# Stochastic calculus in mathematician's vs physicist's view

In summary, the conversation discusses the differences between the approach to stochastics in physics and mathematics. The speaker has studied stochastics in both fields and notes that the topics are the same but the approaches are vastly different. They mention the lack of rigor in the physics approach, with no formal definitions of random variables or probability spaces, and the use of concepts like delta functions. They also ask for book recommendations for a more traditional mathematical approach to stochastics. The conversation concludes with a discussion on the importance of rigor in physics and the use of Langevin equations in the field.
Hello,

I've studied physics at a university previously and actually earned a degree in theoretical physics, but then switched over to mathematics, where I focused on stochastic analysis/calculus/processes (I'll just call it stochastics).

Now, I remember taking a course on stochastics while studying physics, and later I've had several courses on stochastics while studying mathematics. The topic seems to be the same, but the approaches are vastly different.

In mathematics, as a basis, you have measure-theoretic probability, Lebesgue integration, different types of convergences of random variables, some functional analysis, Brownian motion, Ito integral, Ito lemma, stochastic (Malliavin) derivatives and so on. These are the topics I've studied in some detail (wrote a master's thesis on Malliavin calculus)

So recently, I came across my (very) old notes on stochastics from my physics studies, and, suprisingly, it all felt foreign to me. The approach taken there is somewhat strange:
1. Expected values are defined as averages of r.v. ensembles (or something -- there is no formal definition of a r.v. or ensemble)
2. The is absolutely no notion of a probability space (sample space, sigma-field, measure, filtration -- nothing)
3. No mentioning of Ito integration, yet there is a great deal of SDEs, which are all written in a form requiring some notion of a white noise process.
4. Delta function is all over the place (anyone else with more mathematical background annoyed by this when reading physics literature?)

While I understand that it is not always necessary to be mathematically rigorous in physics, still I'm wondering, do you think this was just not a very good course, or is there truly an alternative (and rigorous) setting to work with SDEs, without measure-theoretic probability, I'm not aware of?

Now more importantly, there were also some topics discussed in these lecture notes, that I have never seen discussed in any "mathematical" book on stochastics, i.e. in a more traditional framework (measure theory, probability spaces etc):
1. Kramers-Moyal expansion
2. Fokker-Plank equations (ok, this one is mentioned sometimes, but still in a somewhat different form)

Is this (especially the first point) something that simply has a different name in a "mathematical" approach to stochastics? Can you suggest any books, that explain these topics in a traditional mathematical setting?

Why is it so that Ito calculus is a fundamental notion required to study stochastics for mathematicians, but there is often no mentioning of it in stochastics for physics? Why is anything like Kramers-Moyal expansion never discussed in courses on stochastics for mathematicians? I'm not even sure how one would go about to introduce the related results in a measure theory + Ito calculus approach.

Sorry for not being able to form a more concrete/exact question. This comes from my confusion on the subject.

Hey, what you got against delta functions? ;-)

I can't answer most of your questions. However, I don't think there was anything very unusual about your physics course. Obviously my knowledge is not comprehensive, but I've never seen anything like Ito calculus in the sciences. The Wikipedia entry for "Stochastic differential equation" says, "In physical science, SDEs are usually written as Langevin equations", and that matches my experience.

Finance, now, that's another matter. Financial quants love Ito calculus.

pmsrw3 said:
Hey, what you got against delta functions? ;-)

I can't answer most of your questions. However, I don't think there was anything very unusual about your physics course. Obviously my knowledge is not comprehensive, but I've never seen anything like Ito calculus in the sciences. The Wikipedia entry for "Stochastic differential equation" says, "In physical science, SDEs are usually written as Langevin equations", and that matches my experience.

Finance, now, that's another matter. Financial quants love Ito calculus.

Indeed they are written as Langevin equations, but I've never been exposed to a rigorous treatment of the maths behind it, despite that I've been treated to a lot of Langevin equations.
From my understanding, Langevin equations are mathematically more involved (than the Ito calculus approach): you need white noise theory, delta functions (and this means generalized functions, more functional analysis, more measure theory etc). I have never seen these mathematical prerequisites given to physicists, at least in my university. Does anyone have different experience?

I guess I don't have anything about delta functions themselves, only against their introduction in some physics textbooks/courses. It is usually introduced through it's properties (certain integral identities), and some books actually note that such functions (in normal sense) do not exist! But still we took it as a tool, used it all over the place, "derived" it's properties, found somehow their 1st, 2nd, n-th derivative and so on. That is partially why I switched to mathematics :) (although I still like physics)

Indeed they are written as Langevin equations, but I've never been exposed to a rigorous treatment of the maths behind it, despite that I've been treated to a lot of Langevin equations.
From my understanding, Langevin equations are mathematically more involved (than the Ito calculus approach): you need white noise theory, delta functions (and this means generalized functions, more functional analysis, more measure theory etc). I have never seen these mathematical prerequisites given to physicists, at least in my university. Does anyone have different experience?
Well, you kind of answer your own question. First you say that for Langevin equations you "need white noise theory, delta functions (and this means generalized functions, more functional analysis, more measure theory etc)" . Then you correctly point out that physicists don't get those things. And yet physicists routinely work with Langevin equations all the time. So, obviously, a physicist does NOT need all that other stuff.

