MATH 5763 - Stochastic Processes, Section 001 - Spring 2011
MWF 1:30-2:20 p.m., 117 PHSC

Instructor: Nikola Petrov, 802 PHSC, (405)325-4316, npetrov AT math.ou.edu

Office Hours: Monday 2:30-3:30 p.m., Wednesday 3:30-4:30 p.m., or by appointment.

Prerequisite: Basic calculus-based probability theory at the level of MATH 4733 (including axioms of probability, random variables, expectation, probability distributions, independence, conditional probability). The class will also require knowledge of elementary analysis (including sequences, series, continuity), linear algebra (including linear spaces, eigenvalues, eigenvectors), and ordinary differential equations (at the level of MATH 3113 or MATH 3413).

Course description: The theory of stochastic processes studies systems that evolve randomly in time; it can be regarded as the "dynamical" part of probability theory. It has many important practical applications, as well as in other branches in mathematics such as partial differential equations. This course is a graduate-level introduction to stochastic processes, and should be of interest to students of mathematics, statistics, physics, engineering, and economics. The emphasis will be on the fundamental concepts, but we will avoid using the theory of Lebesgue measure and integration in any essential way. Many examples of stochastic phenomena in applications and some modeling issues will also be discussed in class and given as homework problems.

Text: Mario Lefebvre, Applied Stochastic Processes, 1st edition, Springer, 2006, ISBN-10: 0387341714, ISBN-13: 978-0387341712, which is freely available online to OU students through the OU Library.
(At the end of the course we may also use parts of the book Hui-Hsiung Kuo, Introduction to Stochastic Integration, 1st edition, Springer, 2007, also freely available online to OU students through the OU Library.)

Main topics (a tentative list):

• a brief review of probability theory;
• discrete Markov chains: Chapman-Kolmogorov equations, persistence and transience, generating functions, stationary distributions, reducibility, limit theorems, ergodicity;
• continuous Markov processes: Poisson process, birth-death and branching processes, embedding of a discrete-time Markov chain in a continuous-time Markov processes;
• conditional expectation, martingales;
• stationary processes (autocorrelation function, spectral representation);
• renewal processes, queues;
• diffusion processes, Wiener processes (Brownian motion);
• introduction to stochastic differential equations, Ito calculus;
• Fokker-Planck equation, Black-Scholes option-pricing formula, Ornstein-Uhlenbeck process.

Homework:

• Homework 1, due Friday, January 28.
• Homework 2, due Friday, February 4.
• Homework 3, due Friday, February 18.
• Homework 4, due Friday, February 25.
• Homework 5, due Friday, March 4.
• Homework 6, due Friday, March 11.
• Homework 7, due Friday, April 1.
• Homework 8, due Friday, April 15.
• Homework 9, due Friday, April 22.
• Homework 10, due Friday, April 29.
• Homework 11, due Friday, May 6.

Content of the lectures:

