Stochastic Calculus and Black-Scholes Theory MTH772P Exercises Sheet 1
|
|
- Eleanore Curtis
- 5 years ago
- Views:
Transcription
1 Stochastic Calculus and Black-Scholes Theory MTH772P Exercises Sheet. For ξ, ξ 2, i.i.d. with P(ξ i = ± = /2 define the discrete-time random walk W =, W n = ξ ξ n. (i Formulate and prove the property of independence of increments of (W n, n. (ii Show that (W n, n is a discrete-time Markov chain. (iii Show that (W n, n is a discrete-time martingale. (iv Calculate the covariance Cov(W i, W j. (v Find the limit distribution of W nt / n as n, for t >. Solution (i For any integer times = j < j <... < j k the increments W j W j = ξ ξ j,..., W jk W jk = ξ jk ξ jk are independent. Because ξ, ξ 2,... are independent by the assumption, the vectors (ξ,..., ξ j,..., (ξ jk +,..., ξ jk are independent, hence the increments are independent as functions of the independent random vectors. (ii For any possible path w,..., w n of the random walk we have P(W n = w n W n = w n, W n 2 = w n 2,..., W = w = P(W n W n = w n w n W n = w n, W n 2 = w n 2,..., W = w. In this conditional probability the event {W n W n = w n w n } is the same as {ξ n = w n w n }, while the event {W n = w n, W n 2 = w n 2,..., W = w } can be written in terms of ξ,..., ξ n. It follows from the independence of ξ i s that P(W n = w n W n = w n, W n 2 = w n 2,..., W = w = P(W n W n = w n w n = P(W n W n = w n w n W n = w n = which is the Markov property for discrete-time processes. (iii We have using rules for conditional expectations P(W n = w n W n = w n, E(W n+ W,..., W n = E(W n + ξ n+ W,..., W n = E(W n W,..., W n + E(ξ n+ W,..., W n = W n + Eξ n+. Note that we could also write the conditional expectation E(W n+ W,..., W n as E(W n+ F n, where F n is the σ-algebra generated by the events {W = w,..., W n = w n } with arbitrary w,..., w n. Also, F n is the σ-algebra generated by the random variables ξ,..., ξ n. (iv Assume i j. Since EW i = we have Cov(W i, W j = E(W i W j = E(W i (W i + (W j W i = E(Wi 2 +E(W i E(W j W i = Var(W i = ivarξ = i. Thus Cov(W i, W j = i j for any i, j. (v The limit distribution of (ξ ξ nt / nt (for every fixed t > and n is N (, by the central limit theorem. From this, the limit distribution of W nt / n is N (, t.
2 2. Let B = (B(t, t be a standard Brownian motion. Show that the following processes are standard BM: (i X(t = B(t + s B(s, where s is constant. (ii X(t = B(ct/ c, for any c >, (iii X(t = B( t B(, where t [, ] (this BM is defined on [, ]. Solution (i The process has continuous paths and X( =. The increments of X(t i+ X(t i over intervals of partition = t < t <... < t n are the increments of the BM over the intervals between times s < t + s <... < t n + s, hence they are independent and N (, (t i+ t i -distributed. (ii B(ct is N (, ct-distributed, hence B(ct/ c is N (, t-distributed (check the mean and the variance. We have X( =. The increments of X over the intervals of partition = t < t <... < t n are the increments of BM over the intervals of partition = ct < ct <... < ct n, hence the X-increments are independent. (iii The increments of X(t j+ X(t j over = t < t <... < t n = are the increments of the BM over the intervals of partition t n <... < t. The independence of increments follows. The rest is obvious. 3. For X a random variable with density function f consider the event A = {X }. (i Define G to be the σ-algebra generated by A (i.e. the smallest σ-algebra containing event A. Write down the list of all elements of the σ-algebra G. (ii In terms of integrals with density f, describe the random variable E(X 3 G. (iii Using the formulas you derived in (ii show explicitly that E(E(X 3 G = E(X 3. (iv Make the calculations for (ii, (iii assuming that X has N (, distribution. Solution (i G = {, Ω, {X }, {X > }}. (ii The events {X }, {X > } are disjoint and their union is Ω. Hence we can write E(X 3 G = E(X 3 X (X + E(X 3 X (X <, where ( is the indicator random variable. In particular, E(X 3 G may take two values, depending on whether X or X <. In terms of the density, these values are E(X 3 X = x 3 f(xdx f(xdx, E(X3 X < = x3 f(xdx f(xdx. 4. Let (F t, t be a filtration for BM. That means that {ω Ω : B(s x} F t for s t and x R, and that the increments of BM after t are independent of F t. For < a < b < c show that B(c B(b is independent of F a. Solution We have B(c B(b independent of F b, which means that any event {B(c B(a < x} is independent of any event A F b. But F a F b, hence B(c B(b is also independent of F a. 5. (BM with drift Let X(t = B(t + tµ. Show that (X(t, t is a Markov process and find its transition density. Is the process a martingale? 2
