Selected Topics in Finance
Selected Topics in Finance FINA 8397
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This 47 page Class Notes was uploaded by Miss Quentin Grady on Saturday September 19, 2015. The Class Notes belongs to FINA 8397 at University of Houston taught by Alexei Boulatov in Fall. Since its upload, it has received 77 views. For similar materials see /class/208191/fina-8397-university-of-houston in Finance at University of Houston.
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Date Created: 09/19/15
Continuous Time Finance Bauer College of Business Alex Boulatov Spring 2009 Acknowledgements Some of these notes particularly for the rst half of the course are based on teaching notes of Prof Domenico Cuoco Wharton and Prof Henry Cao UC Berkeley Class 1 Probability theory basic de nitions and results Q Should we have a 10 break N0 Final 2 Midterms Homeworks not graded However they are important Plan make it more like a PhD seminar Purpose stimulate research if possible Later on student presentations Make it exible See where we are and adjust Feedback control welcome Syllabus just a guideline may not cover it all Before we go over the Syllabus see where we are technically a few simple questions Simple examples from Analysis easy Integrals and derivatives Find the optimal of midterms 1 or 2 Assumptions To pass gain an average of 1 points in one or two tests Flat priors ie if single test the probability to pass is 11 The results in both tests are statistically independent The problem of riskneutral student hard What happens Drunkard D and random walks How far would the D go on average given time T Mostly just a few de nitions to x the terminology Nothing too formal in this course except for the 2 intro sessions Why This terminology is commonly used in the eld especially in CT models We have to follow the trend Purpose understand the current literature Why making the Probability Theory so abstract As you may know the rigorous approach to the probability is based on the Measure Theory and Functional Analysis approach pioneered by Kolmogorov Moscow State This approach is accepted in Financial Economics as well why It gives a rigorous basis to the theory In any case we just introduce a few de nitions and refer to the known results Plan definitions and concepts Vector space Topological space Topological vector space Normed space Measure and probability space Random variables Expectation Conditional expectation RadonNikodym derivative Lp space Vector space Essentially a linear space with vectors and scalars De nitiuni A set X is a real imam space or linear space if 1 it contains a null vector 0 2 an additmn functiun maps any a y E X to same 1 y E X 3 a scalar multiplication maps any a E IR and any x E X to some can E X l the foll wing eight conditiuns are satis ed for all I1 yr 2 E X and 13 E E a y y 2 c 2D39 if a 3032 g his ID h 13 5 Some definitions convexity functianal etc De nition A subset A of a wet r space is EWWEE pmvided that a 1F ce39jjy E A whenever I y E A CL39 E 1311 De nitian A yummy ordered imam space is a vectcnr spam I Equipped with an nrder rElati n E that 15 campatible With the vector operatmns in the 433159 that 1 Egg 2 mz gzfcra zei 2 may IIEcly far ELMI39JEQ Da mlitiun In a partially nrdemi vecth space any verity satisfying 3 3 D is called a positive Bantam The mutaticm 2 Umeana x 2 I13 and 1 g5 G The set 0f pasitive ia39 ctm s is referred to as the pasitiw cans of X and d u t d by XII De niticm A millvalued functimn an a main space X is called a mustiami A mctional 95 311 X is linear if ats 3 audit My far all my E I n E R i5 gmsi w if 1 3 0 far all it 3 D and is strictiy pasitg ve if in addition x D fr all I ll Topomgy Let X be an arbitrary set A Eapumgy 31 AI is an ah tl39 Et way of specifymg a sense if damn333 amnng elements 0f X Da ni ti n A awlagy of X is a coli mimi 139 if subsets uf called 0135 33133 Bligh that 1quot A ETandwET 2 3122 C