Introduction to the Mathematics of Lévy Processes

Transcription

1 Iroducio o he Mahemaics of Lévy Processes Kazuhisa Masuda Deparme of Ecoomics The Graduae Ceer, The Ciy Uiversiy of New York, 365 Fifh Aveue, New York, NY maxmasuda@maxmasudacom hp://wwwmaxmasudacom/ February Kazuhisa Masuda All righs reserved i

2 Absrac The goal of his sequel is o provide he foudaios of he mahemaics of Lévy processes for he readers wih udergraduae kowledge of sochasic processes as simple as possible The simpliciy is a key because, for he begiers such as fiace majors wihou he experiece i sochasic processes, some available books o Lévy processes are o accessible Lévy processes cosiue a wide class of sochasic processes whose sample pahs ca be coiuous, coiuous wih occasioally discoiuous, ad purely discoiuous Tradiioal examples of Lévy processes iclude a Browia moio wih drif ie he oly coiuous Lévy processes), a Poisso process, a compoud Poisso process, a jump diffusio process, ad a Cauchy process All of hese are well sudied ad well applied sochasic processes We defie ad characerize Lévy processes usig heorems such as he Lévy-Iô decomposiio ad he Lévy-Khichi represeaio ad i erms of heir ifiie divisibiliies ad he Lévy measures I he las decade ad i he field of quaiaive fiace, here was a explosio of lieraures modelig he log asse prices usig purely o-gaussia Lévy processes which are pure jump Lévy processes wih ifiie aciviy To raise a few examples of purely o-gaussia Lévy processes used i fiace, variace gamma processes, empered sable processes, ad geeralized hyperbolic Lévy moios We cover hese purely o-gaussia Lévy processes i he ex sequel wih a fiace applicaio This is because we like o keep his sequal as simple as possible wih he pourpose of providig he iroducory foudaios of he mahemaics of Lévy processes 005 Kazuhisa Masuda All righs reserved ii

3 Coes 1 Iroducio o he Mahemaics of Lévy Processes 1 Basics of Sochasic Processes 3 1 Fucio 3 11 Fucio 3 1 Lef Limi ad Righ Limi of a Fucio 3 13 Righ Coiuous Fucio ad Righ Coiuous wih Lef Limi RCLL) Fucio 4 14 Lef Coiuous Fucio ad Lef Coiuous wih Righ Limi LCRL) Fucio 5 15 Coiuous Fucio 6 16 Discoiuous Fucio 7 Sochasic Processes 11 1 Covergece of Radom Variables 1 Law of Large Numbers ad Ceral Limi Theorem 13 3 Iequaliies 15 3 Puig Srucure o Sochasic Processes Processes wih Idepede ad Saioary Icremes 17 3 Marigale 0 31 Defiio of Marigale 0 3 Example of Coiuous Marigale 1 33 Marigale Asse Pricig 3 34 Submarigales ad Supermarigales 4 33 Markov Processes Discree Time Markov Chais 5 33 Markov Processes 8 4 Sample Pah Properies of Sochasic Processes 8 41 Coiuous Sochasic Process 9 4 Righ Coiuous wih Lef Limi RCLL) Sochasic Processes Toal Variaio 31 3 Lévy Processes Defiiio of Lévy Processes 34 3 Ifiiely Divisible Radom Variable ad Disribuio Relaioship bewee Lévy Processes ad Ifiiely Divisible Disribuios Lévy-Khichi Represeaio 4 iii

