Semimartingales and stochastic integration Spring 2011 ∗ Sergio Pulido ChrisAlmost† Contents Contents 1 0 Motivation 3 1 Preliminaries 4 1.1 Reviewofstochasticprocesses . . . . . . . . . . . . . . . . . . . . . . . 4 1.2 Reviewofmartingales . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7 1.3 PoissonprocessandBrownianmotion . . . . . . . . . . . . . . . . . . 9 1.4 Lévyprocesses . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12 1.5 Lévymeasures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15 1.6 Localization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20 1.7 Integrationwithrespecttoprocessesoffinitevariation. . . . . . . . 21 1.8 Naïvestochasticintegrationisimpossible . . . . . . . . . . . . . . . . 23 ∗[email protected] †[email protected] 1 2 Contents 2 Semimartingalesandstochasticintegration 24 2.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24 2.2 Stabilitypropertiesofsemimartingales. . . . . . . . . . . . . . . . . . 25 2.3 Elementaryexamplesofsemimartingales . . . . . . . . . . . . . . . . 26 2.4 Thestochasticintegralasaprocess . . . . . . . . . . . . . . . . . . . . 27 2.5 Propertiesofthestochasticintegral . . . . . . . . . . . . . . . . . . . . 29 2.6 Thequadraticvariationofasemimartingale . . . . . . . . . . . . . . 31 2.7 Itô’sformula . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35 2.8 ApplicationsofItô’sformula . . . . . . . . . . . . . . . . . . . . . . . . 40 3 TheBichteler-DellacherieTheoremanditsconnexionstoarbitrage 43 3.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43 3.2 ProofsofTheorems3.1.7and3.1.8 . . . . . . . . . . . . . . . . . . . . 45 3.3 AshortproofoftheDoob-Meyertheorem . . . . . . . . . . . . . . . . 52 3.4 Fundamentaltheoremoflocalmartingales . . . . . . . . . . . . . . . 54 3.5 Quasimartingales,compensators,andthefundamentaltheoremof localmartingales . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 56 3.6 Special semimartingales and another decomposition theorem for localmartingales . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 58 3.7 Girsanov’stheorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61 4 Generalstochasticintegration 65 4.1 Stochasticintegralswithrespecttopredictableprocesses . . . . . . 65 Index 72 Chapter 0 Motivation Whystochasticintegrationwithrespecttosemimartingaleswithjumps? ◦ To model “unpredictable” events (e.g. default times in credit risk theory) oneneedstoconsidermodelswithjumps. ◦ A lot of interesting stochastic processes jump, e.g. Poisson process, Lévy processes. Thiscoursewillcloselyfollowthetextbook,Stochasticintegrationanddifferential equationsbyPhilipE.Protter,secondedition. Wewillnotcovereverychapter,and some proofs given in the course will differ from those in the text. The following numberscorrespondtosectionsinthetextbook. I Preliminaries. 1. Filtrations,stochasticprocesses,stoppingtimes,pathregularity,“func- tional”monotoneclasstheorem,optionalσ-algebra. 2. Martingales. 3. Poissonprocesses,Brownianmotion. 4. Lévyprocesses. 6. Localizationprocedureforstochasticprocesses. 7. Stieltjesintegration 8. Impossibilityofnaïvestochasticintegration(viatheBanach-Steinhaus theorem). II Semimartingalesandstochasticintegrals. 1–3. Definitionofthestochasticintegralwithrespecttoprocessesin(cid:76). 5. Propertiesofthestochasticintegral. 6. Quadraticvariation. 7. Itô’sformula. 