Brownian motion and stochastic calculus

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1. graycatSep 2019
   To get very far in stochastic processes, usually need a good background in measure theory. Here is an overview of about two semesters in a fast, often rare, grad course plus much more:

   The integral of freshman calculus is the Riemann integral. Too soon/easily it was seen to be clumsy to inadequate in theoretical developments. Near 1900 in France, E. Borel student H. Lebesgue improved the situation; he did his Lebesgue integral. Don't worry: For every function integrated in freshman calculus, Lebesgue's integral gives the same answer. Otherwise Lebesgue's integral is much more general and, it turns out, is the solid foundation of probability and stochastic processes.

   Lebesgue's work starts with measure which is just a notion of area. So, yes, on the real line, the Lebesgue measure of the closed interval [0,1] is 1, just the same as we learned in grade school. But with Lebesgue's work, it's tough to think of a subset of the real line that is not measurable (has a measure or area in Lebesgue's work) -- the usual example of a subset of R that is not measurable needs the axiom of choice. With measure defined on R, Lebesgue can define the Lebesgue integral of functions

   f: R --> R

   For this we need a meager assumption (which essentially always holds), but lets charge on:

   For the generalizations, crucial for probability and stochastic processes, start with a non-empty set M. R is an example of a such a set M. Define a measure on M, say, m. Then for some subsets A of M, we have the measure of set A as m(A). That is, we have the area (volume, whatever) of A. But to do much, we need more assumptions: We want the set of all subsets A of M to which we assign measure m(A) to be a sigma algebra, say, F. As at

   https://en.wikipedia.org/wiki/Sigma-algebra

   we want M to be an element of F, for each A in F we want the complement, say, the set M - A (obvious but sloppy notation) to be an element of F, and for A_i in F for i = 1, 2, ..., we want the union of the A_i to be an element of F. So, we want F closed under complements and countable unions. No, we don't permit uncountable unions, a point that keeps popping up in stochastic processes. Permitting uncountable unions results in, in one word, a mess.

   The sigma part of sigma algebra is supposed to refer to the countable unions as in the capital letter sigma used in calculus, etc., for countable sums.

   So, we have the set M, the sigma algebra of subset F, and the measure m. Then the pair (M, F) is a measurable space and the triple (M, F, m) is a measure space. In probability theory and stochastic processes, the measure space used is usually denoted by space, capital omega, the sigma algebra, script F, and the measure, P.

   Right -- I just let the cat out of the bag: Probability P, as we saw in elementary treatments of probability, is a measure on the measure space with Omega, script F, and P.

   So for A a subset of the space Omega and an element of sigma algebra script F, P(A) is the probability of set A. Set A is called an event. Now we are back to what you saw in elementary treatments of probability.

   So, it went too fast -- we mentioned measure theory with (M, F, m) and then took a special case Omega, script F, P, a probability space, and got back to P(A) we saw in elementary treatments.

   This jump from a measure space to a probability space was made by A. Kolmogorov in 1933 and has been the foundation of 99 44/100% (as in Ivory soap!) of advanced work in probability and stochastic processes since then.

   So, back to the case of measure space (M, F, m). Suppose we have a function

   f: M --> R

   We want to integrate f as in freshman calculus. Well, for each subset A of M and an element of sigma algebra F, we have the measure (area) of A, that is, m(A). Well with one more assumption, and what Lebesgue did, that's enough! And it's assuming less than Riemann did and, thus, getting more generality in some quite good ways.

   Then for the case of Lebesgue integrating some

   f: R --> R

   compared with the Riemann integral, Lebesgue partitions on the Y axis and Riemann partitions on the X axis. So, Riemann partitions on the domain of the function and Lebesgue partitions on the range. In both cases we are working with lots of little rectangles that get smaller as we converge. So, Lebesgue lets the domain be much more general, e.g., enough for probability theory.

   So, back to a probability space and a function

   f: Omega --> R

   With the meager assumption (that f is measurable -- to be brief I'm skipping over the details here is only because nearly all functions are measurable -- tough to construct one that isn't, so general is Lebesgue's work), then f is a random variable. It is more common to denote random variables by X, Y, Z, etc. Now, finally, maybe after getting lost in a swamp for some years, you have a solid definition of a random variable. Then each point in Omega is one probability or statistical trial.

   Then, sit down for this amazing fact: The expectation of real valued random variable X, denoted by E[X], is just the Lebesgue integral of the function

   X: Omega --> R

   So, in this way, Kolmogorov borrowed from Lebesgue and got a solid foundation for probability and stochastic processes.

