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This chapter introduces Brownian motion as a model of trait evolution. However, as I demonstrated, Brownian motion can result from a variety of other models, some of which include natural selection. For example, traits will follow Brownian motion under selection is if the strength and direction of selection varies randomly through time. In other words, testing for a Brownian motion model with your data tells you nothing about whether or not the trait is under selection.

• 3.1: Introduction to Brownian Motion
Imagine that you want to use statistical approaches to understand how traits change through time. This requires an exact mathematical specification of how evolution takes place. Obviously there are a wide variety of models of trait evolution, from simple to complex. e.g., creating a model where a trait starts with a certain value and has some constant probability of changing in any unit of time or an alternative model that is more detailed and explicit and considers a large set of individuals.
• 3.2: Properties of Brownian Motion
We can use Brownian motion to model the evolution of a continuously valued trait through time. Brownian motion is an example of a “random walk” model because the trait value changes randomly, in both direction and distance, over any time interval. The statistical process of Brownian motion was originally invented to describe the motion of particles suspended in a fluid.
• 3.3: Simple Quantitative Genetics Models for Brownian Motion
• 3.4: Brownian Motion on a Phylogenetic Tree
We can use the basic properties of Brownian motion model to figure out what will happen when characters evolve under this model on the branches of a phylogenetic tree.
• 3.5: Multivariate Brownian motion
The Brownian motion model we described above was for a single character. However, we often want to consider more than one character at once. This requires the use of multivariate models. The situation is more complex than the univariate case – but not much! In this section I will derive the expectation for a set of (potentially correlated) traits evolving together under a multivariate Brownian motion model.
• 3.6: Simulating Brownian motion on trees
To simulate Brownian motion evolution on trees, we use the three properties of the model described above. For each branch on the tree, we can draw from a normal distribution (for a single trait) or a multivariate normal distribution (for more than one trait) to determine the evolution that occurs on that branch. We can then add these evolutionary changes together to obtain character states at every node and tip of the tree.
• 3.S: Introduction to Brownian Motion (Summary)

## Hyperbolic Dynamics and Brownian Motion: An Introduction

The idea of this book is to illustrate an interplay between distinct domains of mathematics. Firstly, this book provides an introduction to hyperbolic geometry, based on the Lorentz group PSO(1, d) and its Iwasawa decomposition, commutation relations and Haar measure, and on the hyperbolic Laplacian. The Lorentz group plays a role in relativistic space–time analogous to rotations in Euclidean space. Hyperbolic geometry is the geometry of the unit pseudo-sphere. The boundary of hyperbolic space is defined as the set of light rays. Special attention is given to the geodesic and horocyclic flows. . More

The idea of this book is to illustrate an interplay between distinct domains of mathematics. Firstly, this book provides an introduction to hyperbolic geometry, based on the Lorentz group PSO(1, d) and its Iwasawa decomposition, commutation relations and Haar measure, and on the hyperbolic Laplacian. The Lorentz group plays a role in relativistic space–time analogous to rotations in Euclidean space. Hyperbolic geometry is the geometry of the unit pseudo-sphere. The boundary of hyperbolic space is defined as the set of light rays. Special attention is given to the geodesic and horocyclic flows. This book presents hyperbolic geometry via special relativity to benefit from physical intuition. Secondly, this book introduces some basic notions of stochastic analysis: the Wiener process, Itô's stochastic integral and Itô calculus. The book studies linear stochastic differential equations on groups of matrices, and diffusion processes on homogeneous spaces. Spherical and hyperbolic Brownian motions, diffusions on stable leaves, and relativistic diffusion are constructed. Thirdly, quotients of hyperbolic space under a discrete group of isometries are introduced, and form the framework in which some elements of hyperbolic dynamics are presented, especially the ergodicity of the geodesic and horocyclic flows. An analysis is given of the chaotic behaviour of the geodesic flow, using stochastic analysis methods. The main result is Sinai's central limit theorem. Some related results (including a construction of the Wiener measure) which complete the expositions of hyperbolic geometry and stochastic calculus are given in the appendices.