• Lecture 1 (Wed, Jan 19): Review of probability: sample space, examples, events, σ-algebra (σ-field), events, elementary events, operations with events (complement, union, intersection), disjoint events, De Morgan's laws, probability (probability measure), probability space (pages 1-3 of Sec. 1.1).
• Lecture 2 (Fri, Jan 21): Review of probability (cont.): elementary properties of probability measures (including the inclusion-exclusion formula), conditional probability, partitions of the sample space, law of total probability, Bayes' formula, independent events, independent family of events (pages 3, 5-8 of Sec. 1.1).
  Random variables: random variables (RVs), (cumulative) distribution function (c.d.f.) of a RV (pages 8-9 of Sec. 1.2).
• Lecture 3 (Mon, Jan 24): Random variables (cont.): properties of c.d.f.s, important discrete RVs (Bernoulli, binomial, Poisson, geometric), important continuous RVs (uniform, exponential, normal/Gaussian, standard normal); conditional c.d.f. of a RV conditioned on an event, conditional p.m.f./p.d.f. of a RV conditioned on an event; expectation of a discrete RV expressed as a sum over the elements of the sample space S and as a sum over the values in the domain S_X of X, example - expectation of an indicator function of an event (pages 9-16 of Sec. 1.2).
• Lecture 4 (Wed, Jan 26): Random variables (cont.): expectation of a function of a RV, nth moment of a RV, variance of a RV, moment-generating function (m.g.f.) of a RV, computing the moments from the m.g.f., characteristic function of a RV (pages 16-20 of Sec. 1.2).
  Random vectors: definition, (joint) c.d.f. of a random vector, marginal p.m.f.s and p.d.f.s of a random vector, independence of the components of a random vector; conditional distribution functions, conditional expectation, tower rule E[E[X|Y]]=E[X] (pages 21-27 of Sec. 1.3).
• Lecture 5 (Fri, Jan 28): Random processes: definition of a stochastic process (random process), classification of random processes: discrete-time or continuous-time, discrete-state space or continous-state space (pages 47-48 of Sec 2.1).
  Markov chains: Markov property, examples: 1-dimensional and 2-dimensional random walks, Ehrenfests' urn model (pages 73-75 of Sec. 3.1).
• Lecture 6 (Mon, Jan 31): Discrete-time Markov chains: time-homogeneous discrete-time discrete-state space Markov chains, one-step and n-step transition probabilities, one-step and n-step transition matrices P⁽ⁿ⁾, stochastic and doubly-stochastic matrices, Chapman-Kolmogorov equations, matrix form of the Chapman-Kolmogorov equations, an example (pages 77-80 of Sec. 3.2).
• Lecture 7 (Wed, Feb 2): Cancelled due to weather.
• Lecture 8 (Fri, Feb 4): Cancelled due to weather.
• Lecture 9 (Mon, Feb 7): Discrete-time Markov chains (cont.): probability ρ_ij⁽ⁿ⁾ of visiting state j for the first time in n steps starting from state i, probability ρ_ii⁽ⁿ⁾ of first return to state i in n steps, representation of p_ij⁽ⁿ⁾ as a sum over k from 1 to n of ρ_ii^(k)p_ii^(n-k), examples of computing ρ_ij⁽ⁿ⁾ for some simple Markov chains, initial distribution a=(a₀,a₁,a₂,...) of a Markov chain, a_i=P(X₀=i), distribution a⁽ⁿ⁾=(a₀⁽ⁿ⁾,a₁⁽ⁿ⁾,a₂⁽ⁿ⁾,...) at time n, formula for evolution of the probability distribution: a⁽ⁿ⁾=aPⁿ, examples: simple 1-dim random walk on Z, simple 1-dim random walk on Z₊ with reflecting and absorbing boundary condition at 0, a Markov chain coming from sums of i.i.d. random variables (pages 80-85 of Sec. 3.2).
• Lecture 10 (Wed, Feb 9): Cancelled due to weather.
• Lecture 11 (Fri, Feb 11): Properties of Markov chains: accessibility of state j from state i, i→j, communicating states i↔j, properties of the relation ↔ (reflexivity, symmetry, transitivity), ↔ as an equivalence relation, equivalence classes with respect of ↔, closed sets of the state space, irreducible MCs, irreducibility criteria, examples, recurrent and transient states, probability f_ij of eventual visit of j starting from i, probability f_ii of eventual return to state i, expressing f_ij as a sum of the first visit probabilities ρ_ij⁽ⁿ⁾, Decomposition Theorem (pages 85-87 of Sec. 3.2).
• Lecture 12 (Mon, Feb 14): Properties of Markov chains (cont.): an example of identifying closed irreducible sets of recurrent states and of sets of transient states, and structure of the stochastic matrix; a necessary and sufficient criterion of recurrence of a state in terms of the expected value of the number of returns to this state, recurrence is a class property (pages 87-89 of Sec. 3.2).