   Now you know why Lebesgue's measure theory is key to advanced work in probability and stochastic processes -- E[X] is just Lebesgue's integral generalization of the calculus Riemann integral.

   Long standard very careful, polished, treatments of measure theory are in Royden, Real Analysis and the first half of Rudin, Real and Complex Analysis. Continuing with probability is Breiman, Probability, Neveu, Mathematical Foundations of the Calculus of Probability, Loeve, Probability, etc. Uh, Loeve was long at Berkeley, and both Breiman and Neveu were Loeve students.

   Can use this work to make good sense out of Brownian motion, and there are plenty of books there.

   And can do more on stochastic processes, e.g.,

   Ioannis Karatzas and Steven E. Shreve, Brownian Motion and Stochastic Calculus.

   R. S. Lipster and A. N. Shiryayev, Statistics of Random Processes.

   Ronald K. Getoor, Markov Processes and Potential Theory.

   This material is now decades old. With this material it is possible to give high end treatments of some topics in Wall Street type investing, e.g., exotic options and in particular the Black-Scholes formula.

   This material was called by Breiman "graduate probability" as he left the field and did, e.g., CART -- classification and regression trees, maybe the most important seed for the current work in machine learning. Breiman was interested in making sense out of some complicated medical data.

   The potential of such graduate probability for careers in academics and/or applications seems now to be open to question. But, with a larger view, probability and stochastic processes just will NOT go away and will remain some of the most powerful tools for making sense out of reality -- we can be sure that over the horizon we will be glad to have such a solid foundation and WILL encounter important, new applications, ours or those of others.

   There is an old point: Long it was common for the pure mathematicians, yes, to pay close attention to measure theory but essentially to ignore the role in probability theory, saying, that probability theory was just a trivial special case of measure theory for positive measures (can also have measures that have negative or complex values) where the measure of the whole space was 1.

   But: Probability theory does a LOT with the assumption of independence tough to find in texts in just measure theory. Also the Radon-Nikodym theorem (an amazing generalization of the fundamental theorem of calculus) IS treated in measure theory but the applications crucial in probability and stochastic processes to conditioning and sufficient statistics usually are not.

   So, with this overview, you might find Royden, Rudin, Breiman, Neveu (especially elegant) not so hard reading and a lot of fun.

   You will learn about the cases of convergence of sequences of random variables and the classic results of the central limit theorem, the laws, weak and strong, of large numbers, ergodic theory (pour cream into coffee and stir until the cream returns to the distribution it had in the coffee just after pour it in -- Poincare recurrence), and martingales. Margingales are amazing and provide the easiest proof of the strong law of large numbers and some of the strongest inequalities in math.

   So, amazing stuff: Go into a lab, measure something, walk out, and have the value of a random variable -- okay, that's good enough for an elementary course. But typically in math, we can't calculate just what we want in just one step so have to iterate and approximate and hopefully converge in some appropriate sense to just what we want. Well, in those classic results, we have notions of convergence. The most amazing case is of strong convergence where we have a sequence of random variables that converge to some random variable. From the elementary treatments, having a sequence of randomness converge to more randomness seems absurd. But with Kolmogorov's treatment, a random variable is just a function, and from math such as Rudin, Principles of Mathematical Analysis we know quite a lot about convergence, and in measure theory we learn more. Then the case of a sequence of random variables converging to another random variable is just a special case of one case of convergence of sequences of functions. So, we can approximate a random variable X by having a sequence of random variables converge to random variable X. How 'bout that! Thank you Borel, Lebesgue, Kolmogorov, von Neumann, etc.

   Have fun! Get rich???

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2. graycatJul 2018
   To make the study easier and better rounded, just add some material.

   Rudin's Principles is really nice, especially in retrospect once understand it, but going in as a student it can seem quite severe. It's precise but not too severe -- he just makes you go a chapter or two before it becomes clear why he is doing what he is doing.

   To help, my nutshell view is that mostly he is just trying to develop the Riemann (Stieltjes) integral. His main result is, the Riemann integral exists for continuous functions on compact sets. So, then, he needs to say what a compact set is. Well, the most relevant example is just a closed interval of real numbers such as [a,b]. So, why compact? Because every continuous function on a compact set is uniformly continuous, and that lets us know that the Riemann sums converge. What is compact? Every open cover has a finite subcover, and that lets us get uniform continuity. And, in R^n, a set is compact if and only if it is closed and bounded. So, Rudin needs to talk about closed versus open -- he does that on metric spaces although really he needs it only on R^n.

   So, net, he starts with metric spaces and discusses open, closed, and compact. Then he shows that in R^n, compact is the same as closed and bounded. He shows that a continuous function on a compact set is uniformly continuous. Then, presto, he shows that the Riemann (or Riemann-Stieltjes if you wish) sums converge and the Riemann integral exists.

