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# Class Note for FINA 8397 with Professor Boulatov at UH

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## Reviews for Class Note for FINA 8397 with Professor Boulatov at UH

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Date Created: 02/06/15

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 nition A set X is a real teeter Space or linear spaot if 1it contains anullvector D 2 an addition functiun maps any 35 y E X to some 1 y E X 3 a scalar multiplication maps any a E IR and any E X to some at E X 4 the fullewing eight conditions are satis ed far all SE1 3 r E X and if E R I39Dv Some definitions convexity functional etc De nition A subset A of a vector space is cannan providad that as 1 F Ely E A whenever 1733 E A 139 E 01 De nitinn A naminlly ordered nectar 53mm is a vector space X Equipped with an nrd l39 relatinn I that is cmnpatible Will the vector opemtinns in the new that 12 nz 3yzfnrvallzexg a 2i 1 5 1139 E cry far all 11 3 l quot2393 IV 39quot Dellni39tlnn In n partially Jrlewd vector space any vectnr antla iing r 3 D is called a positive warm The nntatinn 17 leenna n 3 El and I 5399 U The set nf pnsitive quotmum s is referred to as the mailing cans of X and dnnntnd l3 X4quot Dn nitinn A railvalued functinn an a main space X is called a functinnnl 431 mcnionnl 95 can I is linear if dim 3 wilt y for all my Equot I a E l is awaiting if 943123 3 I for all it 2 D and is slriclly pnsitivn if in addition I 3 CI in all I it 0 Topology Let X be an arbitrary set A topmer on AC is an abstract way of epeeifgrmg a sense of closeneae among elements ef Definition A topology of X is a eolieetim T of subsets 0f 35 cal16d 010 563 Buflh that 1quot Aquot E T and W E T 2 D E 0 3 I IGEET 3 H 0 I T is an whimemf collection of open gets them LL 0 E T The pair 712 15 called a topological space The weakest tepelog39y ie the ne with quotI X l3 aid the strongest tepelegzr is the one whinh contains all the subsets of X De nition Let A be an arbitrary wheat of e topological space X We ea thet A is closed if its eomplement A E in E X 5 a A is open An element x E A an mime3 pitmt of A if then is an upen set D C 3 Such that SEE I39 Q The i t f 44 denoted by IntM is the set of all interior points of A The aimurea of A denoted by E is the set of all points not in the integior of 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 Sequence In C X is said Ize message he sn 5 e X in the teerelay fTi written l 2 if for s1 epen sets 0 E T containing 1 there is an integer N such that In E G fer all n E N Si l aflj we can use tepelegies m de ne euntinuity Definition Let T and Y be mpelegicsl spaces A map 35 from X to 1quot is seisi to be eu39n i39ra39u u39ue in the given L p iegies if 3quot I implies emu 1quot 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 o toiiology quotT is called a topological vector Spam if addition is a continuous map from if 4 X into Aquot and scalar multipiioutiou is u continuous map from R i X into X De nition If If is a topological vector spaoe the out of oil continuous linear funiztioiiuis on If is called the duo of X and deuo tud by X39 If X j X then X is suici to be re eziuo 39 Theorem 1 Soparating Hyperplano Theorem Lot A our 3 be two convex subueiii of 1 ioooiooiooi ruucioi Spoier X umi assume that A has on l39 iv ri i point If IntM N E ii than there exists 1 uouii iuiui o E Xquot and on o E R such that Moi 5 o F iiiiii is E ii i E B 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 E or S py With XEA yEB X 3 a My ltpX Normed vector space De nition A functiunal v an a vectur spam i5 called 3 mm if it satis ea the fullan pmperties far all ml 3 E X and 2 E 13 1 HE E 0 2quot WM l fllii li 3 His yll S ll 1 llyll 4 Mac 13 a i0 N ormed topology Example 1 The Euclidean norm on E ls de ned by n a in 23 13951 De nition In a nnrmed space X the set I E X Ha ill t 5 where if E X and 5 3 CI is called the ball nf center39 5 and mdins 5 A subset of a normed space is bounded if it is contained in mine ball Normed Spaces come with a natural idea of closeness and hence with a natural topology Definition The worm topology of a