• Lecture 13 (Wed, Feb 16): Properties of Markov chains (cont.): average number μ_i of transitions for first return to state i, positive recurrent and null-recurrent states, criterion for null-recurrence, type of recurrence is a class property, recurrent states of a finite MC are positive recurrent (pages 87-90 of Sec. 3.2). periodic and aperiodic states, remarks about periodicity, examples; limiting probabilities π_i, limiting probability distribution π=(π₀,π₁,π₁,...), ergodic states, Ergodic Theorem (giving conditions for existence and uniqueness of a limiting probability distribution, relation between π_i and μ_i, and an algorithm for computing π), (pages 87-95 of Sec. 3.2).
• Lecture 14 (Fri, Feb 18): Properties of Markov chains (cont.): examples showing the importance of the conditions in the Ergodic Theorem, examples of computing π, proposition giving π for the case of an irreducible and aperiodic MC over a finite set with a doubly stochastic transition matrix, an example of computing π for a MC with infinite state space (a random walk with a partially absorbing boundary) (pages 96-98 of Sec. 3.2).
• Lecture 15 (Mon, Feb 21): Absorption problems: definition of probabilities r_i⁽ⁿ⁾(C) and r_i(C) of absorption by the recurrent class C after exactly n steps and eventual absorption if strating at state i; theorem giving r_i(C) in terms of the (p_ij), example - the gambler's ruin problem; martingales (pages 100-104 of Sec. 3.2)
• Lecture 16 (Wed, Feb 23): Continuous-time discrete-state space MCs: definition, transition functions p_ij(t), stationary (time-homogeneous) MCs, irreducibility, analogue of the condition of being a stochastic matrix for p_ij(t) (pages 121-122 of Sec. 3.3).
• Lecture 17 (Fri, Feb 25): Continuous-time discrete-state space MCs (cont.): embedded chain, Chapman-Kolmogorov equations, evolution of the occupation probabilities p_j(t)=P(X_t=j) expressed in terms of the initial occupation probabilities p_i(0) and the transition probabilities p_ij(t), memorylessness, memorylessness properties of the exponential random variables; definition of a Poisson process (counting process) (pages 109, 110, and 123 of Sec. 3.3, pages 231-232 of Sec. 5.1).
• Lecture 18 (Mon, Feb 28): Poisson process: derivation of the distribution of N(t) for a Poisson process N by induction and by the method of generating functions.
• Lecture 19 (Wed, Mar 2): Poisson process (cont.): properties of the Γ(α,λ) distribution: moment-generating function, proof that the sum of j independent Exp(λ) RVs is a Γ(j,λ) RV, c.d.f. of a Γ(j,λ) RV (pages 115-119 of Sec. 3.3, pages 237-238 of Sec. 5.1).
• Lecture 20 (Fri, Mar 4): Continuous-time Markov chains (cont.): instantaneous transition rates (infinitesimal parameters) ν_ij, generating matrix (generator) G, properties of G (sum of the elements ν_ij in each row of G is zero), stochastic semigroup P_t (pages 124-126 of Sec. 3.3).
• Lecture 21 (Mon, Mar 7): Continuous-time Markov chains (cont.): standard semigroups, Kolmogorov backward and forward equations, boundary coditions for Kolmogorov equations, exponent of a square matrix (possibly infinite), solution of an initial-value problem for a linear first-order system of ordinary differential equations with constant coefficients, the stochastic semigroup in terms of its generator: P_t=exp(tG), detailed solution for a continuous-time, two-state MC (pages 126-131 of Sec. 3.3).
• Lecture 22 (Wed, Mar 9): Continuous-time Markov chains (cont.): Laplace transform, linearity of the Laplace transform, Laplace transform of a derivative of a function, solving the problem of a two-state continous-time Markov process by using Laplace transform method (a sketch); stationary distribution π of a stochastic semigroup P_t, recurrent and transient states, positive recurrent and null recurrent states of a continuous-time Markov chain, irreducible Markov chains, Ergodic Theorem for continuous-time Markov process, remarks, finding stationary distributions from the generator: πG=0 (pages 138-140 of Sec. 3.3).
• Lecture 23 (Fri, Mar 11): Continuous-time Markov chains (cont.): balace equations; birth processes, computing the expectation of the time T_n for a birth process starting at X₀=1 to reach X_t=n for the first time; derivation of the infinitesimal-time evolution probabilities and the Kolmogorov differential equations of a birth process, solving the differential equations by using a generating function,