   He does some nice work on infinite sequences and series, and the main reason is that he uses those tools to show lots of limits exist, e.g., for sines, cosines, and Fourier series.

   There's more of value in Principles, but IMHO I gave you a good start to make the book easier. I wish I'd had been given that outline when I was working through Principles at 1+ hour a page.

   But the Lebesgue integral in Royden is the one to take fully seriously.

   Make Halmos the second or third text on linear algebra. And then look at some quantum mechanics where they discuss eigenvalues, eigenvectors, Hermitian, unitary, and the spectral decomposition! Right, the Halmos book is baby Hilbert space. Then look at some applied connections, e.g.,

   George E.\ Forsythe and Cleve B.\ Moler, {\it Computer Solution of Linear Algebraic Systems,\/}

   Maybe spend an evening on the documentation of LINPACK.

   Some weekend of great fun, take a fast pass through the Gauss, ..., Stokes theorem parts of

   Tom M.\ Apostol, {\it Mathematical Analysis: A Modern Approach to Advanced Calculus,\/}

   where don't take the proofs very seriously but to see how physical science and engineering look at calculus of several variables.

   Neveu is a great last probability text but not a good first text. So, before Neveu, look quickly, not very seriously, at whatever, including in some introductory statistics texts.

   Also, Breiman's Probability is easier to read than Neveu. So, is K. L. Chung's competitive book. And there are others.

   There is some more advanced material, e.g.,

   Ioannis Karatzas and Steven E.\ Shreve, {\it Brownian Motion and Stochastic Calculus, Second Edition,\/}

   Good luck!

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3. graycatDec 2014
   Take the standard path.

   High School

   Algebra I

   Plane geometry (with emphasis on proofs)

   Algebra II

   Trigonometry

   Solid geometry (if can get a course in it -- terrific for intuition and techniques in 3D)

   College

   Analytic geometry (conic sections)

   Calculus I and II

   Linear algebra

   Linear algebra II, Halmos, Finite Dimensional Vector Spaces (baby version of Hilbert space theory)

   Advanced calculus, e.g., baby Rudin, Principles of Mathematical Analysis -- nice treatment of Fourier series, good for signals in electronic engineering. The first chapters are about continuity, uniform continuity, and compactness which are the main tools used to prove the sufficient conditions for the Riemann integral to exist. At the end Rudin shows that the Riemann integral exists if and only if the function is continuous everywhere but on a set of measure zero. But what Rudin does there at the beginning with metric spaces is more general than he needs for the Riemann integral but is important later in more general treatments in analysis. Rudin does sequences and series because they are standard ways to define and work with some of the important special functions, especially the exponential and sine and cosine further on in the book. The material in the back on exterior algebra is for people interested in differential geometry, especially for relativity theory.

   Ordinary differential equations, e.g., Coddington, a beautifully written book, Coddington and Levinson is much more advanced) -- now can do basic AC circuit theory like eating ice cream.

   Advanced calculus from one or several more traditional books, e.g., the old MIT favorite Hildebrand, Advanced Calculus for Applications, Fleming, Functions of Several Variables, Buck, Advanced Calculus -- can now look at Maxwell's equations and understand at least the math. And can work with the gradient for steepest descent in the maximum likelihood approach to machine learning.

   Maybe take a detour into differential geometry so that can see why Rudin, Fleming, etc. do exterior algebra, and why Halmos does multi-linear algebra, and then will have a start on general relativity.

   Royden, Real Analysis. So will learn measure theory, crucial for good work in probability and stochastic processes, and get a start on functional analysis (vector spaces where each point is a function -- good way to see how to use some functions to approximate others). Also will learn about linear operators and, thus, get a solid foundation for linear systems in signal processing and more.

   Rudin, Real and Complex Analysis, at least the first, real, half. Here will get a good start on the Fourier transform.

   Breiman, Probability -- beautifully written, even fun to read. Measure theory based probability. If that is too big a step up in probability, then take a fast pass through some elementary treatment of probability and statistics and then get back to Breiman for the real stuff. Will finally see what the heck a random variable really is and cover the important cases of convergence and the important classic limit theorems. Will understand conditioning, the Radon-Nikodym theorem (von Neumann's proof is in Rudin, R&CA), conditioning, the Markov assumption, and martingales and the astounding martingale convergence theorem and the martingale inequality, the strongest in mathematics. So will see that with random variables, can look for independence, Markov dependence, and covariance dependence, and these forms of dependence, common in practice, can lead to approximation, estimation, etc.