### Bibliographic Information

 Print publication date: 2012 Print ISBN-13: 9780199654109 Published to Oxford Scholarship Online: January 2013 DOI:10.1093/acprof:oso/9780199654109.001.0001

### Authors

#### Affiliations are at time of print publication.

Jacques Franchi, author
Professor of Mathematics, Université de Strasbourg

Yves Le Jan, author
Professor of Mathematics, Université Paris Sud (Orsay) and Institut Universitaire de France

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Financial Engineering is a multidisciplinary field drawing from finance and economics, mathematics, statistics, engineering and computational methods. The emphasis of FE & RM Part I will be on the use of simple stochastic models to price derivative securities in various asset classes including equities, fixed income, credit and mortgage-backed securities. We will also consider the role that some of these asset classes played during the financial crisis. A notable feature of this course will be an interview module with Emanuel Derman, the renowned ``quant'' and best-selling author of "My Life as a Quant". We hope that students who complete the course will begin to understand the "rocket science" behind financial engineering but perhaps more importantly, we hope they will also understand the limitations of this theory in practice and why financial models should always be treated with a healthy degree of skepticism. The follow-on course FE & RM Part II will continue to develop derivatives pricing models but it will also focus on asset allocation and portfolio optimization as well as other applications of financial engineering such as real options, commodity and energy derivatives and algorithmic trading.

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Pricing, Financial Modeling, Financial Risk, Financial Engineering

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Perfect course to grasp the fundamental theories in financial engineering and risk management! I deeply appreciate the videos and lecture notes! Hopefully, I can start the next course soon :)

A very well designed course! I knew some topics prior to the course and it helped me to strengthen my knowledge on derivative market systematically, particularly on interest rate derivatives

## Teaching at the Courant Institute, NYU

Verbal Description: An introduction to the mathematical treatment of random phenomena occurring in the natural, physical, and social sciences. Axioms of mathematical probability, combinatorial analysis, binomial distribution, Poisson and normal approximation, random variables and probability distributions, generating functions, the Central Limit Theorem and Laws of Large Numbers.

Verbal Description: Techniques of integration. Further applications. Plane analytic geometry. Polar coordinates and parametric equations. Infinite series, including power series.

Verbal Description: Systems of linear equations, Gaussian elimination process, Matrices and matrix operations, Determinants, Cramer’s rule. Vectors, Vector spaces, Basis and dimension, Linear transformations, Eigenvalues, Eigenvectors, inner product, orthogonal projection, Gram-Schmidt process, quadratic forms and several applications.

## Brownian motion

Brownian motion is the observed movement of small particles as they are randomly bombarded by the molecules of the surrounding medium. This was first observed by the biologist Robert Brown and was eventually explained by Albert Einstein, for which work he received the Nobel prize.

We can simulate Brownian motion in one dimension by tossing coins. If the sequence of tosses is

then we can move are particle one unit to the right each time we see a 1, and on unit to the left each time we see a 0. This results in the new sequence We haven't dispalyed the starting position (0). We can also display this graphically:

We can simulate Brownian motion using the computer if we have a good generator of uniformly distributed random numbers. This is what the program uni.c does. To copy uni.c to your directory, do this

We copied uni.c from the directory /u/cl/doc/ma217 . Note the period (.) in the copy (cp) command. It stands for your current directory. Now type the program r1.c . Finally, compile it like this: This compiles both your program and uni.c . The two compiled files are then linked together, and you can run them using Recall that a.out is the default name of a compiled C program. You should get this output: Once you have a good way of producing uniformly distributed random numbers in the range [0,1], you can compute other sequences of random numbers. For example, to get a uniformly distributed sequence of zeros and ones, use the rule: where u is a random variable uniformly distributed in the range [0,1]. To get a sequence of uniformly distributed numbers in the range [-1,+1], set

1. Modify r1.c so that you can enter the seed value and the number of random numbers to be printed. You may want to change the output format. Recall that the seed is a long , so you need code like this: scanf("%ld", &iseed) .
2. Test the random number generator uni.c using five "bins" --- five equal intervals which partition the unit interval. You should use your own random seed (perhaps more than once).
3. Devise a program which simulates random walk in one dimension. Do this by setting x = 0, then repeatedly adding uniformly distributed random numbers in the range [-1,1] to x. Random walk models the motion of a small particle in a thin tube of water. It is repeatedly hit, but at random, by the water molecules surrounding it. As your first application of this program, find the distance moved from the origin by the particle in 100 steps.
4. Continuation. Let our particle perform a 10-step random walk 100 times. For each of these "experiments &quot, let Y denote the signed distance from the origin at the end of 100 steps. Make a frequency histogram for Y and find the average value of Y. Is your average value a good approximation of what the expected value is?

Redo the previous problem for 100-step random walks. Study the relative shape of the frequency histograms.