normed space X is this smallest topology containing all the balls in can be veri ed that a set A C X is open in the norm topobgy if far all r E A there 121315113 5 ball B such that 1 E B C A and Llint a sequence In C X convergcn tn an Element in E X if and only if for all e gt El there exists an intnger N Such that Him a m c e for all n a N In thn fnullcrwixixg1 when we talk of named spaces we will always assume that they have been endowed with the norm topology Normad spaces are an important special case 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 nition A Sequence In in e named epeee ie eellecl e Cauchy sequence if fer all e e 0 there exists en integer N thee lien emll eie Vii m 393 N A nenned linear space X ie eempleie if every Cauchy sequence converges te same a E X A eemplete rimmed linear space is called e Banach speee Properties of functionsfunctionals De nitinm A fumtimn f fmm X to ER Ll is 1 51 Emma if HM 1 E My 3 f39EiJ 1 ll y far all my E X and D E 01 b lawn smimntinww if ii 5 111m In far all m E X and all sequenms in Lquot X such that 3 F 2 c camin if m m far 311 Sequences in C X with In air at as n r m Any simple examples of convexnoneonvex and coercivenoncoercive functions Basic result convex analysis Theurem 2 Let X 55 a TE E EiIJE Banach Space and f X r JRUW be cunttt and tamtr temtcanttmmus Than I attains L1 mttttmttm an my rmnempty 3571333 subsist af 1 that ts dated and tauntsd If f ts cigarette that it attatus a minimum an X Measure and probability Spaces Let ll be a cullectian 0139 elements each all which is denote by w DE nitinm A a eld on El clennted by F is a nonempty famin If Subsets Bl fl satisfying tha fm lltmring canditimns L AEFQA GEF 3 Anliid C Fi U311 All E Jr The pair fl F is said to he 3 msasumbls space Ffmm this de ning pmpe ies of a cre elc l it is may tn verify that 1 I E I 2 Anita f a at A e 54 The smallest cr eld is the one that contains Duly El and f the trigi433 tr eld and the largest tra eld is the one that cumming all the subsets mlquot Q the discrete a eld If fl is a topological space than there is a natural in eld 0n ll De nition The Banal sf eld of a topological Space ll1 T is the smallest E elcl that contains all the 0pm sets in T Measure De nitinn Let 53911ng and grfgl be measurable spaces A map f n1 53932 is said tn be mmmmh a if f 1A E F1 in all A E F In parmicular a mallvalued functiun an 51 F j is mmura39blt if it is mammable as a map fmm Q I m IR BEE We will snmetimes mite this as f E F D39E iti ll measu i m on the measurable space 1575 is a mapping fr m F tn EL 90 which 13 mummy additive La if A l C F is such that Aj m At 2 El for all j i then 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 thehility Spam De niti n The triple f LLJI is said ta he a measure space When M51 2 1 we will denote the measure by P end refer to the measure space 11 F P as e prehehe lity space In eur future cliemesien ef eeenemie medele1 we will always take e prehehility space as primitiveT with the follewing interpretation The state Spam3 R is a collection of all the peseihle States hf nature each at which is a complete description Qf a poet31bit reselution ef the exqgenaue uncertain enviwnment The J lfl summarizes the informalen available tn the agents in the eeenemy by describing the events distinguishable h the agents Finally the prehehility measure P represents the belief held by Lhe eeenumiu egente ehuul Lhe liltellheud all the listiuguleliehle events Example 4 The fellewing is a pr babili 39 space 9 MI1LL EFMEFM4 F wll w 3 MEL M4 wlswisw wlsl r39Eaul l waitU41 Ea Pllwwel a a mum Putt gt Random variables Fit from new on a probability space ll F P In static models under uncertainty agents will choose consumption plans that are contingent on the particular state of nature This leads us to model the eonsumption space as a set of random 1sariables De nition A modem testable is a meaemable reeltrainee hanetion on Q Remark in economic terms requiring that a consumption plan be measurable is an informational constraint For example in the eaee of the probability space of Example 4 any random variable must take the same values in the states all and Leg