8. StochasticexponentialandLévy’scharacterizationtheorem. III Bichteler-Dellacherietheorem. (NFLVR)impliesS isasemimartingale (NFLVR)andlittleinvestmentifandonlyifS isasemimartingale IV Stochasticintegrationwithrespecttopredictableprocessesandmartingale representationtheorems(i.e.marketcompleteness). Formoreinformationonthehistoryofthedevelopmentofstochasticintegra- tion,seethepaperbyProtterandJarrowonthattopic. 3 Chapter 1 Preliminaries 1.1 Review of stochastic processes The standard setup we will use is that of a complete probability space (Ω,(cid:70),(cid:80)) and a filtration (cid:70) = ((cid:70)t)0≤t≤∞ of sub-σ-algebras of (cid:70). The filtration can be thought of as the flow of information. Expectation (cid:69) will always be with respect to(cid:80)unlessstatedotherwise. Notation. Wewillusetheconvectionthat t,s,anduwillalwaysberealvariables, notincluding∞unlessitisexplicitlymentioned,e.g.{t|t ≥0}=[0,∞). Onthe otherhand,nandkwillalwaysbeintegers,e.g.{n:n≥0}={0,1,2,3,...}=:(cid:78). 1.1.1Definition. (Ω,(cid:70),(cid:70),(cid:80))satisfiestheusualconditionsif (i) (cid:70) containsallthe(cid:80)-nullsets. (ii) (cid:70)0isrightcontinuous,i.e.(cid:70) =(cid:84) (cid:70) . t s<t s 1.1.2Definition. A stopping time is a random time, i.e. a measurable function T :Ω→[0,∞],suchthat{T ≤t}∈(cid:70) forall t. t 1.1.3Theorem. Thefollowingareequivalent. (i) T isastoppingtime. (ii) {T <t}∈(cid:70) forall t>0. t PROOF: Assume T isastoppingtime. Thenforany t>0, (cid:91) (cid:95) {T <t}= {T ≤t− 1}∈ (cid:70) ⊆(cid:70) . n s t n≥1 s<t Conversely,sincethefiltrationisassumedtoberightcontinuous, (cid:92) (cid:92) {T ≤t}= {T <t+ 1n}∈ (cid:70)s=(cid:70)t+=(cid:70)t, n≥1 s>t so T isastoppingtime. (cid:131) 4 1.1. Reviewofstochasticprocesses 5 1.1.4Theorem. (i) If(Tn)n≥1isasequenceofstoppingtimesthen(cid:86)nTnand(cid:87)nTnarestopping times. (ii) If T andS arestoppingtimesthen T+S isastoppingtime. 1.1.5Exercises. (i) If T ≥S thenis T−S astoppingtime? (ii) ForwhichconstantsαisαT astoppingtime? SOLUTION: ClearlyαT neednotbeastoppingtimeifα<1. Ifα≥1then,forany t≥0, t/α≤t so{αT ≤t}={T ≤t/α}∈(cid:70)t/α⊆(cid:70)t andαT isastoppingtime. Let H be a stopping time for which H/2 is not a stopping time (e.g. the first hittingtimeofBrownianmotionatthelevel1). Take T =3H/2andS=H,both stoppingtimes,andnotethat T−S=H/2isnotastoppingtime. (cid:250) 1.1.6Definition. Let T be a stopping time. The σ-algebras of events before time T andeventsstrictlybeforetime T are (i) (cid:70) ={A∈(cid:70) :A∩{T ≤t}∈(cid:70) forall t}. T t (ii) (cid:70)T−=(cid:70)0∨σ{A∩{t<T}:t∈[0,∞),A∈(cid:70)t}. 1.1.7Definition. (i) AstochasticprocessX isacollectionof(cid:82)d-valuedr.v.’s,(Xt)0≤t<∞. Astochas- ticprocessmayalsobethoughtofasafunctionX :Ω×[0,∞)→(cid:82)d orasa randomelementofaspaceofpaths. (ii) X isadaptedifX ∈(cid:70) forall t. t t (iii) X is càdlàg (Xt(ω))t≥0 has left limits and is right continuous for almost all ω. X iscàglàdifinsteadthepathshaverightlimitsandareleftcontinuous. (iv) X is a modification of Y if (cid:80)[X (cid:54)= Y ] = 0 for all t. X is indistinguishable t t fromY if(cid:80)[X (cid:54)=Y forsome t]=0. t t (v) X−:=(Xt−)t≥0 whereX0−:=0andXt−:=lims↑tXs. (vi) Foracàdlàgprocess X,∆X :=(∆Xt)t≥0 istheprocessofjumpsof X,where ∆Xt :=Xt−Xt−. 