gE Tg r l ggET 3 H On I T is an mbitrmy collection cf Open sets then LJ Om E T The pair 3 T is called a topological space The amalgam t p l g is this ne with quotI X l3 sad the strangest topolugjr39 i5 the one which Inclmains all the subsets Df X De nitian Let A his an arbitrary m m et M a t 1pnlog39ma l space X We say that A is ringed if its mmplement A E 17 E X r 235 A is open An element x E A an mtg111m pamt bf A if there an Q pen Eat 1 r 3 and that 5 v G The i t Eff A denoted by In is the set 0f all interim paints nf A The ciasum 3f A dentth by A is the set if all points not in the intE3501 13f In other words D or U of any countable number of the open sets is still an open set Trivial example segments and intervals in R1 The set is closed if its compliment is open Closure of A all elements of A that do not belong to the interior of the closure of A Any simple examples in R Convergence under topology De nition A Eequence 512 C X is said in sewage to an at Equot X in the topology Ti written it 1 if for Eli open sets 0 E T containing I there 15 an integer Nanci1 that In E 339 for ail n E N Similarlm we can use topologies to de ne continuity De nition Let T and Iii be topological spaces A map 33 om X to iquot is one LU be uu39uii39wuu39ue in the given tUp i gies if 1 a 1 implies ailix i TYv x Convergence compare to the de nition of convergence from the standard analysis What is the point The countable sequence converges to a certain element X if for any no matter how small neighborhood 0 of X there eXists a number 11 starting with which all elements of the sequence belong to O What is the key issue De ne the notion of the neighborhood topology Topological vector space De nition A vector space X with a t pology T is called a iapalagiml vector Spam if addition is a continuous map from X bi X into 1 and scalar multipli ation is a continuous map from ll 13 X into X De nition If X is a topological vector spane the suit of all continuous linear functioth on If is called the dual of X and denotad 1353 If X X then A is said to be re emla Thomson 1 Saparating Hyperplane Theorem LEI A and B be two convex subsets of a topological illicitquot SPIME X and assume that A has an iniainiar point if IntUl i i B than them exists a fi lf llilml Ii E X and an a E H such that mi 5 cr if all iii E M E E Important result how does it work ABCX convex A has an interior point IntA does not intersect B There exists a nontrivial linear functional p st pX S a S py with XEA yeB A X 3 My ltpX Normed vector space De nitiun A functional I w M an a vectur space i5 callasd 3 mm if it satis as the fn mring prapert ies for all ml 3 E X and E E H 1 HI 3 D 2 EEG EH l flli lh 3 WE yll 5 Hill Hull 41 Was 2D E 3 a D Nai med topaiogy Example 1 The Euclidean Harm an E is de ned by I1 3 13951 De nition In a normed space X the set x E X 2 1 EH lt1 5 where if E X and 5 3 CI is called the but ef centEv and mdim 6 A subset of a normed space is bounded if it is contained in some ball Nermed Spaces come with a natural idea of closeness and hence with a natural topology De nition The new topology of a normed Specie X is the smallest topelogy containing all the balls It can be veri ed that a set A C X is open in the norm 03301an if far all 1 E A there exists a ball 5 such Linit u E B E A and Lhei a sequence zen C X converge3 to am element a E X if and only if for all e 3 D there exists an integer N Such that NI m r e for all n 3 N In the following when we talk of named spaces we will always assume that they have been endowed with the Harm topelegy Normed spaces are an important special casa of topological vector spaces 0 Is a ball in a functional space always convex 0 Answer convex in metric Banach or Hilbert spaces Cauchy sequences Banach space De nitien A sequence in in e named space is called s Cauchy sewenee if fer all e 2 0 there exists es integer N such shes In em eis Vs m 3 N A sensed linear space X is senseless if every Cauchy