• Lecture 24 (Mon, Mar 21): Continuous-time Markov chains (cont.): computing the average M(t)=E[X(t)|X(0)=i] of a birth process by deriving and solving a differential equation for M(t); a birth-death-immigration-disaster process, detailed derivation of the short-time transition probabilities of a death-immigration process.

• Lecture 25 (Wed, Mar 23): Compound Poisson processes: definition of a compound Poisson process, mean, variance, and moment generating function of a compound Poisson process; proof that the sum of two independent compound processes coming from Poisson processes with rates λ₁ and λ₂ is a compound process coming from a Poisson process with rate λ₁+λ₂ (pages 254-258 of Sec. 5.3).
• Lecture 26 (Fri, Mar 25): Nonhomogeneous Poisson processes: definition, intensity function λ(t), mean-value function m(t) (such that m(0)=0, m'(t)=λ(t)), proof that the distribution of the increment N_s+t-N_s is Poisson with parameter m(s+t)-m(s), "homogenizing" a nonhomogeneous Poisson process (pages 250, 251, 253, 254 of Sec. 5.2).
• Lecture 27 (Mon, Mar 28): Doubly stochastic Poisson processes: conditional (or "mixed") Poisson process, proof that the conditional Poisson process has stationary, but not independent increments, best estimator of the rate of a Poisson process; doubly stochastic Poisson process ("Cox process") - only mentioned (pages 258, 259, 262 of Sec. 5.4).
  Renewal processes: definition of a renewal process, modified ("delayed") renewal process; relations between the process N_t, the times of the events T_n, and the interevent times τ_n; expression for the p.m.f. of N_t in terms of the c.d.f. of T_n (Proposition 5.6.1) (pages 267-269 of Sec. 5.6).
• Lecture 28 (Wed, Mar 30): Renewal processes (cont.): "honesty" of a renewal process, renewal function m_N(t); definition of the Riemann-Stieltjes integral, particular cases, applications to computing expected values of discrete and continuous random variables; expected value of an N-valued random variable X as a sum (over n from 1 to infinity) of probabilites of X to be greater or equal to n, expected value of a non-negative continuous random variable X as an integral of [1-F_X(x)], geometric meaning; expressing m_N(t) in as a sum of the c.d.f.s of all the T_n's (pages 267-271 of Sec. 5.6).
  