1.1.8Theorem. (i) If X is a modification of Y and X and Y are left- (right-) continuous then theyareindistinguishable. (ii) IfΛ⊆(cid:82)d isopenandX iscontinuousfromtheright(càd)andadaptedthen T :=inf{t>0:X ∈Λ}isastoppingtime. t (iii) If Λ ⊆ (cid:82)d is closed and X is càd and adapted then T := inf{t > 0 : X ∈ t ΛorXt−∈Λ}isastoppingtime. (iv) If X iscàdlàgandadaptedand∆XT1T<∞=0forallstoppingtimes T then ∆X isindistinguishablefromthezeroprocess. PROOF: Readtheproofsofthesefactsasanexercise. (cid:131) 6 Preliminaries 1.1.9Definition. (cid:79) = σ(X : X isadaptedandcàdlàg) is the optional σ-algebra. AstochasticprocessX isanoptionalprocessifX is(cid:79)-measurable. 1.1.10Theorem(Débuttheorem). IfA∈(cid:79) thenT(ω):=inf{t:(ω,t)∈A},the débuttimeofA,isastoppingtime. Remark. Thistheoremrequiresthatthefiltrationisrightcontinuous. Forexample, supposethatT isthehittingtimeofanopensetbyaleftcontinuousprocess. Then youcanprove{T <t}∈(cid:70) foralltwithoutusingrightcontinuityofthefiltration, t butyoucannotnecessarilyprovethat{T =t}∈(cid:70) withoutit. Youneedto“look t intothefuture”alittlebit. 1.1.11Corollary. If X is optional and B ⊆(cid:82)d is a Borel set then the hitting time T :=inf{t>0:X ∈B}isastoppingtime. t 1.1.12Theorem. IfX isanoptionalprocessthen (i) X is((cid:70) ⊗(cid:66)([0,∞)))-measurableand (ii) XT1T<∞∈(cid:70)T foranystoppingtime T. Inparticular,(Xt∧T)t≥0isalsoanoptionalprocess,i.e.optionalprocessesare“sta- bleunderstopping”. 1.1.13Theorem(Monotoneclasstheorem). SupposethatH iscollectionofbounded(cid:82)-valuedfunctionssuchthat (i) H isavectorspace. (ii) 1Ω∈H,i.e.constantfunctionsareinH. (iii) If (fn)n≥0 ⊆ H is monotone increasing and f := limn→∞ fn (pointwise) is boundedthen f ∈H. (In this case H is called a monotone vector space.) Let M be a multiplicative col- lection of bounded functions (i.e. if f,g ∈ M then fg ∈ M). If M ⊆ H then H containsalltheboundedfunctionsthataremeasurablewithrespecttoσ(M). PROOF(OFTHEOREM1.1.12): UsetheMonotoneClassTheorem. Define M :={X :Ω×[0,∞)→(cid:82)|X iscàdlàg,adapted,andbounded} H:={X :Ω×[0,∞)→(cid:82)|X isboundedand(i)and(ii)hold} ItcanbecheckedthatH isamonotonevectorspace, M isamultiplicativecollec- tion, andσ(M)=(cid:79). Ifweprovethat M ⊆H thenwearedone. Let X ∈ M and define ∞ X(n):=(cid:88)k=1X2kn1[k2−n1,2kn). SinceX isrightcontinuous,X(n)→X pointwise. LetB beaBorelset. {X(n)∈B}=(cid:91)∞ (cid:110)X ∈B(cid:111)×(cid:20)k−1, k (cid:19)∈(cid:70) ⊗(cid:66)([0,∞)) k=1 2kn 2n 2n 1.2. Reviewofmartingales 7 ThisprovesthatX satisfies(i). Toprove(ii),let T beastoppingtimeanddefine (cid:168)k if k−1 ≤T < k T := 2n 2n 2n n ∞ if T =∞. Thenthe T ’sarestoppingtimesand T ↓T. n n (cid:91)∞ (cid:110) (cid:111) (cid:26) k (cid:27) {X ∈B}∩{T ≤t}= X ∈B ∩ T = ∈(cid:70) Tn n k=1 2kn n 2n t 2kn≤t ThisshowsthatX ∈(cid:70) . SinceX isrightcontinuous, T T n n XT1T<∞=nl→im∞XTn1Tn<∞. It can be shown that, since the filtration is right continuous, (cid:84)∞ (cid:70) =(cid:70) , so XT1T<∞∈(cid:70)T andX satisfies(ii). ThereforeX ∈H. n=1 Tn T (cid:131) If X is càdlàg and adapted then an argument similar to that in the previous proofshowsthatX|Ω×[0,t]∈(cid:70)t⊗(cid:66)([0,t])forall t,i.e.X isprogressivelymeasur- able. By a similar monotone class argument, it can be shown that every optional processisaprogressiveprocess. 1.1.14Definition. (cid:86) := σ(X : X isprogressivelymeasurable) is the progressive σ-algebra. 1.1.15Corollary. (cid:79) ⊆(cid:86). 