Stochastic calculus, as its name suggests: is an area of mathematics concerned with the calculus (e.g. derivatives, integrals, limits, etc.) of stochastic processes (also known as random processes, effectively sequences of random numbers which possess certain properties). It is a natural extension of ordinary calculus to functions of a random variable, which are not…

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I have a master's in statistics. It has been almost a decade but my knowledge of probability had (once) gone as far as the following concepts : the probability triple, the monotone and dominated convergence theorems, martingales, and very vaguely the radon-nikodym derivative (you get the idea). Probably need a few weeks of quiet study to review all these and get back up to speed.

Anyway, I am now aiming to make a career switch from being a buy-side macro economist to a buy-side quant, and I think learning about stochastic calculus would be helpful (avoided that course back in the day because the professor was a known disaster). My aim is NOT to become so acquainted with the subject so I could begin dong Phd research on the topic. Just want to be solidly grounded enough to be an effective and efficient quant.

Need a book for self-study and I have narrowed down my list to the these two. Which one would you think is better for me? Or would there be a 3rd better option I am not aware of? My goal is to get as much out of a book as quickly as possible, to the point that I am totally comfortable with the key concepts, how they have come about, and how they are applied in finance. Your advice is much appreciated, thank you so much!

1. Shreve: Stochastic Calculus and Finance 2. Michael Steele: Stochastic Calculus and Financial Applications

• 2

i prefer 1 as it has more worked out examples and can make the subject more mechanical like regular calculus

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@bootstrap - I am self-learning and working through the exercises of (1). I'd be glad to speak/connect with you sometime.

• 4

I'd also recommend starting directly with the 2nd volume of Shreve's book - it is self-contained.

• 5

Why has no one mentioned Glasserman (2004)?

• 6

  I'd also recommend starting directly with the 2nd volume of Shreve's book - it is self-contained.

It has no numerics. Even Euler method only gets a one-line mention.

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  It has no numerics. Even Euler method only gets a one-line mention.

So..? OP is asking about a book for stochastic calculus.

• 8

  So..? OP is asking about a book for stochastic calculus

Depends how you define stochastic calculus. Because you will only half understand stochastic calculus by studying theoretical SDEs. It is too one-sided; besides most SDEs don't have analytic solutions.

Ideally, you need to learn Measure Theory and Lebesgue Integration as foundation. How in earth will you understand Girsanov and Radon-Nikodym otherwise?

Last edited: 12/12/21

• 9

  Depends how you define stochastic calculus. Because you will only half understand stochastic calculus by studying theoretical SDEs. It is too one-sided; besides most SDEs don't have analytic solutions.

Ideally, you need to learn Measure Theory and Lebesgue Integration as foundation. How in earth will you understand Girsanov and Radon-Nikodym otherwise?

This is what makes Shreve's volume 2 great, he packages the necessary measure theoretic results into the first two chapters so that one is able to understand results such as the ones you mention. Of course, his treatment is surface level, but nonetheless it is sufficient to understand the motivation behind results like that of Girsanov.

Learning stochastic calculus does not require you to learn numerical methods for SDEs. The main ingredients, e.g. brownian motion and martingales, stochastic integration, Itos lemma, Girsanov and Feynman-Kac, SDEs, in my opinion have nothing to do with numerical methods at the surface. Where numerical methods make sense is in a second course -- a first course in stochastic calculus should focus on the (pure) mathematics, and this is exactly what Shreve volume 2 does.

• 10

  This is what makes Shreve's volume 2 great, he packages the necessary measure theoretic results into the first two chapters so that one is able to understand results such as the ones you mention. Of course, his treatment is surface level, but nonetheless it is sufficient to understand the motivation behind results like that of Girsanov.

Learning stochastic calculus does not require you to learn numerical methods for SDEs. The main ingredients, e.g. brownian motion and martingales, stochastic integration, Itos lemma, Girsanov and Feynman-Kac, SDEs, in my opinion have nothing to do with numerical methods at the surface. Where numerical methods make sense is in a second course -- a first course in stochastic calculus should focus on the (pure) mathematics, and this is exactly what Shreve volume 2 does.

That's why I prefer Glasserman .. he integrates the two approaches. Horses for courses.

• 11

This book is good and maybe a precursor to Shreve.

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  @bootstrap - I am self-learning and working through the exercises of (1). I'd be glad to speak/connect with you sometime.

I think if you want to study stochastic calculus deeply, Hui Hsuing Kuo's Introduction to Stochastic Integration might be a good mathematical spin to the topic. I am going through this book myself, along with David William's Probability with Martingales. Thomas Mikosch's Introduction to Stochastic Calculus might also be a good entry point, if talking through a mathematical stand point.

• 13

My two cents: I think Bjork's "Arbitrage Theory in Continuous Time" is an underrated gem — balances intuition, formal rigor, and applications really well.

• 14

Again, get Glasserman >> all others

and Kloeden/Platen

Study and program them and then proclaim victory.

• 15

  Again, get Glasserman >> all others

and Kloeden/Platen

Study and program them and then proclaim victory.

Lovely books!

• 16

  Again, get Glasserman >> all others

and Kloeden/Platen

Study and program them and then proclaim victory.

I liked the book : A first course in StoCal, by LP Arguin, and have a copy of it on my desk.

It doesn't do handwaving. The proofs are first motivated, and some are left as exercises. There are also numerical projects. Accessible to an undergrad (with just some knowledge of algebra/calculus).

Which book is best for advanced calculus?

Calculus forms all the higher math. Calculus by Spivak is a great rigorous introduction to calculus, and goes in pretty deep. Calculus on Manifolds by the same author is a great book on advanced calculus, though it requires a lot of mathematical experience to really appreciate.

Is it hard to learn stochastic calculus?

Stochastic calculus is genuinely hard from a mathematical perspective, but it's routinely applied in finance by people with no serious understanding of the subject. Two ways to look at it: PURE: If you look at stochastic calculus from a pure math perspective, then yes, it is quite difficult.

What level is stochastic calculus?

Stochastic calculus is not something that you would encounter in an elementary calculus sequence. It is typically a graduate level course for applied mathematics and statistics majors, and requires a firm grasp of real analysis, differential equations, and probability theory, among other subjects.

What is the best calculus book for beginners?

6 Best Basic Calculus Books for Beginners.

Calculus I. Differentiation and Integration. ... .

ADVANCED CALCULUS. By Lynn Harold Loomis - Harvard University Professors + 1 more. ... .

Calculus DeMYSTiFieD, Second Edition. By Steven Krantz. ... .

Calculus Made Easy. By Silvanus P. ... .

The Calculus Primer. By William L. ... .

Calculus. Volume1..