Universitetet i Oslo / Økonomisk institutt / NCF; this version December 3, 2012 ECON5155 autumn 2012: Notes and curriculum This document summarizes key concepts and learning outcome goals for the course, in- cluding typical problems you will be expected to be able to solve. It will be more elaborate on topics where there is no set literature (then this document serves as a reference). Now the exam is open book, so you are allowed to bring any printed or written resource. That of course includes sources which have not been subject to ordinary scientific quality assurance – like your own notes, like this note or like any Wikipedia article. However, just like other mathematics exams, you will be expected to show your calculations from known results. To avoid the issue of trustworthiness of sources which can at any time be changed by anyone without any scientific review (e.g. Wikipedia), I have compiled and attached a few such articles which I have checked, and officially OK’ed for the purposes of this course. 0 The reading list The reading list is taken from the following: • This document (all of it) except the attachment from free sources, which is included as a service. • Tom Lindstrøm’s lecture notes to MAT2400, available at http://www.uio.no/studier/ emner/matnat/math/MAT2400/v12/MathAnalBook-Tom.pdf (with errata here). All ref- erences to this version (note that the 2011 version had completely different enumera- tion). • Atle Seierstad: «Stochastic Control in Discrete and Continuous Time», Springer. Avail- able online for UiO login, as of now at http://link.springer.com/book/10.1007/ 978-0-387-76617-1/. • All the problems assigned are curriculum. From Lindstrøm: • Preliminaries (Chapter 1) • Metric spaces (Chapter 2) • Regarding Chapter 3: – The required topics concerning convergence is covered in this note – in the setup of general topology as well – and Lindstrøm section 3.2 is on the reading list to the extent it supports those. – The required topics concerning convergence is covered in this note – again, in the setup of general topology as well – and Lindstrøm section 3.1 and its special treatment of uniform continuity is only supplementary reading. 1 – Lindstrøm section 3.3: the topic of vector spaces, and thus function spaces, is itself on curriculum, and you must know the space of continuous functions, and the supremum metric (indeed, the supremum norm as well). – Lindstrøm section 3.4 is an application of the Banach fixed-point theorem, and was assigned as a problem. – The rest of Lindstrøm chapter 3 has not been covered per se. • Regarding Chapter 4: – The concepts of limsup / liminf are covered (they are preliminary concepts). – Vector spaces and normed spaces (section 4.5): To read: to and including the Remark on top page 101. You are not required to «recognize» vector spaces by applying each and every part of the definition, but you should be able to see that e.g. the space of positive functions is not a vector space (point out «−f»). You are required to recognize norms using the definition, to test a given func- tional to check whether it is a norm or not – just like you are required to recog- nize a metric in the same manner; you should in particular recognize the L1, L2 and L1 norms just as you should recognize the taxicab, Euclidean and uniform metric. We have however not covered the (Minkowski, Seierstad formula (5.30)) triangle inequality for other Lp spaces, and you can/shall take as granted without justification that the p-norms are indeed norms for any p ≥ 1. – Inner products have only been mentioned as application. It might be worth to know that the inner product with itself is the square of the 2-norm. • Regarding Chapter 5: 1 Note that Lindstrøm restricts himself to the Lebesgue measure (i.e. length in R , area in 2 R etc.) – we have covered more general measures, see e.g. Seierstad chapter 5 below. Also, see this note. – Skip: Outer measures and the construction of measures by way of such; You should however know the fact that measures can be defined by starting with «prototypical sets» (e.g. open intervals for the Borel case). From Seierstad: • From Chapter 1: Sections 1.1–1.2; then 1.5–1.6 as special cases of 1.1–1.2. Furthermore, 1.9 to the extent it covers the problem given on fixed point iteration. Also, refer to section 1.11 for problems. • Chapter 4, except – page 189 is for the curious – regarding Theorem 4.1, you only need to know that it basically works analogous to the deterministic case (which was assigned as application of the Banach fixed- point theorem) – the proof of the Itô formula – although the