3 Solution Let (F t, t be a filtration for the BM. The Markov property of the BM itself means that for s < t E(f(B(t F s = E(f(B(t B(s for any function f. (Note that if we take f(x = (x a the conditional expectation becomes the conditional probability E[(B(t a F s ] = P[B(t a F s ]. Thus from the Markov property of BM E(f(X(t F s = E(f(B(t + tµ F s = E(f(B(t + tµ B(s. But conditioning on B(s is the same as conditioning on X(s = B(s + sµ, because X(s and B(s uniquely determine one another. Hence the above becomes and so E(f(B(t + tµ B(s = E(f(B(t + tµ X(s = E(f(X(t X(s, E(f(X(t F s = E(f(X(t X(s, which is the Markov property for (X(t, t. To compute the transition probability function of X we should reduce to the BM, and there are various equivalent ways to do that. The probability that X moves from X(s = x to some value X(t y is P(X(t y X(s = x = P(B(t + tµ y B(s = x sµ = P(B(t y tµ B(s = x sµ = P(B(t B(s y x (t sµ B(s = x sµ = P(B(t B(s y x µ(t s = y x (t sµ exp{ u 2 /(2t 2s}du The transition density from x to y in time t s is obtained by differentiating this (conditional distribution function in y p(t s, x, y = e (y x (t sµ2 /(2t 2s Another possibility is to use the fact that the transition density satisfies E(f(X(t X(s = x = p(t s, x, yf(ydy for any function f. Using Lemma.9 from the lecture notes E(f(X(t X(s = x = E[f((B(t B(s + B(s + tµ B(s = x sµ] = E[f((B(t B(s + x sµ + tµ] = p(t s,, yf(u + x sµ + tµdu = p(t s, x, yf(ydy, where the last step used change of variable u + x sµ + tµ = y and the transition density p(t s,, y = exp( 2y 2 /(2t 2s/ of the BM. 3
4 Finally, somewhat heuristic but quick way is as follows. Process X moves from X(s = x to X(t [y, y + dy] when the BM moves from B(s = x sµ to B(t [y tµ, y + dy tµ]. The latter is an event of probability p(t s, x sµ, y tµdy = p(t s,, y x (t sµdy. Discarding dy yields the transition density p(t s, x, y = p(t s,, y x (t sµ for the process X. 6. (Geometric BM Let S(t = S( exp(νt + σb(t, where S(, σ, ν are positive constants. Show that (S(t, t is a Markov process and find its transition density. Solution The Markov property is shown as in Exercise 5: if S(s = x then B(s = (log x S νt/σ, so we can compute S(s from B(s and vice versa. Let for shorthand b = (log x S νt/σ. We have E[f(S(t S(s = x] = E[f(S(t B(s = b] = E[f(S(e νt+σ(b(t B(s+σB(s B(s = b] = E[f(S(e νt+σ(b(t B(s+σb ] = Using the change of variable (recall the definition of b f(s(e νt+σu+σb e u2 /(2t 2s du. y = S(e νt+σu+σb, u = log(y/x ν(t s, du = dy σ σy the above integral becomes exp ( (log(y/x ν(t s2 2σ f(y 2 (t s σy dy. Therefore the transition density function of (S(t, t is ˆp(t s, x, y = exp ( (log(y/x ν(t s2 2σ 2 (t s σy 7. (Black-Scholes formula Let S(t = S( exp((r σ 2 /2t + σb(t, where S(, σ, r are positive constants. For K > and T > show that E[e rt (S(T K + ] = S(Φ(d + (T, S( Ke rt Φ(d (T, S(, where Φ is the standard normal distribution function, and d ± (T, S( = ( σ log S( T K σ2 + (r ± 2 T Solution We need to compute the integral integrate E[e rt (S(T K + ] = e rt σ (log(k/s( (r σ2 /2T 4.. ( S(e (r σ2 /2T e +σx x 2 /(2T K dx. T
5 Substituting y = x/ T and using linearity of the integral the above becomes S(e σ2 T/2 S( σ T (log(k/s( (r σ2 /2T σ T σ T (log(k/s( (r σ2 /2T Ke rt σ T (log(k/s( (r σ2 /2T e y2 /2+σ T y dy e y2 /2 dy = e z2 /2 dz Ke rt Φ(d (T, S( = S(Φ(d + (T, S( Ke rt Φ(d (T, S(. 5
1.1 Definition of BM and its finite-dimensional distributions
1 Brownian motion Brownian motion as a physical phenomenon was discovered by botanist Robert Brown as he observed a chaotic motion of particles suspended in water. The rigorous mathematical model of BM
More informationKolmogorov Equations and Markov Processes
Kolmogorov Equations and Markov Processes May 3, 013 1 Transition measures and functions Consider a stochastic process {X(t)} t 0 whose state space is a product of intervals contained in R n. We define
More information1 Simulating normal (Gaussian) rvs with applications to simulating Brownian motion and geometric Brownian motion in one and two dimensions
Copyright c 2007 by Karl Sigman 1 Simulating normal Gaussian rvs with applications to simulating Brownian motion and geometric Brownian motion in one and two dimensions Fundamental to many applications