4 35 Lévy-Iô Decomposiio of Sample Pahs of Lévy Processes Lévy Measure Classificaio of Lévy Processes I Terms of Gaussia or No I Terms of he Behavior of Lévy Measure I Terms of he Toal Variaio of Lévy Process I Terms of he Properies of Lévy Triple A,, γ ) by Sao I Terms of he Sample Pahs Properies of Lévy Processes Lévy Processes as a Subclass of Markov Processes Oher Impora Properies of Lévy Processes 6 4 Examples of Lévy Processes Browia Moio Defiiio of a Browia Moio Sample Pahs Properies of a Browia Moio Equivale Trasformaios of Sadard Browia Moio Characerisic Fucio of Browia Moio Browia Moio as a Subclass of Coiuous Marigale Browia Moio as a Subclass of Markov Processes 75 4 Poisso Process Expoeial Radom Variable 77 4 Poisso Radom Variable ad Poisso Disribuio Relaioship bewee he Sums of Idepede Expoeial Radom Variables ad Poisso Disribuio Poisso Process Properies of Poisso Process 8 46 Poisso Process as a Lévy Process Characerisic Fucio of Poisso Process Lévy Measure of Poisso Process Poisso Process as a Subclass of Markov Processes Poisso Process ad Marigales: Compesaed Poisso Process Compoud Poisso Process Compoud Poisso Process Properies of Compoud Poisso Process Compoud Poisso Process as a Lévy Process Characerisic Fucio of Compoud Poisso Process Lévy Measure of a Compoud Poisso Process 95 5 Sable Processes Sable Disribuios ad Sable Processes 96 5 Selfsimilar ad Broad-Sese Selfsimilar Sochasic Processes Relaioship bewee Sabiliy ad Broad-sese iv

5 Selfsimilariy for Lévy Processes More o Sable Processes Sabiliy Idex α Properies of Sable Disribuios ad Sable Processes wih he Sabiliy Idex 0< α < 105 Appedix 111 A1 Dirac s Dela Fucio 111 Bibliography 114 v

6 [1] Iroducio o he Mahemaics of Lévy Processes Firs of all, we have o say ha Lévy processes are ohig ew because heir properies were origially characerized by Paul Lévy i he 1930s Lévy processes are simply defied as sochasic processes 1) whose icremes are idepede ad saioary, ) hey are sochasically coiuous, ad 3) whose sample pahs are righ coiuous ad lef limi fucios of ime wih probabiliy 1 Thus, Lévy processes cosiue a wide class of sochasic processes whose sample pahs ca be coiuous, coiuous wih occasioally discoiuous, ad purely discoiuous Tradiioal examples of Lévy processes iclude a Browia moio wih drif ie he oly coiuous Lévy processes), a Poisso process, a compoud Poisso process, a jump diffusio process, ad a Cauchy process All of hese are well sudied ad well applied sochasic processes The probabiliy disribuios of all Lévy processes are characerized by ifiie divisibiliy I oher words, here is oe o oe correspodece bewee a ifiiely divisible disribuio ad a Lévy process For example, a Browia moio wih drif which is a Lévy process is a sochasic process geeraed by a ormal disribuio which is a ifiiely divisible disribuio Lévy-Iô decomposiio saes ha every sample pah of Lévy process ca be represeed as a sum of wo idepede processes which are each expressed usig a Lévy riple A,, γ ): Oe is a coiuous Lévy process ad he oher is a compesaed sum of idepede jumps Obviously, a coiuous Lévy process is a Browia moio wih drif Oe rick is ha he jump compoe has o be a compesaed sum of idepede jumps because a sum of idepede jumps a ime may o coverge Followig Lévy-Iô decomposiio, Lévy-Khichi represeaio gives he characerisic fucios of all ifiiely divisible disribuios I oher words, i gives he characerisic fucios of all processes whose icremes follow ifiiely divisible disribuios Lévy processes i erms of a Lévy riple A,, γ ) The Lévy measure of a Lévy process [0, ) ) is defied as a uique posiive measure o R which measures cous) he expeced average) umber of jumps of all sizes per ui of ime I oher words, i is a uique posiive measure o R which measures arrival rae of jumps of all sizes per ui of ime Noe ha by defiiio of he Lévy measure, all Lévy processes have fiie expeced umber of large jumps per ui of ime If a Lévy process has fiie expeced umber of small jumps per ui of ime ie a fiie iegral of ), he, i is said o be a fiie aciviy Lévy process such as a compoud Poisso process If a Lévy process has ifiie expeced umber of small jumps per ui of ime ie a ifiie iegral of ), he, i is said o be a ifiie aciviy Lévy process such as a gamma process Aoher impora characerizaio of Lévy processes is ha Lévy processes are sochasically coiuous Markov processes wih ime homogeeous ad spaially homogeeous rasiio fucios 1