basic idea and the limiting transition proving (dB)2 = dt, is an application of convergence in norm. 2 – regarding the conditions for the Dynkin formula, it suffices to know that you must in the general case bound the stopping times and then apply a limit transition; Limit transitions are per se curriculum, but for exam purposes, you are required to perform this exercise only if asked to. – the Black–Scholes formula, and its proof, are not on curriculum. You will however need some of the arguments: the Itô formula and the heat equation. – Girsanov’s theorem page 207. – For optimal control, like for Dynkin’s formula, you need to be aware that there are limit transition issues as time grows, but you are required to carry them out only if asked to. – Skip the last part of section 4.4, from and including subsection «Soft Terminal Restrictions». – For optimal stopping, you are only required to treat the problems as a special case of optimal control (i.e. the special form of HJB, and the reservation for limit transitions still apply) plus you need to be able to apply the C1 fit (taking the validity for granted). – Skip sections 4.6 and 4.7. • Chapter 5, although Minkowski (5.30) has not been covered. • Also, you could use the backmatter with problem solutions. 1 Stochastic dynamic optimization Reading list here is Seierstad as detailed above. Sample problems: • As assigned, although I think 1.26 would be a bit lengthy. • Example problem set Problem 1. 2 Metric spaces Reading list here is Lindstrøm as assigned. You should in addition know that the concepts of open sets and of compactness as used by Lindstrøm, have different-looking definitions for the general topology case. Sample problems: As assigned earlier. 3 Topology: the concepts and the fixed-point application The treatment of topological topics is somewhat involved. Some topological concepts were defined both in the general case and in the metric (and even more specialized, norm-induced) case. 3 3.1 Before we start on topology itself: sequence, subsequence, countability, complement, union and intersection In this part, Lindstrøm Chapter 1 may also be helpful. 1 A sequence fxngn2N – also frequently denoted simply by «fxng» or by «xn» – is a function whose domain is the natural numbers. That is, it is infinite; sometimes we extend «finite» sequences fx1; : : : ; xN g (i.e. an N-vector) to infinite by putting xn = xN for n > N. A sequence fykg is a subsequence of fxng if there is some strictly increasing (infinite!) se- quence fnkg of natural numbers, so that yk = xnk for all k 2 N. We speak of a sequence in a given set A if the range of this function is contained in A. A set is countable if it can be covered by a sequence – notice that some authors require that the set is infinite, in which case the «finite-or-countable» property is referred to as at most countable. The complement of a set A, is the set of points not in A. The complement is relative to the 2 space X in question; if X = R, then the complement of [0; 1) is (−∞; 0], but if X = R it is the plane with the half-line cut out. To confuse the fxng notation even more: For an indexed family of points or sets – say, fxigi2I , the index set I will not taken to be countable, unless otherwise stated – for example by writing x1; x2;::: (this language signifies that if you continue enumerating this way, you will cover them all), or by calling it a «sequence». The union of a family fAig of sets is the set of points belonging to one or more Ai. The intersection is the set of points which belong to all. Notice again that the family need not be over countably many – therefore, do not write e.g. A1 [ A2 [ ::: unless this is justified! The usual convention when the index set is empty, is to let the union of no sets be the empty set, and the intersection over no sets be the space itself. Need to know The following properties are preliminaries which you need to know, and which you in an exam situation can use without proof: • An interval (a; b) ⊆ R can be written as the union [ (a; b) = f(ar; br); ar > a and br < b are rationalg: Furthermore, since there are only countably many rationals (i.e. no more than can be covered by a sequence), the union is actually taken over countably many invervals. 1Observe the potentially confusing notation: a set containing a single point x¯, is denoted fx¯g, so there might sometimes be an ambiguity of whether «fxng» refers to the singleton set containing the one and only xn, or the set containing them all. Some authors denote sequences with ordinary parentheses, like vectors of infinite dimension (which they are!).