More informationFiltrations, Markov Processes and Martingales. Lectures on Lévy Processes and Stochastic Calculus, Braunschweig, Lecture 3: The Lévy-Itô Decomposition
Filtrations, Markov Processes and Martingales Lectures on Lévy Processes and Stochastic Calculus, Braunschweig, Lecture 3: The Lévy-Itô Decomposition David pplebaum Probability and Statistics Department,
More informationSelected Exercises on Expectations and Some Probability Inequalities
Selected Exercises on Expectations and Some Probability Inequalities # If E(X 2 ) = and E X a > 0, then P( X λa) ( λ) 2 a 2 for 0 < λ
More informationI forgot to mention last time: in the Ito formula for two standard processes, putting
I forgot to mention last time: in the Ito formula for two standard processes, putting dx t = a t dt + b t db t dy t = α t dt + β t db t, and taking f(x, y = xy, one has f x = y, f y = x, and f xx = f yy
More informationELEMENTS OF PROBABILITY THEORY
ELEMENTS OF PROBABILITY THEORY Elements of Probability Theory A collection of subsets of a set Ω is called a σ algebra if it contains Ω and is closed under the operations of taking complements and countable
More informationThe concentration of a drug in blood. Exponential decay. Different realizations. Exponential decay with noise. dc(t) dt.
The concentration of a drug in blood Exponential decay C12 concentration 2 4 6 8 1 C12 concentration 2 4 6 8 1 dc(t) dt = µc(t) C(t) = C()e µt 2 4 6 8 1 12 time in minutes 2 4 6 8 1 12 time in minutes
More informationThe multidimensional Ito Integral and the multidimensional Ito Formula. Eric Mu ller June 1, 2015 Seminar on Stochastic Geometry and its applications
The multidimensional Ito Integral and the multidimensional Ito Formula Eric Mu ller June 1, 215 Seminar on Stochastic Geometry and its applications page 2 Seminar on Stochastic Geometry and its applications
More informationMASSACHUSETTS INSTITUTE OF TECHNOLOGY 6.265/15.070J Fall 2013 Lecture 7 9/25/2013
MASSACHUSETTS INSTITUTE OF TECHNOLOGY 6.65/15.070J Fall 013 Lecture 7 9/5/013 The Reflection Principle. The Distribution of the Maximum. Brownian motion with drift Content. 1. Quick intro to stopping times.
More informationT n B(T n ) n n T n. n n. = lim
Homework..7. (a). The relation T n T n 1 = T(B n ) shows that T n T n 1 is an identically sequence with common law as T. Notice that for any n 1, by Theorem.16 the Brownian motion B n (t) is independent
More information1. Stochastic Processes and filtrations
1. Stochastic Processes and 1. Stoch. pr., A stochastic process (X t ) t T is a collection of random variables on (Ω, F) with values in a measurable space (S, S), i.e., for all t, In our case X t : Ω S
More informationStochastic Calculus for Finance II - some Solutions to Chapter VII
Stochastic Calculus for Finance II - some Solutions to Chapter VII Matthias hul Last Update: June 9, 25 Exercise 7 Black-Scholes-Merton Equation for the up-and-out Call) i) We have ii) We first compute
More informationBernardo D Auria Stochastic Processes /12. Notes. March 29 th, 2012
1 Stochastic Calculus Notes March 9 th, 1 In 19, Bachelier proposed for the Paris stock exchange a model for the fluctuations affecting the price X(t) of an asset that was given by the Brownian motion.
More informationIEOR 4701: Stochastic Models in Financial Engineering. Summer 2007, Professor Whitt. SOLUTIONS to Homework Assignment 9: Brownian motion
IEOR 471: Stochastic Models in Financial Engineering Summer 27, Professor Whitt SOLUTIONS to Homework Assignment 9: Brownian motion In Ross, read Sections 1.1-1.3 and 1.6. (The total required reading there
More informationA D VA N C E D P R O B A B I L - I T Y
A N D R E W T U L L O C H A D VA N C E D P R O B A B I L - I T Y T R I N I T Y C O L L E G E T H E U N I V E R S I T Y O F C A M B R I D G E Contents 1 Conditional Expectation 5 1.1 Discrete Case 6 1.2
More informationVerona Course April Lecture 1. Review of probability
Verona Course April 215. Lecture 1. Review of probability Viorel Barbu Al.I. Cuza University of Iaşi and the Romanian Academy A probability space is a triple (Ω, F, P) where Ω is an abstract set, F is
More informationMA8109 Stochastic Processes in Systems Theory Autumn 2013
Norwegian University of Science and Technology Department of Mathematical Sciences MA819 Stochastic Processes in Systems Theory Autumn 213 1 MA819 Exam 23, problem 3b This is a linear equation of the form
More information(b) What is the variance of the time until the second customer arrives, starting empty, assuming that we measure time in minutes?