This re ects the feet that one cannot distinguish between those two states Note that for all praetical purposes two random teriablee that are equal almost surely lee except on a set of zero probability should be considered the same Thus if two random variables X and l are such that X lquot as then theF will be identi ed and said to belong to the same equivalence class Random variables convergence De nition A sequence Xn ef rendmrr variables is said 130 cenverye in pmbeb ity to e Tandem variable X written Xn 5 X if lim Fw mum ijl 3 e 0 es e T39d m39 The sequence is said to converge elmest surely to X mitten X11 E X1 if Pw 111ng yew e grimy a More generally a sequence XE ef measurable f39unetiene on e measurable space ELF p is said to converge in measure to a function I written X it A if lim um IX mjl w f II We 3 0 Hun The sequence En is said to eeneerge almost everywhere m X written X X1 if uwl1r E2e an5 Xw ID Remark If 52 lt1 ere then eelwerge elmeet everywhere implies converge in mea sure On the ether hand converge in measure implies converge almost everywhere along a suheequence Simple random variables De nitinn A remdnm ariable X is simp e if there exists a nite partitian A y of Q with Aj E 7 for all j and raal numbers af such that MMEMWL where 13 denu tes the macaw functiun nf the set 41 its 3 1 if m E A 1AM 1 otherwise Expectation If X E39si1 3 143 is a simple mnde variable than the expectatian of X1 denated by ELK or In X nip is de ned to be Eur E Ammuniw 3mm Next let X be an arbitrary n mnneegative random vamiable We Extend the de niticm of expectaticn to X by letting ELK supEX 1 X is simple X 1 X Finally for an arbitrary random miablie X we have I 3 quot Xquot where X macaw X and X Inseam X are unnnegativg random variables and we Set Ex 3W 395 unless both EX and EX equal 03 111 the latter case we say that the expec tazim doeg rmt Him 011 the nther hand1 both EX 1 and am finite WE SEW that X is intetgmbfe Properties of expectation 1 Linau iy EM Y a Em ED 2 Ummtubis addi im y If An C 37 is such that A H A5 E in i 3 j than jug X K mm mm mm 3 Manatanim ty I X 23 Y ans h E X 3 EM 711 Properties of expectation II Daminatad mmergence thearem lf X 39s X as IX l i lquot gs far all n and cs 20 than lim Emu 2 EM iDC39 M tane Cammrgeme Theamm If Iquot T X and X E I than 11111 EH Hz Fatou s lemma If In 3 far all n then li EXn I Eli llgf XE Jensen s inequality If 3913 is a convex fun ti and X 34111 lf fj are integrable than Elm 2 ablElel Conditional expectation Iii dynamic medele infermetien ebeut the true etete ef the werld is often gredneliy revealed ever time ThEI E f rE agents will ierm expectations over time by eendiiieiiing on progressively larger iniemetieii sets This leede ie the notion iii 3 cemiitinnel etpecieiien Prepeeitioii 1 Lei g a eubm eid 0139 if end X en iriiegmbie random variable Then iiiei e erieie e g 1 amp iimbl mne em viii iiibie ELK 191 called the eendiiienai eye peetetien ef X given 939 with the peeeerie iliei lemgieez xee viii g This Tandem iieriiiiiie is unique tip in an equivalence eieee Conditional expectation properties Some useful prop rties of conditional expectations are recorded below where X Xquot m1 1 are integrable random variablea and g 571 I am subm alds of f 1 if 9 551 is the trivial U ld than El 5 2 Em 2 If x E than Ewing 2X am End EIA39YIQ XEH IG 35 CF Conditional expectation more Law of itamted mpectotiam If G1 a Q33 than EiEixl ill a a EEXlgz 1 Ema m Lineamy EDI Y E EXG EWIQ 54 Monotomc y X 2 Y 35 2 EXl 3 EYQ 35 Dominated convergence theorem If In 4 3 as and IXn 1139 3quot as for all 111 than 1193 Evan a EMIQ 213 Conditional expectation more I Monotone Commence Theorem HE T 239 ass and X E 0 than I39m EIXHIQ 2 ELEM 35 3 Fatau s lemma If X 3 U for all n then Egg EWRIQ E Eligigi XnIg 33 i W 9 Jensen s in guaiity If 513 a convex function and X and MK am ilitegarablei than El Xilg 2 MEX 5911 Mr Absolutely COI ltlnll llS measures We will often work with ten prehehility measures P and t on the same measurable spsee ll F In this ease we will have te distinguish between enpeetstien under P say EF and expectatinu under Q say EQ between preemies holding Pshneet surely