7 The goal of his sequel is o provide he foudaios of he mahemaics of Lévy processes for he readers wih udergraduae kowledge of sochasic processes as simple as possible The simpliciy is a key because, for he begiers such as fiace majors wihou he experiece i sochasic processes, some available books o Lévy processes are o accessible from our ow experiece) For hose advaced readers who ied o lear Lévy processes usig heorems ad proofs approach, we highly recommed he excelle book by Sao 1999) which is by far he bes amog available books reaig Lévy processes The srucure of his sequel is as follows Secio provides readers wih miimal ecessary kowledge of sochasic processes We especially emphasize he sample pahs properies such as he coceps of coiuiy ad limis ad he oal variaio of sochasic processes Marigale ad Markov properies are also iroduced here Secio 3 defies Lévy processes, he, provides heorems such as he Lévy-Iô decomposiio ad he Lévy-Khichi represeaio Ifiie divisibiliy ad he Lévy measure of Lévy processes are discussed i deail alog wih various ways o caegorize Lévy processes Secio 4 gives hree radiioal examples of Lévy processes Browia moio, a Poisso process, ad a compoud Poisso process Secio 5 deals wih sable processes which are Lévy processes wih he broad-sese selfsimilariy To gai simpliciy, we eeded o limi he scope of he discussio i a couple of ways Oe is ha we oly deal wih oe dimesioal sochasic process i his sequel, o a d - dimesioal vecor of sochasic processes The oher is ha we oly rea he radiioal examples of Lévy processes, amely a Browia moio, a Poisso process, ad a compoud Poisso process I he las decade ad i he field of quaiaive fiace, here was a explosio of lieraures modelig he log asse prices usig purely o-gaussia Lévy processes which are pure jump Lévy processes wih ifiie aciviy To raise a few examples of purely o-gaussia Lévy processes used i fiace, variace gamma processes, empered sable processes, ad geeralized hyperbolic Lévy moios We cover hese purely o- Gaussia Lévy processes i he ex sequel wih a fiace applicaio

8 [] Basics of Sochasic Processes This chaper preses he fudameal coceps of fucios ad sochasic processes which are esseial o he udersadig of Lévy processes [1] Fucio [11] Fucio Defiiio 1 Fucio A fucio f : a f a) or f : A B o R uiquely maps relaes) a se of ipu values a A o a se of oupu values f a) B The domai of a fucio is he se A o which a fucio is defied ad he se of all acual oupus f a) B is called he rage of a fucio A fucio is a may-o-oe mappig ie o oe-o-may mappig) For example, a fucio f a) = a is a oe-o-oe mappig, f a) = a is a wo-o-oe mappig excep for a = 0, ad f a) = si π a) is a may-o-oe mappig fhal a Figure 1: Examples of a fucio f : a f a) a a si HπaL [1] Lef Limi ad Righ Limi of a Fucio Defiiio Lef limi ad Righ limi of a fucio A fucio f : a f a) o R has a lef limi f b ) a a poi a= b if f a ) approaches f b ) whe a approaches b from he below he lef-had side): lim a b f a) = f b ) A fucio f : a f a) o R has a righ limi f b ) a a poi a= b if f a) approaches f b ) whe a approaches b from he above righ-had side): 3

9 lim a b f a) = f b ) [13] Righ Coiuous Fucio ad Righ Coiuous wih Lef Limi RCLL) Fucio Defiiio 3 Righ coiuous fucio A fucio f o R is said o be righ coiuous a a poi a= b if i saisfies he followig codiios: 1) f b) is defied I oher words, a poi b is i he domai of a fucio f ) Righ limi of he fucio as a approaches b from he above righ had side) exiss, ie lim a b f a) = f b ) 3) f b ) = f b) Defiiio 4 Righ coiuous wih lef limi rcll) fucio A fucio f o R is said o be righ coiuous wih lef limi a a poi a= b if i saisfies he followig codiios: 1) f b) is defied I oher words, a poi b is i he domai of a fucio f ) Righ limi of he fucio as a approaches b from he above righ had side) exiss, ie lim a b f a) = f b ) Lef limi of he fucio as a approaches b from he below lef had side) exiss, ie lim a b f a) = f b ) 3) f b ) = f b) The above defiiios imply ha a rcll fucio is righ coiuous, bu he reverse is o rue I oher words, a rcll fucio is more resricive ha a righ coiuous fucio because a rcll fucio eeds lef limi This poi is illusraed i Figure Figure : Relaioship bewee rc fucio ad rcll fucio Cosider a piecewise cosa fucio defied as illusraed i Figure 3): 4