sequence converges ts same I E X A samples nemsd lines specs is eslled s Benseh spaces Properties of functionsfunctionals De nition A fumtmn f frum X to lFL LJ is a game if i fl 1E all 539 f39lIl 1 al y far all ry E X and a EH11 b lamaquot semlmntmwm if far all m E X and all sequenms in r X such that En F is c caerclwe if fw m far all aequencas 51 C X with In anquot m as n r m Any simple examples of convexnoneonvex and coercivenoncoercive functions Basic result convex analysis Theurem 2 Lat X 15 1 re mva Banach Space and f X r JRU lW be canvas and laws Semimmmwus Then I attains 111 minimum 0139 any rmnempiy grammar sum af 11quot that is 3505351 and bemudad If f is mercive than it attains 1 minimum an X Measure and probability spaces Let 5quot be a cullection 3139 elements each ml which danote by w De nitinm A err eld on El dammed by 77 is a nonempty family 3f Subsets atquot fl satisfying this f llcrwing canditicns l A E 3quot 2 AG E F 2 Aiml C F 0114 a 3quot The pair fl F is said tn he a m asumbls space Ffmm the de ning properties of a cs eld it is easy tn verify that 1 y El E I 2 Aggy 2 far 1A 63 The smallest cr eld is the out that contains Duly and fl the traith if eld and the largest in eld is the HIE that all the subsets Inf Q the discrete ar gid If E a topological space than there a natural au eld on El De nition The Banal J eld of a topological Space FLT is the smallest E eld that contains all the apex sets in T easure De nitinn Let b j and gurfgl39 be measurable spaces A map f 11 RE is said tn be meammb e if f 1A E F1 in all A E F In particular a realvalued functian ran 521 F is mammabl if it is m umbl f 35 a map fmm 51 F to IR HEM We will sometimes mite this as f E F De niti n FL mmsure m on the measurable space ELF is a mapping f mm F tn E 33 which is countabiy additive 112 if A fgl C F is such that Aj m At w for all 339 7E 15 then 2quot M An Z Aw3 39r39l n1 Roughly speaking the map is measurable if it can be inverted ie the inverse rnapping takes us back to the original subset argument set for the forward mapping Measure countablyadditive rnapping De niti n The triple 1 pill is said ts be a measure space When Mill l we will deserte the measure by P end refer to the measure specs ll f P as s prehehility specs In eur future disenssien ef eeenemie Inetlels1 we will elwey s take e prehshility speee as primitive with the fellewing interpretation The state Specsquot 52 is s colleetien of all the pessihle states sf nature eseh of which is a complete description if a possible resolution ef the essgeneus uueertein euvirenment The er eld summarises the infermetien available ts the agents in the eeeuemy by describing the Lquotsweetsquot distinguishable by the agents Finally the prehehility measure P represents the belief held by the economic agents shunt Lhe likelihurnl iii the distinguisliehle events Example 4 The fellewing is s pireltsehilit3r specs S w1w2wsgwsa j Hearing m3F p14 M1W2W3 Mia W21W4i ME1W4 p Pushes a u Paws Filed i Randall s variables Fin train new an a pre hahility spans 5 F P In static medals under uncertainty agents will cheese censuniptien plans that are contingent on the particular state at nature This leads us tn meets the eensnrnptien space as a set of tandem variables De nitienr a tandem unstable is a measurable realvalued functan an L Remark In eeenomin terms requiring that a rsensumptien plan be measurable is an infnrmatienal constraint For example in the pane of the probability spans of Example 4 an random variable must take the same values in the states all and as This re ects the feet that nine cannnt distinguish between these ten states Nate that far all practical purpnsesi two random variables that are equal almnsi surely is except an a set of sets probabilityr ahni d he considered the same Thus if two random variables X and V are such that X itquot as then they will be identi ed and said tn belong to the same equtuaiense class Ramd m variables