   Now will be able to understand EE treatments of second order stationary stochastic processes, digital filtering, power spectral estimation, etc.

   Stochastic processes, e.g., Karatzas and Shreve. Brownian Motion and Stochastic Calculus. Now can get started on mathematical finance.

   But there are many side trips available in numerical methods, linear programming, Lagrange multipliers (a surprisingly general technique), integer programming (a way to see the importance of P versus NP), mathematical statistics, partial differential equations, mathematical finance, etc.

   For some ice cream, Luenberger, Optimization by Vector Space Methods or how to learn to love the Hahn-Banach theorem and use it to become rich, famous, and popular with girls!

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4. veyronOct 2011
   You need to be a little more specific regarding your interests and background. If you are interested in theory, there are plenty of books discussing specific models. However, most practical or buzzword (e.g. HFT) books are rubbish (think about it: if you have a profitable operation, why would you write a book? Unless, like me, you have a screw loose :P )

   If you want some practical discussions, "Trading and Exchanges" by Larry Harris is a bit dated, but nothing comes close. Most of the descriptions are still valid today (even if some of the mechanics have changed)

   If you want something more along the lines of what a financial engineer would know:

   - "Options, Futures, and other Derivatives" by John Hull (goes over basic models without delving too deeply into theoretical math aspects)

   - "Stochastic Calculus for Finance II" by Steve Shreve (goes over the basic stuff but has enough stats to keep a grad student happy)

   - "Monte Carlo Methods in Financial Engineering" by Paul Glasserman (much more practical, and goes over subtleties of monte carlo simulation and other stuff like low discrepancy sampling)

   - "Modelling Fixed Income Securities and Interest Rate Options" by Robert Jarrow (walks through how to perform certain simulations, covers lots of little details most theoretical books skip)

   If you want something more theoretical:

   - "Introduction to Stochastic Calculus Applied to Finance" by Lamberton and Lapeyre (Nice little intro)

   - "Arbitrage Theory in Continuous Time" by Tomas Bjork (Slower discussion, larger breadth)

   - "Brownian Motion and Stochastic Calculus" and "Methods of Mathematical Finance" by Karatzas and Shreve (Solid theoretical foundation, for the more mathematically inclined)

   There was a good non-measure theoretic discussion of financial models but the name escapes me ATM.

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5. HilbertSpaceOct 2010
   Your

   "I'm more than willing to spend an hour per page and do all the exercises, but I would like good exposition."

   is essentially necessary and sufficient.

   If an exercise takes more than two hours, then swallow your pride and skip the exercise (it may be misplaced, have an error, or just be way too difficult for effective education).

   For linear algebra:

   (1) Work through a few introductory texts.

   (2) Work carefully through the long time, unchallenged, world-class classic,

   Halmos, Finite Dimensional Vector Spaces.

   and there note near the back his cute ergodic convergence theorem.

   The glory here is the polar decomposition.

   (3) Get some contact with some applications, including in elementary multi-variate statistics, numerical techniques, optimization, etc.

   (4) Pick from

   Horn and Johnson, Matrix Analysis.

   and

   Horn and Johnson, Topics in Matrix Analysis.

   My first course was an "advanced course" from one of Horn, Johnson, and I knocked the socks off all the other students. How'd I do that? Brilliant? Worked hard? Learned a lot? Nope. Instead the key was just my independent work with (1) -- (3).

   So if you do (1) -- (4), then you will be fine.

   For analysis, (Baby Rudin)

   Walter Rudin, Principles of Mathematical Analysis.

   Note in the back that a function is Riemann integrable if and only if it is continuous everywhere except on a set of Lebesgue measure 0.

   Also know cold that a uniform limit of continuous functions is continuous.

   Royden, Real Analysis.

   and the first, real, half of (Papa Rudin)

   Rudin, Real and Complex Analysis.

   Of course, emphasize the Radon-Nikodym theorem; I like the easy steps in Royden and Loeve (below), but see also the von Neumann proof in Papa Rudin.

   For probability based on measure theory and the limit theorems,

   Breiman, Probability.

   Note his result on regular conditional probabilities.

   Neveu, Mathematical Foundations of the Calculus of Probability.

   If you can work all the Neveu exercises, then someone should buy you a La Tache 1961.

   Loeve, Probability Theory.

   Note the classic Sierpinski counterexample exercise on regular conditional probabilities (also in Halmos, Measure Theory),

   Cover the Lindeberg-Feller version of the central limit theorem as well as simpler versions. Do the weak law of large numbers as an easy exercise. Cover the martingale convergence theorem (I like Breiman here) and use it to give the nicest proof of the strong law of large numbers. Cover the ergodic theorem (Garcia's proof) and its (astounding) application to Poincare recurrence. Cover the law of the iterated logarithm and its (astounding) application to the growth of Brownian motion.