Underlying this: physicists don't "need" rigor, at least not at the level mathematicians do. The fundamental standard of truth in physics is not proof, as in math -- it's experimental verification.

I will tell you honestly: if I can, through intuitive means, derive a differential equation to solve a problem, show through numerical simulation that solutions to my DE also solve the problem with 10-digit accuracy, experimentally verify the predictions within experimental error, and come up with an good intuitive understanding of the physical meaning of the main features, I'm satisfied. The lack of rigor in step 1 doesn't much disturb me.

I guess I don't have anything about delta functions themselves, only against their introduction in some physics textbooks/courses. It is usually introduced through it's properties (certain integral identities), and some books actually note that such functions (in normal sense) do not exist! But still we took it as a tool, used it all over the place, "derived" it's properties, found somehow their 1st, 2nd, n-th derivative and so on.
Yes? And your problem with this is ...? :-o

pmsrw3 said:
Underlying this: physicists don't "need" rigor, at least not at the level mathematicians do. The fundamental standard of truth in physics is not proof, as in math -- it's experimental verification.
...

OK, sorry if I made you misinterpret me. I am not trying to criticize a physicists approach (although it made have sounded so). I would prefer to steer away from any kind of 'wars'.

I consider myself somewhat knowledgeable in stochastic processes, but when I came across these old lecture notes on the same topic, but for physicists, I saw a very different framework. I have tried to analyze it as a mathematician, but these notes lack the required rigour.
So I guess the reason for this thread is to ask experienced people why e.g. we need Ito calculus in one field and not the other. Is there 1-to-1 correspondence between the approaches? What Kramers-Moyal expansion corresponds to? Can anyone suggest a reading on 'physicists stochastics' but with usual mathematical rigour/solid framework? Is it worth the time at all?

pmsrw3 said:
Yes? And your problem with this is ...? :-o
I think at some point you have to well understand the objects you are dealing with. The problem otherwise is that with uninformed symbol manipulation you can reach faulty conclusions.
Also note that I am not talking about theory vs. practice. E.g. in mathematical finance (quants) the introduction of stochastic calculus is very rigorous, but (of course) to be a practitioner, you may not need the rigour at all and still do useful things. But I still think the theory is not a waste of time for a quant-to-be. The same, I think, applies for physics, especially theoretical physics. But, as I mentioned, this is a whole different topic.

So I guess the reason for this thread is to ask experienced people why e.g. we need Ito calculus in one field and not the other.
I think a lot of it is just history and familiarity. I personally find Ito calculus a lot easier to follow than Langevin equations, but that's likely because that's the way I first learned it.

I think at some point you have to well understand the objects you are dealing with. The problem otherwise is that with uninformed symbol manipulation you can reach faulty conclusions.
Well, you can reach faulty conclusions with informed symbol manipulation, too.

I think a physicist does feel he understands what delta functions are, better than a mathematician does. He understands it as the idealization of something real -- something physically real. A mathematician might feel, as Plato did, that the physical world is not real, or not as real as the idealized world of mathematical logic. But I am not willing to concede that the mathematician is more right than the physicist.

To take another example, I find the rigorous mathematical definition of a Brownian motion to be somewhat beside the point, if you're interested in applications. There is no physical process that fits this definition (unbounded mean square velocity as time interval -> 0, for instance). It may make a mathematician feel better to have a completely rigorous definition of W(t) -- he may feel he "well understands the objects he's dealing with" only if he has such a definition. But to a physicist, the place where the rigor happens is exactly the place where the understanding stops.

## 1. What is stochastic calculus?

Stochastic calculus is a branch of mathematics that deals with random processes and their analysis. It involves the study of how systems evolve over time in a probabilistic manner, taking into account both deterministic and random elements.

## 2. How is stochastic calculus used in the mathematician's view?

In the mathematician's view, stochastic calculus is used to model and analyze complex systems that involve uncertainty and randomness. It is often used in areas such as finance, economics, and engineering to understand and make predictions about the behavior of these systems.

## 3. How is stochastic calculus used in the physicist's view?

In the physicist's view, stochastic calculus is used to describe and analyze physical processes that involve randomness, such as the motion of particles in a gas or the behavior of a quantum system. It is also used in statistical mechanics to study the behavior of large systems of particles.

## 4. What are some key concepts in stochastic calculus?

Some key concepts in stochastic calculus include Brownian motion, stochastic differential equations, and Ito's lemma. These concepts are used to model and analyze random processes and make predictions about their behavior.

## 5. What are some applications of stochastic calculus?

Stochastic calculus has a wide range of applications in various fields, including finance, economics, physics, and engineering. It is used to model and analyze stock prices, interest rates, and other financial variables, as well as physical systems that involve randomness. It is also used in machine learning and artificial intelligence to develop algorithms that can effectively deal with uncertainty and make predictions based on incomplete information.

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