10 0 if a < 1 f a) = 1 if 1 a < 1) if a<3 The righ limi a a poi a = 1 is equal o he acual value of he fucio a a poi a = 1: f1 ) = f1) = 1, his meas f is righ coiuous a a poi a = 1 Bu he lef limi a a poi equal o he acual value of he fucio a a poi a = 1: f1 ) = 0 f1) = 1, a =1 is o his meas f is o lef coiuous a a poi a = 1 Therefore, his fucio is righ coiuous wih lef limi Ad he jump size is: f1 ) f1 ) = 1 0 = 1 Figure 3: Righ coiuous wih lef limi rcll) fucio [14] Lef Coiuous Fucio ad Lef Coiuous wih Righ Limi LCRL) Fucio Defiiio 5 Lef coiuous fucio A fucio f o R is said o be lef coiuous a a poi a= b if i saisfies he followig codiios: 1) f b) is defied I oher words, a poi b is i he domai of a fucio f ) Lef limi of he fucio as a approaches b from he below lef had side) exiss, ie lim a b f a) = f b ) 5

11 3) f b ) = f b) Defiiio 6 Lef coiuous wih righ limi lcrl) fucio A fucio f o R is said o be lef coiuous wih righ limi a a poi a= b if i saisfies he followig codiios: 1) f b) is defied I oher words, a poi b is i he domai of a fucio f ) Righ limi of he fucio as a approaches b from he above righ had side) exiss, ie lim a b f a) = f b ) Lef limi of he fucio as a approaches b from he below lef had side) exiss, ie lim a b f a) = f b ) 3) f b ) = f b) Cosider a piecewise cosa fucio defied as: 0 if a 1 f a) = 1 if 1< a ) if < a 3 The lef limi a a poi a = 1 is equal o he acual value of he fucio a a poi a = 1: f1 ) = f1) = 0, his meas f is lef coiuous a a poi a = 1 Bu he righ limi a a poi equal o he acual value of he fucio a a poi a = 1: f1 ) = 1 f1) = 0, a =1 is o his meas f is o righ coiuous a a poi a = 1 Therefore, his fucio is lef coiuous wih righ limi Ad he jump size is: f1 ) f1 ) = 1 0 = 1 [15] Coiuous Fucio Defiiio 7 Coiuous fucio A fucio f : a f a) o R is said o be coiuous a a poi a= b if i saisfies he followig codiios: 1) f b) is defied I oher words, a poi b is i he domai of a fucio f ) Righ limi of he fucio as a approaches b from he above righ had side) exiss, ie lim a b f a) = f b ) Lef limi of he fucio as a approaches b from he below lef had side) exiss, ie lim a b f a) = f b ) 3) f b ) = f b ) = f b) 6

12 I oher words, a coiuous fucio is a lef ad righ coiuous fucio which i ur meas ha a coiuous fucio is he mos resricive amog rc, rcll, ad coiuous fucios All he fucios i Figure 1 are coiuous Figure 4: Illusraio of a coiuous fucio Figure 5: Relaioship bewee rc, rcll, ad coiuous fucios [16] Discoiuous Fucio Defiiio 8 Discoiuous fucio A fucio f : a f a) o R is said o be discoiuous a a poi a= b called a poi of discoiuiy) if i fails o saisfy beig a coiuous fucio There are hree differe caegories of pois of discoiuiies Defiiio 9 A fucio wih removable discoiuiy sigulariy) A fucio f : a f a) o R is said o have a removable discoiuiy a a poi a= b if i saisfies he followig codiios: 7