convergence De nitiam A sequence 35 of random variables is said to cmweryr in pmbability to a randum variable X written Jim is X if lim Pw mum mm e 0 re p 0 Ti W 39 The sequence is said to converge almost surely to X written Xn Equot X1 if Pw XWEM inmj U IVIme generally a sequencr 3 Of measurable functions on a measurable space EL F u is said to comlerge in measure to a function X written X it A if pm IXnu3l f l V6 3339 D The sequence 951 is said to converge almost evcmrhrre tr X written X X1 if uw JEEEXHIZM Xl wj 9 Remark If MR lt1 go then converge almusl everywhere implies converge in mea sure On the rather hand converge in measure implies converge almost everywhere along a suhsequence Simple random variables De nitinn A random variable X is simpie if them exists a nite partitian A g of Q with A E 317 far all j and raal numbers EjELI such that i mjl wL where 13 dena tes the indicath functim 0f the set A 7m I r E 1 ifw E A 1AM 0 otherwise Expectation If X ELI 35 143 is a simple random amiable than the expectatth 0131 dem t d by Epf gr E 3139 all is de ned to be ELK z n 31 ij d w 3 Emmi V i1 Next lat X be an arbitrary nunn egative random waxiable We Extend the de nitinn 0f Expectation to X by letting ELK supEX iquot is 51mph iquot g X Finally for an arbitrary random variable X we have 3 quot Xquot where 14 minim X and X maxm X are nonnegative random variables and we Set ELK Ep EM 391 unless bath EX and EX equal 09 In the latter case we say that the expec tatim does rmt MM 011 the other hand bath EX 1 and 39 are nite WE saw that X is integrable Properties of expectation 1 Mammy EM Y a EX Em 2 ammable additim y If An C F is such that 311 Pl A3 E El far 239 a 31 then b11an X 4 WW n ag1m 3L Mam anim39ty I X 3 Y ans BIKE EM 11 Properties of expectation II Damnate convergence thearem If X X as IX l i 3 31 s far all are and Equot c 33 than EWJEEWL Manama g Thearem If X T X and X11 2 I then Ema Em r Fatou s lemma If Xquot 3 D for 31 n then 11313 Em Enimf XE Jensen s inequality If g3 is a convex functinn and X 341d g lf fj are integrable then El ita f a diliEPi 1 Conditional expectation In dynamic madels infurmatimn akanut tha true state 3f the wurld is often gradually revealed war time Theref re agents will farm Exp ctati ns over time by mnditiuning cm progressively larger infurmatign SETS This leads to the nation Di 3 canditinnal expectatiun Pmpmsitinn 1 Lei Q bra a subm e d af F and X cm intEmmi random variable Then were exists a gmm umbk madam va abk ELK 1ng mum the mnditiunal pectatiun of X given 6 with he prayer y at FmJaw 15 gm m c g Thai randam mrufa ie is unique up to an equivalence 135555 Conditional expectation properties Some useful properties of conditional expectations are recorded below where X it and i are integrable random variableg and g 91 Q 313 subur ilcis of f 1 If 939 6 is the trivial U ld than ELI i9 2 EW 2 UK E Q then Epihg EX ELS nd Eixriil mm is ILquot Conditional expectation more Law of itemtad expectations If G1 a g than ElEiXIQ l a a EEXQz 1 ED W1 m Mammary EX my 2 EX 3mg 345 Monotom39city X 2 Y 35 a Bung 3 E Yg 35 Dominated convergence theamm If If ma X39 as and IXn i 3quot ans for all 11 than 1 51ch EX E EXQ 334 Conditional expectation more Monotone Cannergence Tilemm IfX T I ans and X E 0 than EW9 a ELEM 33 E Fatau s lemma If EL 1 I for all n then 1139 EXn 3 Eng 13mg as 5quot 9 Jensen s inequality If 9513 a convex function and X and q X am integrablei than Emirng a mam asquot Absolutely continuous measures We will often work with two probability measures P and Q on the same measurehle spsee El F In this ease we will have to distinguish between espeetetion under P so EP and expeetetion under 2 say EQ between properties holding Pelniost surely and properties holding Qalmost surely and so on In feet we will see that economic considerations will often lead us to considEI 1139o12nelzailitjt measures that have the some measure eero sets that is equieslent probability