Problems 4 and 5 of the last problem set dealt with random variables that are themselves sums of random variables:

Here the X_i are independent and have identical probability distributions. The central limit theorem, which we have mentioned before, says that as N gets larger and larger, the probability distribution of Y becomes closer and closer to a fixed distribution, the normal distribution. Its general shape is that of a bell curve, with the center of the bell at This bell can be a narrow one or a broad one. About 68% of the area under the bell lies above the interval where is the standard deviation. Let us call this the core of the bell. Note that about 95% of the area under the bell lies above the two standard deviation interval.

1. In the histograms of the previous problem set, find the area of the core and compare it to its expected value.
2. Find the histogram for a random walk of 100 steps, repeated 1000 times. You will want to set up an array to hold the statistics and have your program compute them. Graph the results and compare them with what theory predicts.
3. Brownian motion explains processes as diverse as diffusion of a salt in water and conduction of heat. Image that a lump of salt is placed in the center of a long thin tube. Individual salt ions dissolve and are subject to brownian motion. The random walks of distinct ions are independent. Consequently, if we record the final positions of 1000 ions which "walk " 100 steps, we get the same kind of results as when repeatedly make a single ion walk from the origin. Thus the core in our simulation represents the densest part of region of salty water.

Suppose that after one second the core is one millimeter wide. How wide will it be afer 100 seconds? How wide will it be after 10,000 seconds (almost three hours later). If the concentration is 10 units at one second, what will it be at 100 seconds and at 10,000 seconds?

Make a sketch of the width of the core as a function of time.

Conclude by make a brief, labeled sketch of the diffusion process we have studied. Feature bell-shaped blobs getting wider and shorter, and explain what is going on.

The authors strive to develop statistical intution with as little mathematics as possible.

Markov processes are among the most important stochastic processes for both theory and applications. This book develops the general theory of these processes and applies this theory to various special examples. The initial chapter is devoted to the most important classical example&mdashone-dimensional Brownian motion. This, together with a chapter on continuous time Markov chains, provides the motivation for the general setup based on semigroups and generators. Chapters on stochastic calculus and probabilistic potential theory give an introduction to some of the key areas of application of Brownian motion and its relatives. A chapter on interacting particle systems treats a more recently developed class of Markov processes that have as their origin problems in physics and biology.

This is a textbook for a graduate course that can follow one that covers basic probabilistic limit theorems and discrete time processes.

## Teaching in Spring 2019

431/001, Intro-Theory of Probability 09:55 - 10:45 MWF
(Syllabus, course materials, homework are also posted on Canvas.)

Course Subject, Number and Title
Introduction to Probability Theory MATH/STAT 431 001
Credits 3
Canvas Course URL
https://canvas.wisc.edu/courses/134139
Meeting Time and Location
VAN VLECK B115
09:55 am - 10:45 am MWF.

INSTRUCTORS AND TEACHING ASSISTANTS
Instructor Name: Hao Shen
Instructor Availability Mon 11am-12pm Tue 4-5pm or by appointment
Office: Van Vleck 619
Instructor Email/Preferred Contact [email protected]
Teaching Assistant: Xiao Shen ([email protected])
The teaching assistant will hold general office hours in Van Vleck 101 at the following time:
Tuesday 4:00pm to 6:00pm
Saturday 3:00pm to 5:00pm
Sunday 3:00pm to 5:00pm

The teaching assistant will also hold dedicated office hours.
This means they will have prepared examples or problems to go over.
These are optional to attend. There will not be any essential new material (which would otherwise cause imbalance for those who do not attend these office hours.) These will be held:
Wednesday 4:30-6pm in Van Vleck B337.

OFFICIAL COURSE DESCRIPTION
Math 431 is an introduction to the theory of probability, the part of mathematics that studies random phenomena. Topics covered include axioms of probability, random variables, the most important discrete and continuous probability distributions, expectations, how and when to estimate probabilities using the normal or Poisson approximation, moment generating functions, conditional probability and conditional expectations, multivariate distributions, Markov's and Chebyshev's inequalities, laws of large numbers, and the central limit theorem.
Requisites MATH 234 or 376 or graduate/professional standing or member of the Pre-Masters Mathematics (Visiting International) Program

REQUIRED TEXTBOOK, SOFTWARE & OTHER COURSE MATERIALS Anderson, Seppalainen, Valko: Introduction to Probability.

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### Review

"An attractive text…written in [a] lean and precise style…eminently readable. Especially pleasant are the care and attention devoted to details… A very fine book."