13 1) f b ) is defied or f b) is o defied ) Lef limi lim a b f a) = f b ) exiss Righ limi lim a b f a) = f b ) exiss 3) f b ) = f b ) f b) This meas ha a removable discoiuiy a a poi a= b looks like a dislocaed poi as show by Figure 6 where he example is a fucio: a 5 if a 3 f a) = 5 if a = 3 This fucio has a lef limi which is equal o he righ limi a a poi a = 3: f3 ) = f 3 ) =, bu hese limis are o equal o he acual value ha his fucio akes a a poi a = 3: f3 ) = f3 ) = f3) = 5, which idicaes ha f is discoiuous a a poi a= b Figure 6: Example of a removable discoiuiy wih he defied discoiuiy poi f 3) = 5 Or, cosider a fucio: a 5 f a) =, a 5 which is udefied a a poi equal: a = 5 Bu is lef limi ad righ limi exis ad hey are 8

14 f5 ) = f5 ) = 10 Therefore, i is a fucio wih removable discoiuiy, oo Figure 7: Example of a removable discoiuiy wih he udefied discoiuiy poi f 5) Defiiio 10 A fucio wih discoiuiy of he firs kid jump discoiuiy) A fucio f : a f a) o R is said o have a jump discoiuiy a a poi a = b if i saisfies he followig codiios: 1) f b) is defied I oher words, a poi b is i he domai of a fucio f ) Lef limi f b ) exiss Righ limi f b ) exiss 3) f b ) f b ) The, he jump is defied by he amou f b ) f b ) Cosider a fucio: 1 if a > 1 f a) = 0 if a= 1 3) 1 if a < 1 This fucio has a lef limi -1 which is o equal o he righ limi 1 a a poi a =1: ad he jump size is: f1 ) = 1 f1 ) = 1, f1 ) f1 ) = 1 1) = 9

15 Figure 8: Example of a jump discoiuiy Defiiio 11 A fucio wih discoiuiy of he secod kid esseial discoiuiy) A fucio f : a f a) o R is said o have a esseial discoiuiy a a poi a= b if eiher or boh) of lef limi f b ) or righ limi f b ) does o exis The ypical example of a esseial discoiuiy give i mos exbooks is he fucio: si1/ a) if a 0 f a) =, 4) 0 if a = 0 which does o have boh lef limi f b ) ad righ limi f b ) as show by Figure fhal a Figure 9: Example of a esseial discoiuiy Figure 10 illusraes he relaioship bewee rcll, coiuous, ad discoiuous fucios 10

16 Figure 10: Relaioship bewee rcll, coiuous, ad discoiuous fucios [] Sochasic Processes A sochasic process is a collecio of radom variables: ), [0, T] where he idex deoes ime Noe ha we are ieresed i he coiuous ime sochasic process where he ime idex akes ay value i he ierval [0, T] or i could be a ifiie horizo) Discree ime sochasic process ca be defied usig a couable idex se N : N ) A sochasic process is defied o a filered probabiliy space Ω, F [0, T], P) where Ω is a arbirary se ad is a probabiliy measure o is called a filraio P F [0, T] F [0, T] which is a icreasig family of σ -algebras of a subse of Ω which saisfy for 0 s : F F s Iuiively speakig, a filraio is a icreasig iformaio flow abou progresses [0, T] ) We ca aleraively sae ha a real valued coiuous ime sochasic process is a radom fucio: :[0, T ] Ω R as ime 11