1.2 Review of martingales 1.2.1Theorem. Let X bea(sub-, super-)martingaleandassumethat(cid:70)satisfies theusualconditions. ThenX hasarightcontinuousmodificationifandonlyifthe function t (cid:55)→ (cid:69)[X ] is right continuous. Furthermore, this modification has left t limitseverywhere. PROOF(SKETCH): TheprocessX(cid:101)t :=lims↓t Xs isthecorrectmodification. (cid:131) s∈(cid:81) 1.2.2Corollary. Everymartingalehasacàdlàgmodification, uniqueuptoindis- tinguishability. 1.2.3Theorem. LetX bearightcontinuoussub-martingalewithsup (cid:69)[X+]< t≥0 t ∞. ThenX∞:=limt→∞Xt existsa.s.andX∞∈L1. 1.2.4Definition. Acollectionofrandomvariables(Uα)α∈Aisuniformlyintegrable oru.i.if nl→im∞sαu∈pA(cid:69)[1|Uα|>nUα]=0. 8 Preliminaries 1.2.5Theorem. Thefollowingareequivalentforafamily(Uα)α∈A. (i) (Uα)α∈Aisu.i. (ii) supα∈A(cid:69)[|Uα|]<∞ and for all (cid:34) >0 there is δ >0 such that if (cid:80)[Λ]<δ thensupα∈A(cid:69)[1ΛUα]<(cid:34). (iii) (delaVallée-Poussincriterion)Thereisapositive,increasing,convexfunc- tionG on[0,∞)suchthatlimx→∞ G(xx) =∞andsupα∈A(cid:69)[G(|Uα|)]<∞. 1.2.6Theorem. ThefollowingareequivalentforamartingaleX. (i) (Xt)t≥0 isu.i. (ii) X isclosable,i.e.thereisanintegrabler.v. Z suchthatX =(cid:69)[Z|(cid:70) ]. t t (iii) X convergesin L1. (iv) X∞=limt→∞Xt a.s.andin L1 andX∞ closesX. Remark. (i) If X is u.i. then X is bounded in L1 (but not vice versa), so Theorem 1.2.3 implies that Xt → X∞ a.s. Theorem 1.2.6 upgrades this to convergence in L1,i.e.(cid:69)[|Xt−X∞|]→0. (ii) If X is closed by Z then (cid:69)[Z|(cid:70)∞] also closes X, where (cid:70)∞ := (cid:87)t≥0(cid:70)t. Furthermore,X∞=(cid:69)[Z|(cid:70)∞]. 1.2.7Example(Simplerandomwalk). Let(Zn)n≥1 bei.i.d.with 1 (cid:80)[Z =1]=(cid:80)[Z =−1]= . n n 2 Take(cid:70) :=σ{Z :k≤t}andX :=(cid:80)(cid:98)t(cid:99) Z . ThenX isnotaclosablemartingale. t k t k=1 k 1.2.8Example. Let M :=exp(B − 1t), the stochastic exponential of Brownian t t 2 motion,amartingale. Then M →0a.s.byTheorem1.2.3becauseitisapositive t valued martingale (hence −M is a sub-martingale with no positive part). But (cid:69)[|M |]=1forall t so M isnotu.i.byTheorem1.2.6. t 1.2.9Theorem. (i) If X is a closable (sub-, super-) martingale and S ≤ T are stopping times then(cid:69)[X |(cid:70) ]=X (≥,≤). T S S (ii) If X is a (sub-, super-) martingale and S ≤ T are bounded stopping times then(cid:69)[X |(cid:70) ]=X (≥,≤). T S S (iii) IfX isarightcontinuoussub-martingaleand p>1then (cid:18) p (cid:19) (cid:107)supX (cid:107) ≤ sup(cid:107)X (cid:107) . t≥0 t Lp p−1 t≥0 t Lp Inparticularif p=2then(cid:69)[sup X2]≤4sup (cid:69)[X2]. t≥0 t t≥0 t (iv) (Jensen’sinequality)Ifϕisaconvexfunction, Z isanintegrabler.v.,and(cid:71) isasub-σ-algebraof(cid:70) thenϕ((cid:69)[Z|(cid:71)])≤(cid:69)[ϕ(Z)|(cid:71)]. 1.3. PoissonprocessandBrownianmotion 9 1.2.10Definition. Let X be a process and T be a random time. The stopped processisXtT :=Xt1t≤T +XT1t>T,T<∞=XT∧t. If T isastoppingtimeandX iscàdlàgandadaptedthensoisXT. 1.2.11Theorem. (i) IfX isau.i.martingaleandT isastoppingtimethenXT isau.i.martingale. (ii) IfX isamartingaleand T isastoppingtimethenXT isamartingale. Thatis,martingalesare“stableunderstopping”. 1.2.12Definition. LetX beamartingale. (i) IfX ∈L2 forall t thenX iscalledasquareintegrablemartingale. t (ii) If(Xt)t∈[0,∞) isu.i.andX∞∈L2 thenX iscalledan L2-martingale. 