and properties holding IQalmost surely1 and en on In feet we will see that economic eensiderstiens will eften lead us to consider nrtihehilitP measures that have the same measure sere sets that is equivalent prnbsbillty measures De nitien A messnre Q is said tn he steelmer sentiment with respect to a measure P mitten Q e P if PM implies QM 0 Ear any A E F The measures l5 end Q ere seltl te be equivalent written P s 2 if P i Q and Q cs P HOW to construct ACM Malice that equivalent prabability measures have the same measure Earn seats 50 that a pmperty holds P almnst surely if and 0111quot if it lmlds Qalmosl 5111613 Given a pmbability measme P it is away to canslmct plubabllily magma that are abaulutely cantinu us with respect m it Let X be any random variable with X 3 J as and EH 1 nd de ne QM Axlwdle 1 far all A E F It is than immediate to that Q is a n mnegatlve sauntably addlll ve set funclion with Q l 1 and hence a probability mmsure Momma Phil 0 implies A 0 5 that Q is absMullah continuous with respem m P RadonNikodym Propusitimn 2 RadunNikudym Lei P and Q be Ewe pmbabi ity measum an ELF with Q i P Then there exists a randam variable X with X 2 U P 23 and mm 1 with the pmpe y that QM mm v4 e y Mareaver P a Q if and only if X 3 CI 415 RadonNikodym derivative The random variable X in the above propnsltion is unique up 130 a Peequivalenne 31333 It is called the RainaNiles le derivative if Q with IEEPEB E to P and is denoted by g Following are some useful pmperties 1 If P Q than i 1 cm LP quot 2 If X is any lmnnegative random variable then r I A 559 39 1 wJdQLLU A mmdP may 3 A randum variable X is Q integrable if and only if is Pdmegmblez in Which case for all A E 3439 A Kind dQlw xingm am 4 Ganditianal Bayes Rules If X is a Q integrable random variable and Q is a subcr alcl of F then I EPl f l l EQIX39Q Lp space Da nitinn Let 24in be a pmbability space For any 33 E 0033 a rande variable X is said tn belting t0 the space HIIE ZJZ P if EHX m aquot Do EL random variable X is said tn being in the space Lm F P if 355 sup I lt2 no where awn a a is lasssup Xw i11f E 1R denutes the essential swimmith Elf Xv 393 LP umfm meqmali eg Prapnsiti n 3 H lder Inequality If M E 11322 are such that 1 E MP and Y E MP than XY E LIL and HXYIIL i KHAN Pmp sitiau 4 Iviinlmwski Inequality Let p E 1 m and X F E L13 Then X Yllr E I IXHp HY It f H WE rm Minkwaki inequality that far p E 1133 H ll de nas 3 mm fur LFP prmricied that as 115113 we iderntify madam variables that are equal 31mm Surely Tlmmef39maI MP is a 119mm vemr space far all 19 E 1 ca In the fn nwing unleSs atherwise mated we will TEStI i t lli SElVES ta 3 E 1 m Completeness of Lp Propusitien 15 The LPIfP spaces are Banach emcee The INF spaces are of in nite dimeneien unless there exiete e nite eelleetien Auk ef measurable sets with strictly positive meeeure such that any B E J is the mien ei sets fmm An eed measure eere sets The positive erthent ef HfPJ demented by Li Pi is the set ef all rendem variables X with X 3 U ae Prepeeition 5 Fee e E Lee Lille has an empty interim unless LHP is 0f nite deme eien 0n ihe other fiend LEW eiweye has a nenemp y interim Lp commodity space Prices positive linear functionals Proposition 739 A p ti imam funciimmi an Hip is commwas This leada us to exploring in more detail the dual of Um Proposition 3 Let Y E Um Then the functional M fnxon tmmwi is an element of Phiquot where l l i In addition if It 3 D 12 than if is a positive it39an functional and if Y9 2 ma than 95 is a strictly paganv5 linear functional Theorem 3 Riesz Representation Theorem Let be a continuous linearfzmc tionoi art Um where p 6 100 Then there exists a unique Y 6 BOA where 1 such that we ammable 39VX e we Corollary 1 The U bu spaces mthp 6 100 are re exive The next result is useful in establishmg the existence of optimal consumption plans Theorem 4 Let flip be a complete measure space with M9 lt 00 and let f L1Qfo gt R U 00 be convex and lower semicontinuous in the topology T of canvergenee m measure Then f attains a minimum on any nonempty comer subset of lef u that is Tclosed and normbounded

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