• Lecture 29 (Fri, Apr 1): Renewal processes (cont.): derivation of an integral equation for the renewal function m_N(t)=E[N_t], solving renewal-type equations by using Laplace transform, recursive formula for the c.d.f. of the event times T_n expressed through Riemann-Stieltjes integrals (pages 271, 273-276 of Sec. 5.6).
• Lecture 30 (Mon, Apr 4): Renewal processes (cont.): another derivation of the formula for the renewal function m_N(t) by performing Laplace transformation on the formula representing m_N(t) as a sum of the c.d.f.s of all the T_n's, relation between the Laplace transform of the p.d.f. of a random variable and the moment generating function of the random variable; an example of application of renewal-type problems - crossing a street.
  Queues: set-up of the problem, examples of queues (queues with baulking, multiple servers, airline check-in, FIFO, LIFO, group servise, "student discipline", "continental queueing").
• Lecture 31 (Wed, Apr 6): Queues (cont.): A/B/k/s/... classification of the queues, where A and B are deterministic (D), or have Markovian (M - with exponentially distributed interrarival/service times), Erlang (or Gamma, Γ) or general (G) distributions; trafic intensity, stability of a queue; M(λ)/M(μ)/1 queue - see problem in handout; M(λ)/G/1 queue - constructing of a discrete-time Markov chain embedded in the queueing process and derivation of the transition probability matrix of this Markov chain.
  Please read the handout on sigma-fields.
• Lecture 32 (Fri, Apr 8): General properties of stochastic processes: cumulative distribution function F(x₁,...,x_k;t₁,...,t_k), probability mass function p(x₁,...,x_k;t₁,...,t_k), and probability density function f(x₁,...,x_k;t₁,...,t_k) of order k of a stochastic process X={X_t:t∈[0,∞)}; mean m_X(t)=E[X_t], autocorrelation function R_X(t₁,t₂)=E[X_t1X_t2], autocovariance function C_X(t₁,t₂)=R_X(t₁,t₂)-m_X(t₁)m_X(t₂), variance var(X(t))=C_X(t,t), and autocorrelation coefficient ρ_X(t₁,t₂) of a stochastic process; processes with indepent increments, processes with stationary increments, strict-sense stationary (SSS, strongly stationary) processes, wide-sense stationary (WSS, weakly stationary) processes, an example of a WSS stochastic process that is not SSS (Sec. 2.1, pages 52-53 of Sec. 2.2).
• Lecture 33 (Mon, Apr 11): General properties of stochastic processes (cont.): average power E[X_t²] of a stochastic process, E[X_t²] of a WSS stochastic process does not depend on t; spectral density S_X(ω) of a WSS process, properties of S_X(ω) (pages 53-54 of Sec. 2.2).
  Gaussian and Markov processes: multinormal distribution of a random vector X=(X₁,...,X_n)∼N(m,K), vector of the means m, covariance matrix K=(cov(X_i,X_j)); characteristic function φ_X(ω)=E[exp(iωX)] of a random variable X, (joint) characteristic function φ_X(ω)=E[exp(iωX)] of a multinormal random variable X (Proposition 2.4.1); if two components of X=(X₁,...,X_n)∼N(m,K) are uncorrelated, then they are independent, Gaussian process {X_t} - a continuous-time stochastic process with (X_t1,...,X_tn) being multinormal for any n and times t₁,...t_n; if {X_t} is a Gaussian process such that its mean m_X(t) does not depend on t and its autocovariance function C_X(t₁,t₂) depends only on t₂-t₁, then the process is SSS (Proposition 2.4.2); definition of a Markov (or Markovian) processes, examples (random walk, Poisson process) (pages 58-61 of Sec. 2.4).
• Lecture 34 (Wed, Apr 13): Gaussian and Markov processes (cont.): (first-order) density function f(x;t), conditional transition density function p(x,x₀;t,t₀)=f_Xt|Xt0(x|x₀); integrals of f(x;t) and p(x,x₀;t,t₀) over x are equal to 1; expressing f(x;t) as in integral of f(x₀;t₀)p(x,x₀;t,t₀) over x₀; more on the meaning of the p.d.f. of a continuous RV: P(X∈(x,x+Δx])≈f_X(x)Δx, generalization for jointly continuous random vectors P(X∈A)≈f_X(x)vol(A) where A is a small domain in R^k containing x; application to kth order p.d.f.'s of a random process: P(X_t1∈(x₁,x₁+Δx₁],...,X_tk∈(x_k,x_k+Δx_k])≈f_{(Xt1,...,Xtk)}(x₁,...,x_k)Δx₁...Δx_k; Chapman-Kolmogorov equations for the conditional transition density function p(x,x₀;t,t₀)=f_Xt|Xt0(x|x₀); Dirac δ-function; time-homogeneous process (pages 62-64 of Sec. 2.5).
  The Wiener process: definition of a Wiener process (Brownian motion) W_t∼N(0,σ²t) and a standard Wiener process B_t∼N(0,t); p.d.f. of order k of a Wiener process; autocorrelation function R_B(t,s)=E[B_tB_s]=min(t,s) of a Wiener process (pages 175, 177, 178 of Sec. 4.1).
• Lecture 35 (Fri, Apr 15): A digression on generalized functions (distributions): test functions (infinitely smooth compactly supported functions), Dirac δ-function δ_a, generalized derivatives, derivatives of δ_a; example: generalized derivative of the Heaviside (unit step) function: H_a'=δ_a, representing δ(x) as a limit of (1/ε)χ_[0,ε] as ε→0+, representing δ_a as a limit of (2πε)^-1/2exp{-x²/(2ε)} as ε→0+.
  Gaussian and Markov processes (remarks): derivation of the Chapman-Kolmogorov equations, consistency conditions among the conditional density functions.