### From the Back Cover

A highly readable introduction to stochastic integration and stochastic differential equations, this book combines developments of the basic theory with applications. It is written in a style suitable for the text of a graduate course in stochastic calculus, following a course in probability.

Using the modern approach, the stochastic integral is defined for predictable integrands and local martingales then Itô’s change of variable formula is developed for continuous martingales. Applications include a characterization of Brownian motion, Hermite polynomials of martingales, the Feynman-Kac functional and Schrödinger equation. For Brownian motion, the topics of local time, reflected Brownian motion, and time change are discussed.

New to the second edition are a discussion of the Cameron-Martin-Girsanov transformation and a final chapter which provides an introduction to stochastic differential equations, as well as many exercises for classroom use.

This book will be a valuable resource to all mathematicians, statisticians, economists, and engineers employing the modern tools of stochastic analysis.

The text also proves that stochastic integration has made an important impact on mathematical progress over the last decades and that stochastic calculus has become one of the most powerful tools in modern probability theory.

Journal of the American Statistical Association

An attractive text…written in [a] lean and precise style…eminently readable. Especially pleasant are the care and attention devoted to details… A very fine book.

## Fission Product Release and Transport

### Agglomeration of Aerosols

Agglomeration occurs due to particle collisions arising from their differing velocities. Particle motion is induced by Brownian diffusion , sedimentation, and turbulence (shear and inertial effects) where other influences such as electrical forces and acoustic influences are less relevant in the present context (see the argument for limited charging of particles in the RCS under Section 5.4.4 ). Particles combine due to Van der Waals forces, changes in surface free energies, and/or chemical reactions where codes generally assume that the sticking efficiency is unity (i.e., colliding particles always stick together). In the RCS, it is the Brownian mechanism that is of most importance since, once embryo particles have formed, this phenomenon will rapidly lead to larger (and fewer) particles. Once the aerosols have grown to a greater size, the other agglomeration mechanisms (properly termed kinematic agglomeration) will come into play. However, the generally short residence times of aerosols and turbulent conditions in the RCS mean that sedimentary agglomeration is usually insignificant.

A particular point that should not be overlooked is that if agglomeration is a significant mechanism, then the numerical treatment of the aerosol population is critical in reproducing what the models intend. In severe accident codes, most commonly the so-called sectional method is employed. According to this method, the particle spectrum is discretized and divided into a number of sections (particle-size bins), thereby approximating the particle-size distribution by a histogram. The aerosol population is then solved for each section. If the particle size bins remain fixed then a high number (50 or more) of size bins is usually required to avoid significant spurious diffusion in the re-sizing scheme: a newly formed particle will not find a size in the discretized scheme that suits it perfectly, and fractioning is needed between two adjacent size classes. The coarser the discretization is, the worse the spurious diffusion.

An important uncertainty in this area is related to particle shape. It is common to associate two shape factors with an aerosol particle, one affecting its mobility (or dynamic) properties, the other its collision properties. Spheres are the most compact particle form possible, and so any deviation from this has some impact on the resistance to movement and the probability of colliding with another particle. Particles in the presence of high steam humidity tend to collapse to compact forms under the influence of water surface tension. However, generally RCS conditions are highly superheated, and compaction due to steam is likely to occur only near the breach for particular sequences producing saturated or near-saturated conditions in this region (cold-leg break or in the steam-generator tube in the case of a steam-generator tube rupture). Nevertheless, perhaps other condensing species are abundant enough to cause a compacting effect since there is evidence from representative experiments that particles are, in fact, fairly compact despite superheated steam conditions. This means that high values for the shape factors, such as for chain-like agglomerates, can probably be excluded. Nevertheless, the shape factors and their evaluation remain a significant uncertainty. Evaluation techniques are often empirical where, in relation to an arbitrary particle, there are no reliable analytical techniques for estimating the appropriate values for use in an agglomeration model. Nonetheless, it is desirable that review of the most representative experiments is undertaken with the objective of proposing more realistic values for the shape factors, these values becoming the default values (rather than unity as is now assumed) in nuclear-safety computer codes.

Brownian agglomeration is most significant for small particles where the free-molecular regime (Knudsen number ≫ 1) and transition regime (Knudsen number of the order of 1) must be considered. The mobility of small particles is very large, but this effect is tempered by the reduced target area that they present. Brownian agglomeration is most effective between very small and larger particles. In general, models derive from Brownian diffusion theory with correction factors for the free-molecular regime and nonspherical particles. The VICTORIA code uses a summation covering the free-molecular (the classic Brownian diffusion theory with the correction for the free-molecular regime proposed by Fuchs  ) and continuum regimes. In contrast, ASTEC uses the Davies model  for the free-molecular and transition regimes and classic Brownian diffusion theory (including the two shape factors for nonspherical corrections) for the continuum regime.