17 Afer he realizaio of he radomessω, a sample pah of A sochasic process F or F [0, T] ) -adaped if he value of [1] Covergece of Radom Variables ω): R or ω) : ω) [0, T] ) is defied as: is said o be oaicipaig wih respec o he filraio is revealed a ime for each [0, T] Defiiio 1 Poiwise covergece Le ω) ) N be a sequece of real valued radom variables o a space Ω, F, P) uder a sceario ie eve or radomess) ω Ω A sequece N ω) ) is said o coverge piwisely o a radom variable if: lim ω) = Poiwise covergece is he sroges oio of covergece because i requires covergece o a radom variable for all scearios samples)ω Ω, ie eve for hose scearios wih zero probabiliy Defiiio 13 Almos sure covergece Le ω) ) N be a sequece of real valued radom variables o a space Ω, F, P) uder a scearioω Ω A sequece N ω) ) is said o coverge almos surely o a radom variable if: ω ) P lim ) = = 1 Almos sure covergece is weaker ha poiwise covergece sice hose samples ω Ω wih o covergece lim ω) have zero probabiliy: ω ) ω ) P lim ) = P lim ) = 1 0 = 1 Almos sure covergece is used i he srog law of large umbers Almos sure covergece implies covergece i probabiliy which i ur implies covergece i disribuio Defiiio 14 Covergece i probabiliy Le N ) be a sequece of real valued radom variables o a space Ω, F, P) A sequece N ) is said o coverge i probabiliy o a radom variable if for everyε R : 1

18 ε ) lim P > = 0, or equivalely: ε ) lim P = 1 Iuiively speakig, covergece i probabiliy meas ha he probabiliy of geig closer o rises ad eveually coverges o 1) as we ake larger ad larger Covergece i probabiliy is used i he weak law of large umbers Covergece i probabiliy implies covergece i disribuio Defiiio 15 Covergece i mea square Le N ) be a sequece of real valued radom variables o a space Ω, F, P) A sequece ) mea square o a radom variable if for everyε R : ) lim E = 0 N is said o coverge i Covergece i mea square implies covergece i probabiliy followig Chebyshev s iequaliy Defiiio 16 Covergece i disribuio Weak covergece) Le N ) be a sequece of real valued radom variables o a space Ω,, P) said o coverge i disribuio o a radom variable z) = z) lim P P N F A sequece ) is if for z R : Loosely speakig, covergece i disribuio meas ha oly whe we ake sufficiely large, he probabiliy ha is i he ierval [ ab], approaches he probabiliy ha is i he ierval [ ab], Covergece i disribuio is he weakes defiiio of covergece i he sese ha i does o imply ay oher covergece bu implied by all oher oios of covergece lised above [] Law of Large Numbers ad Ceral Limi Theorem Defiiio 17 Weak law of large umbers Le 1,, 3 be iid radom variables from a disribuio wih mea µ ad variaceσ < Defie is sample mea as: 1 = 13

19 The, he sample mea for ay ε R Or equivalely: coverges i probabiliy o he populaio) mea µ : µ ε) lim P > = 0, µ ε) lim P < = 1 Defiiio 18 Srog law of large umbers Le 1,, 3 be iid radom variables from a disribuio wih mea µ < Defie is sample mea as: 1 = The, he sample mea for ay ε R coverges almos surely o he populaio) mea µ : µ ) P lim = = 1, Defiiio 19 Ceral limi heorem Le 1,, 3 be iid radom variables from a disribuio wih mea µ < ad variaceσ < Defie he sum of a sequece of radom variables as: We kow he followigs: The, iformally, he sum S = 1 [ ] = [ 1] [ ] [ ] =, [ ] [ ] [ ] [ ] E S E µ Var S = Var 1 Var Var = σ S mea µ ad variace σ as : lim P S b) = P Y b) coverges i disribuio o a ormal disribuio wih µ ) b 1 1 Y = πσ σ exp dy, 14

20 where Y is a ormal radom variable, ie Y N µ, σ ) Formal ceral limi heorem is a sadardizaio of he above iformal oe Defie a radom variable Z as: Z S µ = σ The, Z coverges i disribuio o he sadard ormal disribuio as : Z b) = Z ) lim P P b 1 1 = π b exp Z dz, where Z is he sadard ormal radom variable, ie Z N 0,1) [3] Iequaliies Defiiio 0 Markov s iequaliy for ayb R : Le be a oegaive radom variable The, [ ] E P b) b Proof b [ ] = P ) = P ) P ) E d d d 0 0 b This meas: E[ ] dp ) bdp ) = b dp ) = bp b) b b b Thus: [ ] E b b) P 15