IEOR 3106: Introduction to Operations Research: Stochastic Models Fall 2006, Professor Whitt SOLUTIONS to Final Exam Chapters 4-7 and 10 in Ross, Tuesday, December 19, 4:10pm-7:00pm Open Book: but only
More informationLecture 17 Brownian motion as a Markov process
Lecture 17: Brownian motion as a Markov process 1 of 14 Course: Theory of Probability II Term: Spring 2015 Instructor: Gordan Zitkovic Lecture 17 Brownian motion as a Markov process Brownian motion is
More informationUniversal examples. Chapter The Bernoulli process
Chapter 1 Universal examples 1.1 The Bernoulli process First description: Bernoulli random variables Y i for i = 1, 2, 3,... independent with P [Y i = 1] = p and P [Y i = ] = 1 p. Second description: Binomial
More informationPROBABILITY: LIMIT THEOREMS II, SPRING HOMEWORK PROBLEMS
PROBABILITY: LIMIT THEOREMS II, SPRING 218. HOMEWORK PROBLEMS PROF. YURI BAKHTIN Instructions. You are allowed to work on solutions in groups, but you are required to write up solutions on your own. Please
More information1 IEOR 6712: Notes on Brownian Motion I
Copyright c 005 by Karl Sigman IEOR 67: Notes on Brownian Motion I We present an introduction to Brownian motion, an important continuous-time stochastic process that serves as a continuous-time analog
More information6. Brownian Motion. Q(A) = P [ ω : x(, ω) A )
6. Brownian Motion. stochastic process can be thought of in one of many equivalent ways. We can begin with an underlying probability space (Ω, Σ, P) and a real valued stochastic process can be defined
More informationBrownian Motion and Stochastic Calculus
ETHZ, Spring 17 D-MATH Prof Dr Martin Larsson Coordinator A Sepúlveda Brownian Motion and Stochastic Calculus Exercise sheet 6 Please hand in your solutions during exercise class or in your assistant s
More informationMATH4210 Financial Mathematics ( ) Tutorial 7
MATH40 Financial Mathematics (05-06) Tutorial 7 Review of some basic Probability: The triple (Ω, F, P) is called a probability space, where Ω denotes the sample space and F is the set of event (σ algebra
More informationBasic Definitions: Indexed Collections and Random Functions
Chapter 1 Basic Definitions: Indexed Collections and Random Functions Section 1.1 introduces stochastic processes as indexed collections of random variables. Section 1.2 builds the necessary machinery
More information1.1 Review of Probability Theory
1.1 Review of Probability Theory Angela Peace Biomathemtics II MATH 5355 Spring 2017 Lecture notes follow: Allen, Linda JS. An introduction to stochastic processes with applications to biology. CRC Press,
More informationStochastic Differential Equations
Chapter 5 Stochastic Differential Equations We would like to introduce stochastic ODE s without going first through the machinery of stochastic integrals. 5.1 Itô Integrals and Itô Differential Equations
More information1 Solution to Problem 2.1
Solution to Problem 2. I incorrectly worked this exercise instead of 2.2, so I decided to include the solution anyway. a) We have X Y /3, which is a - function. It maps the interval, ) where X lives) onto
More informationADVANCED PROBABILITY: SOLUTIONS TO SHEET 1
ADVANCED PROBABILITY: SOLUTIONS TO SHEET 1 Last compiled: November 6, 213 1. Conditional expectation Exercise 1.1. To start with, note that P(X Y = P( c R : X > c, Y c or X c, Y > c = P( c Q : X > c, Y
More informationLecture 4: Ito s Stochastic Calculus and SDE. Seung Yeal Ha Dept of Mathematical Sciences Seoul National University
Lecture 4: Ito s Stochastic Calculus and SDE Seung Yeal Ha Dept of Mathematical Sciences Seoul National University 1 Preliminaries What is Calculus? Integral, Differentiation. Differentiation 2 Integral
More informationSTAT331 Lebesgue-Stieltjes Integrals, Martingales, Counting Processes
STAT331 Lebesgue-Stieltjes Integrals, Martingales, Counting Processes This section introduces Lebesgue-Stieltjes integrals, and defines two important stochastic processes: a martingale process and a counting
More informationBrownian Motion. 1 Definition Brownian Motion Wiener measure... 3
Brownian Motion Contents 1 Definition 2 1.1 Brownian Motion................................. 2 1.2 Wiener measure.................................. 3 2 Construction 4 2.1 Gaussian process.................................