  The Wiener process (cont.): characteristic functions, characteristic function of W_t, the Wiener process as a limit of simple random walk; historical remarks (Robert Brown, Albert Einstein, Marian Smoluchowski, Norbert Wiener, Andrey Kolmogorov).
• Lecture 36 (Mon, Apr 18): The Wiener process (cont.): m.g.f. and moments of B_t: E[B_t²ⁿ⁺¹]=0, E[B_t²]=t, E[B_t⁴]=3t², E[B_t⁶]=15t³; short-time behavior: E[ΔB_t]=0, E[(ΔB_t)²]=Δt; nondifferentiability of B_t: E[ΔB_t/Δt]=0, E[(ΔB_t/Δt)²]=1/Δt→∞ as t→0; (Gaussian) white noise ξ_t=dB_t/dt, making sense of ξ_t=dB_t/dt by treating it as a functional (on test functions φ) taking value ξ(φ) in the space of random variables, moments of ξ(φ).
• Lecture 37 (Wed, Apr 20): The Wiener process (cont.): more on the meaning of ξ(φ) as a measurement "smeared by φ", writing the facts about moments of ξ(φ) as consequences of E[ξ_t]=0 and E[ξ_tξ_s]=δ(t-s).
  Stochastic differential equations (SDEs) and Ito integrals: discussion of the concept of a stochastic differential equation and the meaning of its solution, Fokker-Planck equation for the conditional transition density function p(x,x₀;t,t₀), an example (the Fokker-Planck equation and its solution for the Wiener process B_t), discretizing the SDE, Ito's way of defining the integral as a limit of left Riemann sums, main reason for using left Riemann sums - independence of B_t and the increment ΔB_t:=B_t+Δt-B_t for any positive Δt (due to the independence of the increments of the Wiener process).
• Lecture 38 (Fri, Apr 22): SDEs and Ito integrals (cont.): mean-square (L²-) convergence of functions, examples, more on the meaning of the definition of Ito's integral.
• Lecture 39 (Mon, Apr 25): SDEs and Ito integrals (cont.): derivation of the formula ∫ B_tdt=(B_t²-t)/2.
• Lecture 40 (Wed, Apr 27): Ito formula: more on the meaning of stochastic integrals, sketch of the derivation of the Ito formula, examples.
• Lecture 41 (Fri, Apr 29): Conditional expectation and martingales: probability spaces (Ω,F,P), random variables, σ-algebra σ(X) generated by a random variable X, σ-algebra σ(F₁,...,F_n) generated by a by a collection of σ-algebras, σ-algebra σ(X₁,...,X_n) generated by a family of random variables, distribution (c.d.f.) F_X(x)=P(X≤x) of a random variable, integrable (L¹-) random variables, expectation E[X] of a random variable X, conditional expectation E[X|A] of a random variable X conditioned on an event A, conditional expectation E[X|F] of a random variable X conditioned on a σ-algebra F, conditional expectation E[X|Y] of a random variable X conditioned on another random variable Y.
• Lecture 42 (Mon, May 2): Conditional expectation and martingales (cont.): filtration F₁,F₂,... of σ-algebras, discussion of the meaning of F_n in the context of "coin tossing" (F_n is our knowledge at time n), filtration F_n=σ(Y₁,...,Y_n) of σ-algebras generated by a sequence Y₁,Y₂,... of random variables ("coin tosses"), a sequence {X_n} of random variables adapted to a filtration {F_n} of σ-algebras, martingale with respect to a filtration of σ-algebras, example - if X_n is a symmetric one-dimensional random walk, then X_n and X_n^2-n are martingales; filtration {F_t} of σ-algebras and martingales X_t in the case of continuous time, example - exponential martingale exp(αB_t-α²t/2), obtaining a family of polynomial martingales from the Taylor expansion of the exponential martingale: exp(αB_t−α²t/2)=1+B_tα+(1/2)(B_t²−t)α²+(1/6)(B_t³−3tB_t)α³+(1/24)(B_t⁴−6tB_t²+3t²)α⁴+(1/120)(B_t⁵−10tB_t³+15t²B_t)α⁵+...
  SDEs and Ito integrals (cont.): Ito isometry.
• Lecture 43 (Wed, May 4): SDEs and Ito integrals (cont.): Ito integrals are martingales.
  Ornstein-Uhlenbeck process: deterministic motion of a body in a viscous fluid (the resistance force is proportional to the speed and has a direction opposite to the velocity); random fluctuations in the rate of change of the velocity (due to molecular collisions), Ornstein-Uhlenbeck process, Langevin equation, derivation of the explicit form of X_t, mean and variance of X_t, behavior of the variance in the short-time and long-time limits and in the limit of disappearing a random force.
• Lecture 44 (Fri, May 6): Ornstein-Uhlenbeck process (cont.): Fokker-Planck equation for the conditional transition density p(x,x₀;t,t₀) of a stochastic process {X_t}, solution of the Fokker-Planck equation for the Ornstein-Uhlenbeck process, deriving the equation for the moment generating function M(θ,t|X_t0=x₀) of the Ornstein-Uhlenbeck process and solving it, identifying the type of distribution of X_t (conditioned on the event {X_t0=x₀}.
  Simple population growth at a noisy rate: derivation of the differential equation in the deterministic and the stochastic cases, solving the SDE dX_t=rX_tdt+αX_tdB_t by using Ito's formula, computing the expected value of the solution, discussion of the behavior of the population and its average in the weak noise (α²<2r) and in the strong noise (α²<2r) cases.

