Quadratic variation in Stochastic Calculus

[kdmfe]

Kindly, if anyone can provide me guidance/material to read on the below topics



Why is computing of Quadratic variation or knowing about it is important in Stochastic Calculus (for Financial modelling)?


I am reading Steven Shreve's Book 2 along with the help of a Phd Mathematics student.

I have watched several videos on the above topic, where I am not clear/or struggling to register in my mind is 'what is that I am missing if the above is not computed'
Be it in Itos integral, jump process or the entire book of Steven Shreve's Stochastic calculus.

Thank you very much in advance for all the time and knowledge.

 

[Qui-Gon]

Check out Kuo’s “Introduction to Stochastic Integration”. The first chapter showcases how nonzero quadratic variation prevents one from defining the Itô integral as a path-wise Riemann-Stieltjes integral. More broadly, quadratic variation is key for establishing Itô’s formula/lemma and additionally for linking continuous martingales with Brownian motion through the theorem of Dambis, Dubins and Schwarz. Some people like to think of quadratic variation as acting as a clock that tracks the “rate at which the stochastic process accumulates randomness”.

Kuo’s text has a far more rigorous treatment of the basic mechanics of stochastic calculus than Shreve 2 does and so it may be a book you and your colleague should consider checking out.

 

[PepeQuant]

@kdmfe you are wasiting your time bro. if you ask quesitons such as why QV matters then you are not ready for this. And why are you jumping everywhere? one day you are in Jump diffusion and suddenly back to QV.

I would recommend you to start from the basic stuff. (fk chapter 1 & 2 and pay real effort starting from chapter 3).

if chapter 3 is difficult for you, then go back to basic probability/statistics/calculus courses

 

[Quasar Chunawala]

Shreve makes one thing explicit - Continuous functions have zero quadratic variation, whereas the standard Brownian motion has non-zero QV. So, the differential form of Ito's formula must have an extra second-order term.

 

[kdmfe]

Hello Qui-Gon,

Thank you for your time and guidance.

In amazom India, the following two books showed for the title 'Introduction to Stochastic Integration'

Book One:

Introduction to Stochastic Integration (Universitext) eBook : Kuo, Hui-Hsiung: Amazon.in: Kindle Store

Introduction to Stochastic Integration (Universitext) eBook : Kuo, Hui-Hsiung: Amazon.in: Kindle Store

www.amazon.in

Book two:

Amazon.in

I thnk the book you were referring was book one.

Kindly confirm so that i can purchase.

Thankyou once again

 

[Qui-Gon]

Hello Qui-Gon,

Thank you for your time and guidance.

In amazom India, the following two books showed for the title 'Introduction to Stochastic Integration'

Book One:

Introduction to Stochastic Integration (Universitext) eBook : Kuo, Hui-Hsiung: Amazon.in: Kindle Store

Introduction to Stochastic Integration (Universitext) eBook : Kuo, Hui-Hsiung: Amazon.in: Kindle Store

www.amazon.in

Book two:

Amazon.in

I thnk the book you were referring was book one.

Kindly confirm so that i can purchase.

Thankyou once again

Yes, I was referring to the first book.