   Of course apply the Radon-Nikodym theorem and conditioning to sufficient statistics and note that order statistics are always sufficient. Show that sample mean and variance are sufficient for i.i.d. Gaussian samples and extend to the exponential family.

   Give yourself an exercise: In Papa Rudin, just after the Radon-Nikodym theorem, note the Hahn decomposition and use it to give a quite general proof of the Neyman-Pearson lemma.

   To appreciate the law of large numbers in statistics, read the classic Halmos paper on minimum variance, unbiased estimation.

   For tools for research in statistics, might want to get going in stochastic processes. So, for elementary books, look for authors Karlin, Taylor, and Cinlar and touch on some applications, e.g., Wiener filtering and power spectral estimation. Note the axiomatic derivation of the Poisson process and the main convergence theorem in finite Markov chains (also a linear algebra result). Then for more, note again the relevant sections of Breiman and Loeve and then:

   Karatzas and Shreve, Brownian Motion and Stochastic Calculus.

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6. graycatDec 2021
   (1) Financial Math. Some of the comments here have to do with mathematical finance, e.g., the Black-Scholes model.

   Mathematical finance done with theorems and proofs has some strong prerequisites of measure theory with sigma algebras, Markov processes, Brownian motion, the Radon-Nikodym theorem, conditional expectation, stochastic integration, e.g., Ioannis Karatzas, Steven E. Shreve, Brownian Motion and Stochastic Calculus, Robert M. Blumenthal and Ronald K. Getoor, Markov Processes and Potential Theory. Can also want parabolic partial differential equations and stochastic dynamic programming.

   In my experience, with a pure/applied math Ph.D. and as a prof in an MBA program, nearly no MBA students or profs have those prerequisites. And with my experience in finance, nearly no one in finance has those prerequisites, either.

   The people in MBA programs, as students or professors, may have gotten such a background from Cinlar, Shreve, or one of a few more.

   Bluntly, not many people, a tiny fraction even in pure math, and a good background in pure math is a prerequisite, actually study stochastic processes and stochastic optimal control with a measure theory prerequisite.

   Anyone X with anything like that math background might want to be careful or that background will scare nearly all potential employers, in finance or anything else, and result in X being ostracized and fired or not hired.

   (2) Math in MBA Programs. When I was a prof in an MBA program, all the math used was undergraduate level, e.g., with hardly even a calculus prerequisite.

   (3) Advanced STEM Educations for Non-Academic Jobs. It appears that an advanced education, e.g., in computer science, can help in getting a non-academic job, but the real help is just passing a filter, having proved oneself, and not aiding in doing the job.

   This situation should not be a surprise: Having exceptionally good qualifications will make a person X exceptional and rare, that is, one out of 100+. Then the other 99 can be afraid and resentful and gang up on X, sabotage X with gossip, keep X out of the loop, etc.

   (4) Jobs and People with Exceptional Qualifications. It is common for someone X with exceptional qualifications to attempt to use those qualifications to get and do well in a job as an employee. That would mean that person X has a supervisor Y who created the job. But as we can anticipate, it is rare for a supervisor Y to have the exceptional qualifications or to create a job that needs such qualifications. Moreover Y could be uncomfortable, feel threatened by, a subordinate X having and using qualifications Y did not have. In simple terms, commonly Y hires X to apply routine muscle to work conceived by Y. To borrow from a James Bond movie, X is not hired to think and, instead, is hired to do what they are told.

   (5) Potential of Exceptional Qualifications. For a person with exceptional qualifications that might be powerful and valuable for a career, don't look for a corresponding job created by someone else and, instead, create the corresponding job for yourself. E.g., do a startup.

   (6) Financial Success. If person X works as an employee, then the supervisor of X knows to the last penny just how much money X is making and usually will do their best to keep down the amount. However, if person X owns a business, then for X the customers replace the supervisor, and it's common for the customers not to know even roughly how much money X is making.

   So, net, for X with exceptional qualifications that seem powerful and valuable, start and run a successful business and make use of those qualifications. There X can be making a nice fortune, with no supervisor at all and with customers who neither know nor much care how much X is making.

   (7) Academic Research Jobs. In academic research, the prof is essentially begging a little money from sources with big money, working hard, and then begging journals to let them give away the results of the work for free. So, we might anticipate that such jobs won't pay very well. I concluded that broadly academic jobs were, for someone who wanted to support a family with financial security, financially irresponsible.

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