Attendance: You are required to attend class on those days when an examination is being given; attendance during other class periods is also strongly encouraged. You are fully responsible for the material covered in each class, whether or not you attend. Make-ups for missed exams will be given only if there is a compelling reason for the absence, which I know about beforehand and can document independently of your testimony (for example, via a note or a phone call from a doctor or a parent).

Grading: Your grade will be determined by your performance on the following coursework:

Homework: It is absolutely essential to solve the assigned homework problems! Homework assignments will be given regularly throughout the semester and will be posted on this web-site. The homework will be due at the start of class on the due date. Each homework will consist of several problems, of which some pseudo-randomly chosen problems will be graded. Your lowest homework grade will be dropped. All homework should be written on a 8.5"×11" paper with your name clearly written, and should be stapled. No late homework will be accepted!

You are encouraged to discuss the homework problems with other students. However, you have to write your solutions clearly and in your own words - this is the only way to achieve real understanding! It is advisable that you first write a draft of the solutions and then copy them neatly. Please write the problems in the same order in which they are given in the assignment.

Exams: There will be one take-home midterm and a comprehensive final. All tests must be taken at the scheduled times, except in extraordinary circumstances. Please do not arrange travel plans that will prevent you from taking any of the exams at the scheduled time.
Tentative date for the midterm exam: March 11 (Friday). The final is scheduled for May 9 (Monday), 8:00-10:00 a.m.

Academic calendar for Spring 2011.

Course schedule for Spring 2011.

Policy on W/I Grades : Through February 25 (Friday), you can withdraw from the course with an automatic "W". In addition, from February 28 (Monday) to May 6 (Friday), you may withdraw and receive a "W" or "F" according to your standing in the class. Dropping after April 4 (Monday) requires a petition to the Dean. (Such petitions are not often granted. Furthermore, even if the petition is granted, I will give you a grade of "Withdrawn Failing" if you are indeed failing at the time of your petition.) Please check the dates in the Academic Calendar!

The grade of "I" (Incomplete) is not intended to serve as a benign substitute for the grade of "F". I only give the "I" grade if a student has completed the majority of the work in the course (for example everything except the final exam), the coursework cannot be completed because of compelling and verifiable problems beyond the student's control, and the student expresses a clear intention of making up the missed work as soon as possible.

Academic Misconduct: All cases of suspected academic misconduct will be referred to the Dean of the College of Arts and Sciences for prosecution under the University's Academic Misconduct Code. The penalties can be quite severe. Don't do it!
For details on the University's policies concerning academic integrity see the Student's Guide to Academic Integrity at the Academic Integrity web-site. For information on your rights to appeal charges of academic misconduct consult the Academic Misconduct Code. Students are also bound by the provisions of the OU Student Code.

Students With Disabilities: The University of Oklahoma is committed to providing reasonable accommodation for all students with disabilities. Students with disabilities who require accommodations in this course are requested to speak with the instructor as early in the semester as possible. Students with disabilities must be registered with the Office of Disability Services prior to receiving accommodations in this course. The Office of Disability Services is located in Goddard Health Center, Suite 166: phone 405-325-3852 or TDD only 405-325-4173.

Good to know:

• The Greek_alphabet.
• Some useful notations.
• Basic principles of counting.