21 Markov s iequaliy provides a upper boud of he probabiliy ha a oegaive radom variable is greaer ha a arbirary posiive cosa b by relaig a probabiliy o a expecaio A varia of Markov s iequaliy is called Chebyshev s iequaliy Defiiio 1 Chebyshev s iequaliy Le be a radom variable o R ie boh R ad R ) wih mea µ < ad variaceσ < The, for ay k R : Proof Sar wih Markov s iequaliy: σ P µ k) k P b) E[ ] Replace a radom variable wih a radom variable µ ) ad b wih k : which i ur idicaes: E µ ) σ P µ ) k ) =, k k b σ P µ k), k 1 P µ kσ) k Chebyshev s iequaliy provides bouds of radom variables from ay disribuios as log as heir meas ad variaces are kow For example, whe k = : 1 P µ σ) 4 1 P µ σ, µ σ) 4 1 P µ σ, µ σ) 4 3 P µ σ µ σ) 4 16

22 This ells us ha he probabiliy ha ay radom variable lies wihi wo sadard deviaios is a leas 75 Defiiio Cauchy-Schwarz s iequaliy Le ad Y be joily disribued radom variables o R wih each havig fiie variace The: Proof Refer o Abramowiz 1993) ) E[ Y ] E[ ] E[ Y ] [3] Puig Srucure o Sochasic Processes The purpose of ay mahemaical saisical) modelig regardless of he field is o fi less complicaed models o he highly complicaed real world pheomea as accurae as possible Mahemaical models are less complicaed i he sese ha hey make some simplifyig assumpios or pu some simplifyig srucures resricios) o he real world pheomea for he purpose of gaiig racabiliy There are some popular depedece srucures pu o sochasic processes which mahemaicias have developed ad used for years [31] Processes wih Idepede ad Saioary Icremes: Imposig Srucure o a Probabiliy Measure P Before givig he defiiio of processes wih idepede ad saioary icremes, we mus kow he basics Defiiio 3 Codiioal probabiliy The codiioal probabiliy of a arbirary eve A give a eve wih posiive probabiliy B is: Whe P B ) = 0, P A B) is udefied P A B) P AB) = P B) Defiiio 4 Saisical Sochasic) idepedece are said o be idepede, if ad oly if: Two arbirary eves A ad B P A B) = P A) P B) This defiiio of idepedece has wo advaages Firsly, i is symmeric i A ad B I oher words, a eve A s idepedece of a eve B implies a eve B s idepedece of a eve A Secodly, his defiiio holds eve whe a eve B has zero probabiliy, ie P B ) = 0 17

23 Whe wo arbirary eves A ad B are idepede, from he defiiio of a codiioal probabiliy: P A) P B) P AB) = = P A) P B) I is impora o oe ha his is a resul of saisical idepedece ad o he defiiio This is because he above equaio is o rue ie udefied) whe P B ) = 0 ad i is o symmeric i ha P A B ) = P A ) does o ecessarily imply P BA) = P B) Defiiio 5 Muual saisical idepedece Arbirary eves A1, A,, A are said o be muually idepede, if ad oly if: P A A A ) = P A) P A ) P A ) ) 1 1 Defiiio 6 Processes wih Idepede ad Saioary Icremes sochasic process wih values i R o a filered probabiliy space Ω, F, P) [0, T] [0, T] is said o be a process wih idepede ad saioary icremes if i saisfies he followig codiios: 1) Is icremes are idepede I oher words, for 1 < < < : A P ) = P ) P ) P ) ) Is icremes are saioary: ie for h R, h has he same disribuio as h I oher words, he disribuio of icremes does o deped o ie emporal homogeeiy) Cosider a icreasig sequece of ime 0 < 1 < < < < < u< where is he prese As a resul of idepede icremes codiio: P,,, ) u P u 1 0,,, ) 1 = P,,, ) P u ) P 1 0,,, ) 1 = P,,, ) = P ), u