More information{σ x >t}p x. (σ x >t)=e at.
3.11. EXERCISES 121 3.11 Exercises Exercise 3.1 Consider the Ornstein Uhlenbeck process in example 3.1.7(B). Show that the defined process is a Markov process which converges in distribution to an N(0,σ
More informationBrownian motion. Samy Tindel. Purdue University. Probability Theory 2 - MA 539
Brownian motion Samy Tindel Purdue University Probability Theory 2 - MA 539 Mostly taken from Brownian Motion and Stochastic Calculus by I. Karatzas and S. Shreve Samy T. Brownian motion Probability Theory
More informationLecture 12: Diffusion Processes and Stochastic Differential Equations
Lecture 12: Diffusion Processes and Stochastic Differential Equations 1. Diffusion Processes 1.1 Definition of a diffusion process 1.2 Examples 2. Stochastic Differential Equations SDE) 2.1 Stochastic
More informationLecture 9. d N(0, 1). Now we fix n and think of a SRW on [0,1]. We take the k th step at time k n. and our increments are ± 1
Random Walks and Brownian Motion Tel Aviv University Spring 011 Lecture date: May 0, 011 Lecture 9 Instructor: Ron Peled Scribe: Jonathan Hermon In today s lecture we present the Brownian motion (BM).
More informationTheory and Applications of Stochastic Systems Lecture Exponential Martingale for Random Walk
Instructor: Victor F. Araman December 4, 2003 Theory and Applications of Stochastic Systems Lecture 0 B60.432.0 Exponential Martingale for Random Walk Let (S n : n 0) be a random walk with i.i.d. increments
More informationBrownian Motion. Chapter Definition of Brownian motion
Chapter 5 Brownian Motion Brownian motion originated as a model proposed by Robert Brown in 1828 for the phenomenon of continual swarming motion of pollen grains suspended in water. In 1900, Bachelier
More informationB8.3 Mathematical Models for Financial Derivatives. Hilary Term Solution Sheet 2
B8.3 Mathematical Models for Financial Derivatives Hilary Term 18 Solution Sheet In the following W t ) t denotes a standard Brownian motion and t > denotes time. A partition π of the interval, t is a
More informationPROBABILITY: LIMIT THEOREMS II, SPRING HOMEWORK PROBLEMS
PROBABILITY: LIMIT THEOREMS II, SPRING 15. HOMEWORK PROBLEMS PROF. YURI BAKHTIN Instructions. You are allowed to work on solutions in groups, but you are required to write up solutions on your own. Please
More informationFE 5204 Stochastic Differential Equations
Instructor: Jim Zhu e-mail:zhu@wmich.edu http://homepages.wmich.edu/ zhu/ January 20, 2009 Preliminaries for dealing with continuous random processes. Brownian motions. Our main reference for this lecture
More informationExercises in stochastic analysis
Exercises in stochastic analysis Franco Flandoli, Mario Maurelli, Dario Trevisan The exercises with a P are those which have been done totally or partially) in the previous lectures; the exercises with
More informationp 1 ( Y p dp) 1/p ( X p dp) 1 1 p
Doob s inequality Let X(t) be a right continuous submartingale with respect to F(t), t 1 P(sup s t X(s) λ) 1 λ {sup s t X(s) λ} X + (t)dp 2 For 1 < p
More informationLECTURE 2: LOCAL TIME FOR BROWNIAN MOTION
LECTURE 2: LOCAL TIME FOR BROWNIAN MOTION We will define local time for one-dimensional Brownian motion, and deduce some of its properties. We will then use the generalized Ray-Knight theorem proved in
More informationGaussian, Markov and stationary processes
Gaussian, Markov and stationary processes Gonzalo Mateos Dept. of ECE and Goergen Institute for Data Science University of Rochester gmateosb@ece.rochester.edu http://www.ece.rochester.edu/~gmateosb/ November
More informationMATH 6605: SUMMARY LECTURE NOTES
MATH 6605: SUMMARY LECTURE NOTES These notes summarize the lectures on weak convergence of stochastic processes. If you see any typos, please let me know. 1. Construction of Stochastic rocesses A stochastic
More informationStochastic Differential Equations
CHAPTER 1 Stochastic Differential Equations Consider a stochastic process X t satisfying dx t = bt, X t,w t dt + σt, X t,w t dw t. 1.1 Question. 1 Can we obtain the existence and uniqueness theorem for
More informationBernardo D Auria Stochastic Processes /10. Notes. Abril 13 th, 2010
1 Stochastic Calculus Notes Abril 13 th, 1 As we have seen in previous lessons, the stochastic integral with respect to the Brownian motion shows a behavior different from the classical Riemann-Stieltjes
More information4th Preparation Sheet - Solutions
Prof. Dr. Rainer Dahlhaus Probability Theory Summer term 017 4th Preparation Sheet - Solutions Remark: Throughout the exercise sheet we use the two equivalent definitions of separability of a metric space
More informationStochastic Integration and Stochastic Differential Equations: a gentle introduction
Stochastic Integration and Stochastic Differential Equations: a gentle introduction Oleg Makhnin New Mexico Tech Dept. of Mathematics October 26, 27 Intro: why Stochastic? Brownian Motion/ Wiener process
More informationStochastic Differential Equations.
