics (e.g. wave equation).

My students are expected to do independent background research as well.

I will be able to use the notes from the online course to complete writing my notes on your book.
What does this mean?

 

[kdmfe]

I will be able to use the notes from the online course to complete writing my notes on your book.

Ok, here is one instance I have encountered today:


Take for instance under section 2.6 Stiff ODE

The First order Linear Differential equation ‘FLDE’ along with its initial condition is given in equation 2.40

I could arrive at the exact solution ie y(t) = A. Exp{-at}

Next the book gives an Ordinary Differential Equation dy/dt = f(t,y(t)) in equation 2.41 for which the solution is given in equation 2.42 using Theta method

Then the book gives definition of Y subscript (n+1) and value of theta in equation 2.44

I tried to derive these two values 1) Y subscript (n+1), and 2) value of theta and I could not.

Now this is a very specific situation I have mentioned above.

Now, what I am trying to understand is how does the online course help me overcome the above knowledge gap.

I am not assuming that the online course will be a solution for all my knowledge gap. If it helps me identify where to refer to solve these equations, then it will help me make progress.

So the question is how does the online course support me understanding the content in the book?

Thank you for your prompt response and being supportive.

 

[Daniel Duffy]

EQ. 2.42 is standard finite difference method for ODEs, Take theta = 0,1/2, 1 to see which specific schemes (see page 326). The book by Dahlquist is a good background if you don't know. Or Liniger and Willoughly 1970.
2.43 shows how to use it in a simple linear case. It's fundamental. And it is/was common undergrad level?

Long story short: students of ODE/PDE course are supported in this way when they have questions. I usually screen students to see if they the necessary background.

 

[Daniel Duffy]

I will be able to use the notes from the online course to complete writing my notes on your book.
What does this mean?

 

[kdmfe]

I will be able to use the notes from the online course to complete writing my notes on your book.
What does this mean?

Ok, here is one instance I have encountered today:


Take for instance under section 2.6 Stiff ODE

The First order Linear Differential equation ‘FLDE’ along with its initial condition is given in equation 2.40

I could arrive at the exact solution ie y(t) = A. Exp{-at}

Next the book gives an Ordinary Differential Equation dy/dt = f(t,y(t)) in equation 2.41 for which the solution is given in equation 2.42 using Theta method

Then the book gives definition of Y subscript (n+1) and value of theta in equation 2.44

I tried to derive these two values 1) Y subscript (n+1), and 2) value of theta and I could not.

Now this is a very specific situation I have mentioned above.

Now, what I am trying to understand is how does the online course help me overcome the above knowledge gap.

I am not assuming that the online course will be a solution for all my knowledge gap. If it helps me identify where to refer to solve these equations, then it will help me make progress.

So the question is how does the online course support me understanding the content in the book?

Thank you for your prompt response and being supportive.

So, when I am stuck in a situation like the one i posted above, and based on your guidance, I will be able to continue to write my notes.

This is what I wanted to know from the below statement:

I will be able to use the notes from the online course to complete writing my notes on your book.
What does this mean?

Ok, since I am trying to understand how best to leverage your online course to complete writing my notes on your Numerical methods book, Ca I please know if there is an online repository of all the queries and answers, topic wise, in your online portal. The reason I am asking, I do not want to repeat the question which you have answered in the past to any other or previous students.

Thank you

 

[Daniel Duffy]

Ok, since I am trying to understand how best to leverage your online course to complete writing my notes on your Numerical methods book, Ca I please know if there is an online repository of all the queries and answers, topic wise, in your online portal. The reason I am asking, I do not want to repeat the question which you have answered in the past to any other or previous students.

That's not the way it works. Students send soutions and I give feedback within a few days. I expect them to do the legwork and research.
If you really don't solve it, I will send the solution. It is your learning process.

BTW what is your academic goal now?
Someone wrote

one does not learn mathematics by posting on a forum every time they have a question. Sometimes you need to consult multiple different sources, read and reread many times and think hard. The last point is the most important, and no, thinking hard does not mean thinking for 3 hours. I'm not saying you are not already doing this, but learning a subject like stochastic calculus rigorously takes endurance and stubborness. QuantNet is not a substitute for this.

I can see his point.

