All known motion is described by Hamilton's equations, and the time evolution is called a Hamiltonian flow.Brownian (aka Weiner) is defined as a random walk and its process is a stochastic flow.Hamiltonian flows and stochastic flows are mutually exclusive (very well known). So why would anyone call a process "Brownian motion" when no process can be both stochastic and Hamiltonian at the same time?Please, give me some reasons why I shouldn't think that those who teach stochastic calculus are total idiots...

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Because "Brownian flow" sounds totally disgusting. I do wonder if physical Brownian motion isn't Hamiltonian, though. At a fine enough time-scale, the "shocks" of the molecular collisions would resolve as continuous time-varying electrostatic forces between particles. Even in the markets, do we really have true instantaneous stochastic shocks? The distributed nature of the information flows and heterogeneous latencies by decision makers would seem to convert delta-function events into near-continuous variations in supply/demand or bid/ask price changes.

QuoteOriginally posted by: NSo why would anyone call a process "Brownian motion" when no process can be both stochastic and Hamiltonian at the same time?sometimes, it hurts to know too much math.Brown actually observed pollen and dust floating in liquids under microscope. these things move in water. hence, motion. you can observe time and distance of their drift and draw graphs. i have vague recollections of doing this in the lab. if u happen to study physics, they make u do those weird things like rolling balls on the inclined surface trying to repeat Galileo's experiments

sometimes, it hurts to know too much physics too.I did study n-body problems many years ago. As I recall the movement of dust in air actually a Toda problem that was made famous by Fermi in the mid-50s. Why famous? It showed that statistical mechanics is 100% deterministic (and therefore does not involve probability).To this day, I claim my statistical mechanics prof (PhD Harvard) knew only enough math to do his income taxes.

QuoteOriginally posted by: NAs I recall the movement of dust in air actually a Toda problem that was made famous by Fermi in the mid-50s. Why famous? It showed that statistical mechanics is 100% deterministic (and therefore does not involve probability).I don't recall that anyone involved in developing statistical mechanics thought otherwise. As I remember it, probability applied only as a means of solving problems that depend only on the aggregate behavior rather than the specific detailed behavior of individual molecules. So I don't really see what point you are trying to make here.

QuoteOriginally posted by: N As I recall the movement of dust in air actually a Toda problem that was made famous by Fermi in the mid-50s.i didnt live that long to remember 50s i have no clue about tada problem, but assume that Brown wouldnt be able to observe his motion in the air at normal temperatures. the air is probably too thin for that. you need something floating in it, and the thing would be either very large and fluffy, or very small and jumping like carzy. in either case it would be tough to measure its movements. just a guess

QuoteOriginally posted by: FermionQuoteOriginally posted by: NAs I recall the movement of dust in air actually a Toda problem that was made famous by Fermi in the mid-50s. Why famous? It showed that statistical mechanics is 100% deterministic (and therefore does not involve probability).I don't recall that anyone involved in developing statistical mechanics thought otherwise. As I remember it, probability applied only as a means of solving problems that depend only on the aggregate behavior rather than the specific detailed behavior of individual molecules. So I don't really see what point you are trying to make here.My point is that physicists have it exactly backwards! Aggregate behavior is *always* deterministic (and quantized), whereas specific detailed behavior of individual molecules is always quite random looking requiring a probability approach.The toda experiment shows that there is no probability assignment for states in statistical mechanics (all aggregate states are determimistic).

QuoteOriginally posted by: NMy point is that physicists have it exactly backwards! Aggregate behavior is *always* deterministic (and quantized), whereas specific detailed behavior of individual molecules is always quite random looking requiring a probability approach.that's a very general statement. in classic theory of gases molecules are deterministic. there was a notion of this demon who, if told about the state of each molecule in the gas, could compute the exact state of the whole system. the problem was we didn't know exact state of each molecule due to "technical difficulties", but if we could then... they were using statistical approach to overcome that difficulty, not because they thought that individual molecules are indeterministic. imho, that's what i got from general phys course.i'm not sure what folks do now, maybe there's a bunch of mathematicians in the field stirring the pot and complicating the subject

Quote Originally posted by:NAs I recall the movement of dust in air actually a Toda problem that was made famous by Fermi in the mid-50s. You may be speaking about how the Toda lattice is a completely integrable Hamiltonian system, and is closely related to the Fermi-Pasta-Ulam lattice (developed 1953/54), which is not a ciHs, but close eough to retain some of the predictability of those systems.But regarding your original question ("So why would anyone call a process "Brownian motion" when no process can be both stochastic and Hamiltonian at the same time?"), I'd say some things are named because of a particular characteristic, the name sticks, and good luck changing that afterwards (e.g. Native Americans being called Indians). No need to blame that on today's teachers.