Chapter 3 Stochastic Differential Equations. 3.1 Existence and Uniqueness. One of the ways of constructing a Diffusion process is to solve the stochastic differential equation dx(t) = σ(t, x(t)) dβ(t)
More informationExam Stochastic Processes 2WB08 - March 10, 2009,
Exam Stochastic Processes WB8 - March, 9, 4.-7. The number of points that can be obtained per exercise is mentioned between square brackets. The maximum number of points is 4. Good luck!!. (a) [ pts.]
More informationDefinition: Lévy Process. Lectures on Lévy Processes and Stochastic Calculus, Braunschweig, Lecture 2: Lévy Processes. Theorem
Definition: Lévy Process Lectures on Lévy Processes and Stochastic Calculus, Braunschweig, Lecture 2: Lévy Processes David Applebaum Probability and Statistics Department, University of Sheffield, UK July
More informationMath 635: An Introduction to Brownian Motion and Stochastic Calculus
First Prev Next Go To Go Back Full Screen Close Quit 1 Math 635: An Introduction to Brownian Motion and Stochastic Calculus 1. Introduction and review 2. Notions of convergence and results from measure
More informationA Class of Fractional Stochastic Differential Equations
Vietnam Journal of Mathematics 36:38) 71 79 Vietnam Journal of MATHEMATICS VAST 8 A Class of Fractional Stochastic Differential Equations Nguyen Tien Dung Department of Mathematics, Vietnam National University,
More informationLecture 4: Introduction to stochastic processes and stochastic calculus
Lecture 4: Introduction to stochastic processes and stochastic calculus Cédric Archambeau Centre for Computational Statistics and Machine Learning Department of Computer Science University College London
More informationPreliminary Exam: Probability 9:00am 2:00pm, Friday, January 6, 2012
Preliminary Exam: Probability 9:00am 2:00pm, Friday, January 6, 202 The exam lasts from 9:00am until 2:00pm, with a walking break every hour. Your goal on this exam should be to demonstrate mastery of
More informationBrownian Motion. An Undergraduate Introduction to Financial Mathematics. J. Robert Buchanan. J. Robert Buchanan Brownian Motion
Brownian Motion An Undergraduate Introduction to Financial Mathematics J. Robert Buchanan 2010 Background We have already seen that the limiting behavior of a discrete random walk yields a derivation of
More information1. Stochastic Process
HETERGENEITY IN QUANTITATIVE MACROECONOMICS @ TSE OCTOBER 17, 216 STOCHASTIC CALCULUS BASICS SANG YOON (TIM) LEE Very simple notes (need to add references). It is NOT meant to be a substitute for a real
More informationLecture 19 : Brownian motion: Path properties I
Lecture 19 : Brownian motion: Path properties I MATH275B - Winter 2012 Lecturer: Sebastien Roch References: [Dur10, Section 8.1], [Lig10, Section 1.5, 1.6], [MP10, Section 1.1, 1.2]. 1 Invariance We begin
More informationP (A G) dp G P (A G)
First homework assignment. Due at 12:15 on 22 September 2016. Homework 1. We roll two dices. X is the result of one of them and Z the sum of the results. Find E [X Z. Homework 2. Let X be a r.v.. Assume
More informationMFE6516 Stochastic Calculus for Finance
MFE6516 Stochastic Calculus for Finance William C. H. Leon Nanyang Business School December 11, 2017 1 / 51 William C. H. Leon MFE6516 Stochastic Calculus for Finance 1 Symmetric Random Walks Scaled Symmetric
More informationExercises Measure Theoretic Probability
Exercises Measure Theoretic Probability 2002-2003 Week 1 1. Prove the folloing statements. (a) The intersection of an arbitrary family of d-systems is again a d- system. (b) The intersection of an arbitrary
More informationStochastic differential equation models in biology Susanne Ditlevsen
Stochastic differential equation models in biology Susanne Ditlevsen Introduction This chapter is concerned with continuous time processes, which are often modeled as a system of ordinary differential
More informationOn Reflecting Brownian Motion with Drift
Proc. Symp. Stoch. Syst. Osaka, 25), ISCIE Kyoto, 26, 1-5) On Reflecting Brownian Motion with Drift Goran Peskir This version: 12 June 26 First version: 1 September 25 Research Report No. 3, 25, Probability
More informationStochastic Calculus February 11, / 33
Martingale Transform M n martingale with respect to F n, n =, 1, 2,... σ n F n (σ M) n = n 1 i= σ i(m i+1 M i ) is a Martingale E[(σ M) n F n 1 ] n 1 = E[ σ i (M i+1 M i ) F n 1 ] i= n 2 = σ i (M i+1 M