Here's a nice quote

Don't just read it; fight it! Ask your own question, look for your own examples, discover your own proofs. Is the hypothesis necessary? Is the converse true? ... Where does the proof use the hypothesis?

Paul Halmos

 

[kdmfe]

BTW what is your academic goal now?

Thank you for all the support/guidance you are giving me.

I do not think I will come across another author or well learned person who will take some much of time and pain to understand my journey or endeavours. Thank you once again.

I am an accountant and holder of CFA

I worked in the Big 4 consulting firms in the UK in banking and capital markets division.

I quit job about 6 yrs ago to take care of my personal front and in the mean time I took interest in understanding Stochastic calculus.

So I studied mathematics from high school to Post graduation. I have hired two Phd Students, one in Mathematics and the other in Statistics.

I made good progress with their help. I thank them for that.

With their help I wrote my own notes for Stochastic Calculus-Continuous time models by Steven Shreve. I have written my notes for most of the chapter and also did most of the end of chapter questions.

While going through the Chapter 6, it mentioned about PDE and Monte Carlo approach. It is this at this stage that you have been helping me understand that these two topics. I took interest in the PDE approach as my Phd sir was aware of most of the topics and said he was interested in working on these topics with me.

I purchased your book and started to write my notes.

I wrote my notes for Chp 1,4,5 and now I am in the process of writing notes for Chp2. Hence these posts, where the ODE and initial condition is given and the solution (approximate not the exact one) is given. We were wondering how the solution was derived. Yes, admittedly you have referred to several books in various sections in the chapter 2. But we have a limitation, I do not have much time left to refer to these books.

I am quite sure your online course will be of great value to help us understand the Numerical methods. But the task I have on hand is I need to write my notes for this book. So we have the challenge that we are not able to make much progress or just turning the pages. This is an issue we are facing in Chp 2.



Also, my Phd Sir told me that we may not face issues from Chp 6-9. This is just his analysis with his prior knowledge he has.

So I hope it is abundantly clear that I am trying to gain this knowledge sitting in my house in my village.

I need to go back to Europe and join back the job market in Jan 2023. If possible, I will be joining evening college (or a part-time course) to formalise this education.

So my target was to do Stochastic calculus, which I think to a reasonable extent I have made progress and now if I can cover a few topics from the Numerical methods to show that I have used my time enhancing my skill set. The remaining chapters, I can cover once I am back in the job market. I am aware this is a vast subject and it cannot be overdone overnight.

So the next point is, if I take the online ODE/PDE course, firstly, what is its content-online teaching or just material and secondly will it help me understand the topics in the Numerical methods better.

So my target is to gain this understanding and join back the job market and show how I have gained this rare skill.

Hope it is clear now.

Thank you

 

[kdmfe]

one does not learn mathematics by posting on a forum every time they have a question. Sometimes you need to consult multiple different sources, read and reread many times and think hard. The last point is the most important, and no, thinking hard does not mean thinking for 3 hours. I'm not saying you are not already doing this, but learning a subject like stochastic calculus rigorously takes endurance and stubborness. QuantNet is not a substitute for this.

Just a glance of the stochastic book shows there are several terms and nuances. I do not think I have posted so many questions in this forum.

For instance, let us take te solution on Pg 307 Steven Shreve bk 2 on Stochastic calculus, the text book does not provide the steps for the solution V(t). this is one of the lengthiest derivations I have written. I did not find this solution by stream side in my village. I had to write it with the help of Phd sirs. so is the case of writing the derivation of d+ and d- using convolution, heat dispersion PDE. So I can go on with a list where I have not posted not just this but any other forum. I found the text books/online mater and prepared my notes.

So with utmost respect to yourself and the gentlemen who posted the above comment, I wish to park it aside. The above comment was posted to on my original posts in this forum.

I do not think posting a thread seeking guidance for books/study material is bad.

But I thank this forum as it has helped me immensely.

Thank you all once again.

 

[Daniel Duffy]

Your background (Accountancy) is not exactly quantitative. My book and courses are mainly for MFE MSc Stem students.
I suggest enrolling at a local university and do an applied maths degree. You are not yet ready for this book.

I have hired two Phd Students, one in Mathematics and the other in Statistics.
why?
Statistics is important, but useless in PDE and stuff like that.