1.2.13Exercise. Any L2-martingale is a square integrable martingale, but not conversely. SOLUTION: Let X be an L2-martingale. Then X∞ ∈ L2 and so, by the conditional versionofJensen’sinequality, (cid:69)[Xt2]=(cid:69)[((cid:69)[X∞|(cid:70)t])2]≤(cid:69)[(cid:69)[X∞2|(cid:70)t]]=(cid:69)[X∞2]<∞. Therefore X ∈ L2 forall t. Wehavealreadyseenthatthestochasticexponential t of Brownian motion is not u.i. (and hence not an L2-martingale) but it is square integrablebecausethenormaldistributionhasafinitevaluedmomentgenerating function. (cid:250) 1.3 Poisson process and Brownian motion 1.3.1Definition. Suppose that (Tn)n≥1 is a strictly increasing sequence of ran- dasosmoctiaimteedswwiitthh(TT1n)>n≥01.aT.sh.eTrhanedpormocetismseNTt =:=(cid:80)sun≥p1n≥11TnT≤ntisisththeeecxopulonstiionngtpimroec.esIsf T =∞a.s.thenN isacountingprocesswithoutexplosion. 1.3.2Theorem. A counting process is an adapted process if and only if T is a n stoppingtimeforalln. PROOF: If N isadaptedthen{t < Tn}={Nt <n}∈(cid:70)t forall t andall n. There- fore, for all n, {T ≤ t}∈(cid:70) for all t, so T is a stopping time. Conversely, if all n t n the T arestoppingtimesthen,forall t,{N ≤n}={t≤T }∈(cid:70) foralln. Since n t n t N takesonlyintegervaluesthisimpliesthatN ∈(cid:70) . (cid:131) t t 1.3.3Definition. AnadaptedprocessN iscalledaPoissonprocessif () N isacountingprocess. (i) N −N isindependentof(cid:70) forall0≤s<t<∞(independentincrements). t s s 10 Preliminaries (ii) Nt−Ns=(d) Nt−s forall0≤s<t<∞(stationaryincrements). Remark. ImplicitinthisdefinitionisthataPoissonprocessdoesnotexplode. The definition can be modified slightly to allow this possibility, and then it can be proved as a theorem that a Poisson process does not explode, but the details are verytechnicalandrelativelyunenlightening. 1.3.4Theorem. SupposethatN isaPoissonprocess. (i) N iscontinuousinprobability. (ii) N =(d)Poisson(λt)forsomeλ≥0,calledtheintensityorarrivalrateofN. In t particular,N hasfinitemomentsofallordersforall t. t (iii) (Nt−λt)t≥0 and((Nt−λt)2−λt)t≥0 aremartingales. (iv) If(cid:70)tN :=σ(Ns:s≤t)and(cid:70)N =((cid:70)tN)t≥0 then(cid:70)N isrightcontinuous. PROOF: Letα(t):=(cid:80)[Nt =0]for t≥0. Foralls<t, α(t+s)=(cid:80)[Nt+s=0] =(cid:80)[{Ns=0}∩{Nt+s−Ns=0}] non-decreasingandnon-negative =(cid:80)[Ns=0](cid:80)[Nt+s−Ns=0] independentincrements =(cid:80)[N =0](cid:80)[N =0] stationaryincrements s t =α(t)α(s) If t ↓ t then {N =0} (cid:37){N =0}, so α is right continuous and decreasing. It n t t follows that eithenr α ≡ 0 or α(t) = e−λt for some λ ≥ 0. By the definition of countingprocessN =0,soαiscannotbethezerofunction. 0 (i) Observethat,given(cid:34)>0small,foralls<t, (cid:80)[|Nt−Ns|>(cid:34)]=(cid:80)[|Nt−s|>(cid:34)] stationaryincrements =(cid:80)[Nt−s>(cid:34)] N isnon-decreasing =1−(cid:80)[Nt−s=0] N isintegervalued =1−e−λ(t−s) →0ass→t. Therefore N is left continuous in probability. The proof of continuity from therightissimilar. (ii) First we need to prove that limt→0 1t (cid:80)[Nt = 1] = λ. Towards this, let β(t):=(cid:80)[Nt ≥2]for t≥0. Ifwecanshowthatlimt→0β(t)/t=0thenwe wouldhave (cid:80)[N =1] 1−α(t)−β(t) lim t =lim =λ. t→0 t t→0 t It is enough to prove that limn→∞nβ(1/n) = 0. Divide [0,1] into n equal subintervals and let S be the number of subintervals with at least two ar- n rivals. ItcanbeseethatS =(d)Binomial(n,β(1/n))becauseofthestationary n andindependentincrementsof N. Inthedefinitionofcountingprocessthe