More informationNested Uncertain Differential Equations and Its Application to Multi-factor Term Structure Model
Nested Uncertain Differential Equations and Its Application to Multi-factor Term Structure Model Xiaowei Chen International Business School, Nankai University, Tianjin 371, China School of Finance, Nankai
More information(B(t i+1 ) B(t i )) 2
ltcc5.tex Week 5 29 October 213 Ch. V. ITÔ (STOCHASTIC) CALCULUS. WEAK CONVERGENCE. 1. Quadratic Variation. A partition π n of [, t] is a finite set of points t ni such that = t n < t n1
More informationSurvival Analysis: Counting Process and Martingale. Lu Tian and Richard Olshen Stanford University
Survival Analysis: Counting Process and Martingale Lu Tian and Richard Olshen Stanford University 1 Lebesgue-Stieltjes Integrals G( ) is a right-continuous step function having jumps at x 1, x 2,.. b f(x)dg(x)
More informationECE534, Spring 2018: Solutions for Problem Set #4 Due Friday April 6, 2018
ECE534, Spring 2018: s for Problem Set #4 Due Friday April 6, 2018 1. MMSE Estimation, Data Processing and Innovations The random variables X, Y, Z on a common probability space (Ω, F, P ) are said to
More informationA numerical method for solving uncertain differential equations
Journal of Intelligent & Fuzzy Systems 25 (213 825 832 DOI:1.3233/IFS-12688 IOS Press 825 A numerical method for solving uncertain differential equations Kai Yao a and Xiaowei Chen b, a Department of Mathematical
More informationMS&E 321 Spring Stochastic Systems June 1, 2013 Prof. Peter W. Glynn Page 1 of 7
MS&E 321 Spring 12-13 Stochastic Systems June 1, 213 Prof. Peter W. Glynn Page 1 of 7 Section 9: Renewal Theory Contents 9.1 Renewal Equations..................................... 1 9.2 Solving the Renewal
More informationTMS165/MSA350 Stochastic Calculus, Lecture on Applications
TMS165/MSA35 Stochastic Calculus, Lecture on Applications In this lecture we demonstrate how statistical methods such as the maximum likelihood method likelihood ratio estimation can be applied to the
More informationBrownian Motion and Conditional Probability
Math 561: Theory of Probability (Spring 2018) Week 10 Brownian Motion and Conditional Probability 10.1 Standard Brownian Motion (SBM) Brownian motion is a stochastic process with both practical and theoretical
More informationLet (Ω, F) be a measureable space. A filtration in discrete time is a sequence of. F s F t
2.2 Filtrations Let (Ω, F) be a measureable space. A filtration in discrete time is a sequence of σ algebras {F t } such that F t F and F t F t+1 for all t = 0, 1,.... In continuous time, the second condition
More informationMath 735: Stochastic Analysis
First Prev Next Go To Go Back Full Screen Close Quit 1 Math 735: Stochastic Analysis 1. Introduction and review 2. Notions of convergence 3. Continuous time stochastic processes 4. Information and conditional
More informationRandom Process Lecture 1. Fundamentals of Probability
Random Process Lecture 1. Fundamentals of Probability Husheng Li Min Kao Department of Electrical Engineering and Computer Science University of Tennessee, Knoxville Spring, 2016 1/43 Outline 2/43 1 Syllabus
More informationSTAT 331. Martingale Central Limit Theorem and Related Results
STAT 331 Martingale Central Limit Theorem and Related Results In this unit we discuss a version of the martingale central limit theorem, which states that under certain conditions, a sum of orthogonal
More informationLecture 11. Multivariate Normal theory
10. Lecture 11. Multivariate Normal theory Lecture 11. Multivariate Normal theory 1 (1 1) 11. Multivariate Normal theory 11.1. Properties of means and covariances of vectors Properties of means and covariances
More informationP i [B k ] = lim. n=1 p(n) ii <. n=1. V i :=
2.7. Recurrence and transience Consider a Markov chain {X n : n N 0 } on state space E with transition matrix P. Definition 2.7.1. A state i E is called recurrent if P i [X n = i for infinitely many n]
More informationMSH7 - APPLIED PROBABILITY AND STOCHASTIC CALCULUS. Contents
MSH7 - APPLIED PROBABILITY AND STOCHASTIC CALCULUS ANDREW TULLOCH Contents 1. Lecture 1 - Tuesday 1 March 2 2. Lecture 2 - Thursday 3 March 2 2.1. Concepts of convergence 2 3. Lecture 3 - Tuesday 8 March
More information1 Brownian Local Time
1 Brownian Local Time We first begin by defining the space and variables for Brownian local time. Let W t be a standard 1-D Wiener process. We know that for the set, {t : W t = } P (µ{t : W t = } = ) =