 

[kdmfe]

Hello Daniel, I am getting your time. you are an eminent personality. Hope I will the best use of this opportunity.

1. our background (Accountancy) is not exactly quantitative.: not just 'exactly', it is not even remotely related to Maths. As a kid, I scored 35/100 in Mathematics in O level (around age 16). This means the education system has been kind enough to promote me to the next class. I have three professional qualifications in the field of accounting.

2. My book and courses are mainly for MFE MSc Stem students: the most precise reason why I am planning to join evening class in a UK university for MFE. But yes, I will raise to this challenge to complete this field of study successfully.

3. suggest enrolling at a local university and do an applied maths degree. : I will join MFE. But i will be happy if you can let me know which part of Applied Mathematics I am missing or I need to hone my skills on?

4. You are not yet ready for this book.: I am aware it will take a lot of effort but i will do my best to raise to the challenge and write the relevant (from job perspective so limited) notes on your book.

5. I have hired two Phd Students, one in Mathematics and the other in Statistics.
why?
Based on the curriculum in various US universities for MFE, I initially thought Stochastic calculus was the toughest of all of the subjects. I also remember seeing a few Chinese reading Stochastic calculus in London Underground tube during their commute. I then found the prerequisites for stochastic calculus. I understood it was Real Analysis, Measure Theory and Probability. But anyway I studied most of the topics in Graduation level and a few topics from Post graduation level. I did not mention earlier but I have covered Numerical Integration ( Newton Quadrature formula, Trapezoidal rule, Simpson's 1/8, 3/8 and Weddles rule) , also Numerical solutions of ODE (Taylor series, Picards methods, Euler's method, Euler's modified method, Runga Kutta, first order second order, third order and fourth order method)
I also covered Finite Difference methods (forward and backward), Gauss central difference, Sterling difference, interpolation with equal intervals, Newton's formula for interpolation, interpolation with unequal intervals, Lagrange's interpolation formula'

I covered some portion/relevant portion from Probability theory book by Ross Sheldon (quite useful for stochastic calculus-continuous time models) Hence required a Phd Statistics Student for Probability theory.


Statistics is important, but useless in PDE and stuff like that.: i humbly agree and disagree with you.
Agree: from the very very very very limited level of knowledge I have yes it is not relevant for ODE/PDe and numerical methods
Disagree: (this is purely from my experience) If I did not read Stochastic calculus, then I would not have known PDE methods along with numerical methods are used to solve the interest rate curve problems (I am writing here from my understanding in Steven Shreve's book)

But my basic question is or I am still wondering is how will your online course help me understand your book well. I mean will it explain concepts or work problems, provides reference material?

I am sure it will be robust, but I am curious to know before I enrol.


Thank you for your time. I am happy to interact with you. Hope I have not been rude anywhere. I also hope I am able to establish, though I do not have a formal degree in MAthematics I am making the best of the resources available at my disposal to learn this fantastic subject.


NB: Between Monte carlo and PDE, I chose PDE as I like the subject. There are several prerequisites for Monte Carlo, for which I do not have time (now) and also the financial resources.

 

[Daniel Duffy]

But my basic question is or I am still wondering is how will your online course help me understand your book well. I mean will it explain concepts or work problems, provides reference material?

In the short term, probably not because the stuff is not useful but more because I feel you need to get up to speed.
Wait a year or two and then ask me again.
Like getting a black belt in judo..

 

[DDG]

Hi @Daniel Duffy

I am interested in course "The finite difference method in ODE ,PDE and SDE"
Along with basic in ODE, PDE

before taking this course what perquisites are required?

Do i need to have background in Stochastic calculus before taking this course ?

coding requirements: Is C++ Programming for Financial Engineering sufficient?

 

[Daniel Duffy]

Hi @Daniel Duffy

I am interested in course "The finite difference method in ODE ,PDE and SDE"
Along with basic in ODE, PDE

before taking this course what perquisites are required?

Do i need to have background in Stochastic calculus before taking this course ?

coding requirements: Is C++ Programming for Financial Engineering sufficient?

I recommend ODE/PDE before the next courses

Online Courses :: Datasim

www.datasim.nl

Then moving to FDM is more seamless.

For ODE/PDE, some knowledge of Python and/or C++ does no harm.