I've read a fairly extensive amount of Wiener's work and he never defined Brownian motion as a random walk to my knowledge. Replace "walk" with "process" and you got it. Actually, you need to go way back to the joint work of Paley and Wiener and their work on the "fundamental random function" which is continuous almost everywhere but not differentiable in the usual sense. Such things are constructed, not merely hypothesized to exist, it's fairly involved and technical. Also, nearly all of Wiener's stuff(that I've read) was very applied and practical due to his work on automatic fire control(guns.. a lot of the stuff was originally classified), cybernetics, radar systems, tracking, etc. Also, individual points on this manifold/space/whatever are not described in detail, but counts of particles in boxes/intervals/cubes/hypercubes/etc of varying sizes are, and then you can define in-flow out-flow, "weak directional derivatives", for each of these boxes into each other and calculate the evolution deterministically. The "randomness" only appears in the actual sampling of the process and that goes to inference theory. Actually, now that I think about it, the density of the dust particles in space are defined by the parameters of the densities in each of these successively refined and nested boxes where the size of the boxes approaches 0 as the number of boxes approaches infinity. Some connection to Ricci tensors? E.g. uniform density would be Euclidean and space would be "flat" but a non-uniform density of dust particles could be described by some analog of a "stochastic ricci tensor" since the BM is only weakly differentiable.. actually, this looks interesting, only briefly glanced. http://arxiv.org/pdf/hep-th/9812254For all intents and purposes, BM just means gaussian distribution with variance given by |t-s| and the implications of such a process. Anything else besides that definition is not something I pay attention to. It's also well known that the BM does not define any financial series that I'm aware of either, but it does provide a very useful tool to project other more complicated processes onto for algorithmic/computation purposes.

Last edited by crowlogic on May 2nd, 2007, 10:00 pm, edited 1 time in total.

QuoteOriginally posted by: MCarreiraQuote Originally posted by:NAs I recall the movement of dust in air actually a Toda problem that was made famous by Fermi in the mid-50s. You may be speaking about how the Toda lattice is a completely integrable Hamiltonian system, and is closely related to the Fermi-Pasta-Ulam lattice (developed 1953/54), which is not a ciHs, but close eough to retain some of the predictability of those systems..Well yes, just like for completely integrable nonlinear lattices, Kac-van Moerbeke is a modified Toda using a Backlund transform. I blame teachers because "stochastic" theory was generally dismissed over 30 years ago based on the theory associated with toda lattices, inverse scattering, Lax pair formalism and Kac-Moody algebras. When I finish my PhD in the late 70s, the word "stochastic" was dead in the operations research world and dead in the physics world of the day. I actually thought I'd never hear the words "Riemannian manifold" again. During the next 35 years I was involved with computer science. What a surprise it was when I got back into OR and financial engineering and saw all the pure bogus crap taught/written about stochastic processes. Apparently, teachers couldn't do the math so the concepts were dummied-down. How many physics profs understand the role Calabi-Yau manifolds in Vertex Algebras (aka Kac-Moody)? It's easier to spout stuff about inner products of wave functions than take the time to understand underlying math. Do you know any good teachers?

QuoteOriginally posted by:NDo you know any good teachers?I'm not such a connoisseur of modern finance (notwithstanding the amount of books I buy) that I can criticize the roads it has taken, or the people that are steering the ship.As the little I know has been through reading books and articles, and I still have trouble distinguishing a martingale from a nightingale (the only thing I remember from my statistic classes in college is that 17 chairs was the answer to one of the questions of an exam - but I don't remember the question), I could answer that by saying which books I enjoyed more reading, but I think that your question may go deeper.Why do people focus on stochastic calculus and its application to finance ?Mainly because people who who hire people to do modelling stuff today want/need them do use it; if they constitute an adequate representation of the world is another thing. But the sudden demand for these kind of skills will create an offer of professionals to teach those tools. They could end up being inadequate, but they have also become a kind of lingua franca for the professionals of the area.And this may be the unintended and most relevant consequence of this development: maybe more important (in a monetary benefits measure) than modeling reality is to have a common language/grammar so you can build, model and market products. Reality in the financial markets may not be captured in models anyway. So if you convince enough people that a certain procedure/model is good enough for you to price and hedge a financial instrument, that on itself may create a market for it (file under gaussian copulas and credit derivatives) - as long as you keep the revenues flowing no one is going to pay attention to semantics or philosophical discussions, they'll look at the revenues as proof of the validity of the models.So the teachers keep on teaching.

Stochastic calculus has made alot of people on Wall Street rich, don't you know N.

QuoteOriginally posted by: TraderJoeStochastic calculus has made alot of people on Wall Street rich, don't you know N.TJ,What bothers me it that students pay big bucks for an education at top rated universities. And one would think that top rated universities select their teachers and courses based on solid fundamental math and science. Unfortunately this is not the case. Fake mathematics such as 'stochastic calculus' is not only taught but is required for degrees like the MSFE. The students get hit from several angles - first the math is useless in quantitative finance (someone please argue with me on this assertion), second the process of learning fake math blocks the rigor of real mathematics, and finally the student wastes time and money that could be used on other courses that are of value in quantitative finance.This fake math also hurts the quality of Universities. There are just too many math departments that teach junk math. And what's with Columbia math? They appear to be second counting axiom (aka axiom of choice) centric. Bad news...

Last edited by N on May 3rd, 2007, 10:00 pm, edited 1 time in total.

QuoteOriginally posted by: NFake mathematics such as 'stochastic calculus' is not only taught but is required for degrees like the MSFE. The students get hit from several angles - first the math is useless in quantitative finance (someone please argue with me on this assertion)Shouldn't the burden of proof lie with you on that assertion ?If we define quantitative finance as what is practiced today, shouldn't your assertion be: "... the math should be useless in what should constitute proper quantitative finance" ?

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