More information1 Stat 605. Homework I. Due Feb. 1, 2011
The first part is homework which you need to turn in. The second part is exercises that will not be graded, but you need to turn it in together with the take-home final exam. 1 Stat 605. Homework I. Due
More informationWeak solutions of mean-field stochastic differential equations
Weak solutions of mean-field stochastic differential equations Juan Li School of Mathematics and Statistics, Shandong University (Weihai), Weihai 26429, China. Email: juanli@sdu.edu.cn Based on joint works
More informationItô s formula. Samy Tindel. Purdue University. Probability Theory 2 - MA 539
Itô s formula Samy Tindel Purdue University Probability Theory 2 - MA 539 Mostly taken from Brownian Motion and Stochastic Calculus by I. Karatzas and S. Shreve Samy T. Itô s formula Probability Theory
More informationMarkov Chains. X(t) is a Markov Process if, for arbitrary times t 1 < t 2 <... < t k < t k+1. If X(t) is discrete-valued. If X(t) is continuous-valued
Markov Chains X(t) is a Markov Process if, for arbitrary times t 1 < t 2
More information2. Metric Spaces. 2.1 Definitions etc.
2. Metric Spaces 2.1 Definitions etc. The procedure in Section for regarding R as a topological space may be generalized to many other sets in which there is some kind of distance (formally, sets with
More informationA Short Introduction to Diffusion Processes and Ito Calculus
A Short Introduction to Diffusion Processes and Ito Calculus Cédric Archambeau University College, London Center for Computational Statistics and Machine Learning c.archambeau@cs.ucl.ac.uk January 24,
More informationWiener Measure and Brownian Motion
Chapter 16 Wiener Measure and Brownian Motion Diffusion of particles is a product of their apparently random motion. The density u(t, x) of diffusing particles satisfies the diffusion equation (16.1) u
More informationMan Kyu Im*, Un Cig Ji **, and Jae Hee Kim ***
JOURNAL OF THE CHUNGCHEONG MATHEMATICAL SOCIETY Volume 19, No. 4, December 26 GIRSANOV THEOREM FOR GAUSSIAN PROCESS WITH INDEPENDENT INCREMENTS Man Kyu Im*, Un Cig Ji **, and Jae Hee Kim *** Abstract.
More informationStat 366 A1 (Fall 2006) Midterm Solutions (October 23) page 1
Stat 366 A1 Fall 6) Midterm Solutions October 3) page 1 1. The opening prices per share Y 1 and Y measured in dollars) of two similar stocks are independent random variables, each with a density function
More informationMarch 1, Florida State University. Concentration Inequalities: Martingale. Approach and Entropy Method. Lizhe Sun and Boning Yang.
Florida State University March 1, 2018 Framework 1. (Lizhe) Basic inequalities Chernoff bounding Review for STA 6448 2. (Lizhe) Discrete-time martingales inequalities via martingale approach 3. (Boning)
More informationNon-Essential Uses of Probability in Analysis Part IV Efficient Markovian Couplings. Krzysztof Burdzy University of Washington
Non-Essential Uses of Probability in Analysis Part IV Efficient Markovian Couplings Krzysztof Burdzy University of Washington 1 Review See B and Kendall (2000) for more details. See also the unpublished
More informationLecture two. January 17, 2019
Lecture two January 17, 2019 We will learn how to solve rst-order linear equations in this lecture. Example 1. 1) Find all solutions satisfy the equation u x (x, y) = 0. 2) Find the solution if we know
More informationStochastic Calculus (Lecture #3)
Stochastic Calculus (Lecture #3) Siegfried Hörmann Université libre de Bruxelles (ULB) Spring 2014 Outline of the course 1. Stochastic processes in continuous time. 2. Brownian motion. 3. Itô integral:
More informationExpectation, variance and moments
Expectation, variance and moments John Appleby Contents Expectation and variance Examples 3 Moments and the moment generating function 4 4 Examples of moment generating functions 5 5 Concluding remarks
More informationOn the martingales obtained by an extension due to Saisho, Tanemura and Yor of Pitman s theorem
On the martingales obtained by an extension due to Saisho, Tanemura and Yor of Pitman s theorem Koichiro TAKAOKA Dept of Applied Physics, Tokyo Institute of Technology Abstract M Yor constructed a family
More informationFormulas for probability theory and linear models SF2941
Formulas for probability theory and linear models SF2941 These pages + Appendix 2 of Gut) are permitted as assistance at the exam. 11 maj 2008 Selected formulae of probability Bivariate probability Transforms
More information