measures De nition A messere Q is said to he obsoleter continuous with respect to s meesure P written Q cs P it PM l implies QM l for any A E F The measures P end Q ere said to be equitiolent written P s 41 if P s Q and Q s P HOW to consimct ACM Netiee that equivalent prebeiiiiity measures have the same measure eerie sets ea that e preperty imide F eimeet surely if end 0111 if it helde Qeimeei surely Given a probability measure F ii is easy rue eerieiruei prebebiliir measures that are ribeeiliieijrr eentinueue with respect m ii Let X be any random variable with X 3 ii are and EM 1 end de ne are Lemme 1 fer eii A E F ii is iiren immediate to verify that Q is e nn mnegative eeuntabijy additive set funeiien with em 1E and hence 51 probability measure irioreever FU i 0 impiiee QM 0 so that Q is ebeeiuteiy eentinueue with respect Le P RadonNikodym Pmpusitiun 2 RaanNikadymL Let P and Q be Ema pmbabi ity measures on ELF with Q 31 P Then there exists a madam variable X with X 2 P h3m and EPXj 1 with the pmpe y that QM mum w e a Hammer P a Q if and an y ifX 3 U 15 RadonNikodym derivative The random variable X in the above propnaition is unique up 130 a P equivaience class It is called the Radon Nikadym derivative of Q with respect t0 F and i5 denntad by g Fallawing are some useful pmperties 1 If P w 62 than 2 If X is any ll nnegative madam variable than cit t f m dam fa A w w mm 3 A rand m variable X is Qintegrable if and only if is Pdmegmblfaz in Which case for all A E 3 L X u dQw Kiwgtw m 4 Ganditianai Bayes RUEE If X is a Q integmble random variable and Q is a sub cra amplcl of F then EPL f lQ Lp Space De nitinn Let 155in be a pmhability space For any 1 E mag a randem vari ble X is said ta 1131qu to the space FHA P if EHX F 6 no A random variable X is said tn belong in the space Lm 39F P if 355 sup IE urn an where v 95531113 Xw imfa E 1R Xwj d a 515 danntes the essentmi supmmum cf X v39 a g Msz mgfm i mqwj lmtwg Pmpnsiti n 3 H lder Inequality If Mr E 111m are sum that 1 E DTP and Y E 15le thaw391 XY E ELF and HEY 1 HM My P I p s itiam 4 Mnknwakj Inequality Let p E L m and X itquot E WP Then X YHP E X Flip It mum ram Minknwski inequality that far 1339 E 1213 H ll dE 3 mm fur LPF prwided that as 1131131 we identify madam variablea that are equal awash Surely Therefnrg MP is a named vastth space far all 19 E 1 ml 111 th fDHDWi E unless atherwise noted we will restrict nurselves ta 1 E 1 can Cumpleteness of Lp Propositien E The INF spaces are Banach epaeee The DTP spaces are of in nite dimension unless there exists e nite eelleetien nk of measurable eete with strictly positive measure such that any B E j 13 the nnien ei sets from An and measure mm 5355 The peeitiire erthent of HE dennteel by LHP L is the eet hi all tandem verinblee X with 3 U 35 Pmp sm m 5 Fm 3339 E Lee hue an empty interim nineteen LPP is 0f nite dimeneien On the other fiend L P39 nieieye has n nnnempisy interim Lp commodity space Prices positive linear functionals Proposition 739 A positive imam functional in Hip is continuous This leads us to exploring in more detail the dual of mi Proposition B Let Y E Lq i Then this functioniii ism fnxiwwcwww is an element of Muir where l l i In addition if i 3 0 no then ii is Fquot i 1 positive linear functional and if Y9 r EII ma than air is a strictly pomwe imamquot functional Theorem 3 Riesz Representation Theorem Let be a continuous linemfunc time an U39u where p 6 100 Then there exists a unique Y E Um where 1 such that X n XuYudpw 39VX e we Corollary 1 The Um spaces mthp 6 100 are re exive The next result is useful in establishmg the existence of optimal consumpzion plans Theorem 4 Let ip be a complete measure space with M9 lt 00 and let f 11162 M gt IR U 00 be convex and lower semicontinuous in the topology 7 of canvargence in measure Then f attains a minimum on any nonempty convex subset of LIME u that is Tclosed and normbounded
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