Gravitational agglomeration is most clearly understood in terms of particle terminal velocities showing the phenomenon to be proportional to the difference in the velocities of the two particles and the sum of their projected areas (the target). Disparity arises in a factor termed the collision efficiency (see ,  ) where this constitutes a correction from the ideal situation in which the larger particle sweeps and collects with perfect efficiency all the smaller particles in its projected cylinder during free fall. The correction reduces the efficiency due to hydrodynamic effects where smaller particles tend to flow around the larger particles, allowing some to avoid collection. In the RCS, the limited impact of gravitational agglomeration means that exploration of the different efficiencies is not required here where, in any case, ASTEC and VICTORIA are in agreement in using the Pruppacher and Klett formulation  .

Turbulent agglomeration arises due to the relative particle velocities induced by the shearing flow field and particle drift relative to the flow arising from inertial differences. This latter contribution is zero for particles of the same size, and turbulent agglomeration reaches a minimum in this case. The Saffman and Turner approach  is used in both the ASTEC and VICTORIA codes. An extremely important parameter of this model is the rate of energy dissipation per unit mass of the fluid due to turbulence. All codes refer to a correlation attributed to Laufer  :

where Re is the flow Reynolds number, D is the hydraulic diameter of the pipe, U is the mean flow velocity, and the rate of energy dissipation ɛ per unit mass of fluid is in J·kg·s −1 . While from the point of view of dimensional arguments the above expression is satisfactory, its real origin is somewhat obscure and requires review. A second area of uncertainty arises from how to add together the two different turbulent contributions and how to add these to the other contributions to agglomeration. It has been recommended  that the turbulent shear, turbulent inertial, and sedimentation contributions be added in quadrature and then this combined contribution be added linearly to the Brownian contribution.

In ASTEC this is the approach used, which, in terms of agglomeration/coagulation kernels K (see Eq. (24) below), means:

where Ktot, KBrown, Kturb.shear, Kturb.inertia, and Ksedim represent, respectively, the total agglomeration kernel, the agglomeration kernel due to Brownian diffusion, the agglomeration kernel due to turbulent shear, the agglomeration kernel due to turbulent inertia, and the agglomeration kernel due to sedimentation.

In VICTORIA only the turbulent terms are added in quadrature then this is added linearly to the Brownian and gravitational contributions. The small contribution of the gravitational term in the RCS means that this divergence is slight where investigating and justifying the expression used for the turbulent-energy dissipation rate is of a higher priority.

## Introduction to Stochastic Processes

The objective of this book is to introduce the elements of stochastic processes in a rather concise manner where we present the two most important parts &mdash Markov chains and stochastic analysis. The readers are led directly to the core of the main topics to be treated in the context. Further details and additional materials are left to a section containing abundant exercises for further reading and studying.

In the part on Markov chains, the focus is on the ergodicity. By using the minimal nonnegative solution method, we deal with the recurrence and various types of ergodicity. This is done step by step, from finite state spaces to denumerable state spaces, and from discrete time to continuous time. The methods of proofs adopt modern techniques, such as coupling and duality methods. Some very new results are included, such as the estimate of the spectral gap. The structure and proofs in the first part are rather different from other existing textbooks on Markov chains.

In the part on stochastic analysis, we cover the martingale theory and Brownian motions, the stochastic integral and stochastic differential equations with emphasis on one dimension, and the multidimensional stochastic integral and stochastic equation based on semimartingales. We introduce three important topics here: the Feynman&ndashKac formula, random time transform and Girsanov transform. As an essential application of the probability theory in classical mathematics, we also deal with the famous Brunn&ndashMinkowski inequality in convex geometry.

This book also features modern probability theory that is used in different fields, such as MCMC, or even deterministic areas: convex geometry and number theory. It provides a new and direct routine for students going through the classical Markov chains to the modern stochastic analysis.

• Preface to the English Edition
• Preface to the Chinese Edition
• Markov Processes:
• Discrete-Time Markov Chains
• Continuous-Time Markov Chains
• Reversible Markov Chains
• General Markov Processes
• Martingale
• Brownian Motion
• Stochastic Integral and Diffusion Processes
• Semimartingale and Stochastic Integral