24 which meas ha here is o correlaio probabilisic depedece srucure) o he icremes amog he pas, he prese, ad he fuure For example, idepede icremes codiio meas ha whe modelig a log sock price l S as a idepede icreme process, he probabiliy disribuio of a log sock price i year is idepede of he way he log sock price icreme has evolved over he years ie sock price dyamics), ie i does maer if his sock crushes or soars i year ): P l S l S,l S l S,l S l S,l S l S ) = Pl S l S ) Usig he simple relaioship ) for a icreasig sequece of ime 0 < 1 < < < < < u< : u u P,,,,, ) = P ),,,,, ) u 0 u = P ), u which holds because a icreme u ) is idepede of by defiiio ad he value of depeds o is realizaio ω ) This is a srog probabilisic srucure imposed o a sochasic process because his meas ha he codiioal probabiliy of he fuure value depeds oly o he previous realizaio ω ) ad o o he eire u pas hisory of realizaios, 0,,,, 1 ie called Markov propery which is discussed soo) Alhough his codiio seems oo srog, i imposes a very racable propery o he process Because if wo variables ad Y are idepede: E[ Y ] = E[ ] E[ Y ], Var[ Y ] = Var[ ] Var[ Y ], Cov[, Y ] = 0 ie Corr[, Y ] = 0) Saioary icremes codiio meas ha he disribuios of icremes h do o deped o he ime, bu hey deped o he ime-disace h of wo observaios ie ierval of ime) I oher words, he probabiliy desiy fucio of icremes does o chage over ime For example, if you model a log sock price l S as a process wih saioary icremes, he disribuio of icreme i year is he same as ha i year : l S l S d l S l S

25 There is o doub ha he above idepede ad saioary icremes codiios impose a srog srucure o a sochasic process [0, T] ), as a resul of hese resricios, he mea ad variace of he process is racable: E [ ] = µ 0 µ 1, Var[ ] = σ σ, 0 1 where µ 0 = E [ 0], µ 1 = E [ 1] µ 0, σ 0 = E[ 0 µ 0) ], ad σ 1 = E[ 1 µ 1) ] σ0 [3] Marigale: Srucure o Codiioal Expecaio [31] Defiiio of Marigale Origially, he word marigale comes from a Frech acroym of a gamblig sraegy Imagie a coi flip gamble i which you wi if a head urs up ad you lose if a ail urs up Marigale sraegy requires a gambler o double his be afer every loss Followig marigale sraegy, a gambler ca recover all he losses he made ad ed up wih a iiial amou of his wealh plus a iiial be Table 1 gives a sample pah of a marigale sraegy i which a gambler iiially ows $00 of wealh, sar beig wih a sake of $, ad due o his bad luck his firs wi comes a he seveh rial As you ca see, he basically eds up where he sared, ie his iiial wealh of $00 plus a iiial be of $) Thus, a marigale sraegy ells ha afer gamblig may hours a gambler gais ohig loses ohig) ad his wealh remais cosa o average Table 1 Marigale Gamblig Sraegy Trial Resul Loss Loss Loss Loss Loss Loss Wi Be $ $4 $8 $16 $3 $64 $18 Ne Gai -$ -$4 -$8 -$16 -$3 -$64 $18 Wealh $00 $198 $194 $186 $170 $138 $74 $0 I probabiliy heory, a sochasic process is said o be a marigale if is sample pah has o red Formally, a marigale is defied as he follows Defiiio 7 Marigale Cosider a filered probabiliy space Ω, F [0, T], P) A rcll sochasic process ) is said o be a marigale wih respec o he filraio F [0, T ] ad uder he probabiliy measure P if i saisfies he followig codiios: 1) is oaicipaig ) E[ ] < for [0, T] Fiie mea codiio 3) E [ F ] = for u> u 0

26 I oher words, if a sochasic process is a marigale, he, he bes predicio of is fuure value is is prese value Noe ha he defiiio of marigale makes sese oly whe he uderlyig probabiliy measure ad he filraio F ) have bee specified P [0, T ] The fudameal propery of a marigale process is ha is fuure variaios are compleely upredicable wih he filraio : F u> 0, Ex [ x F] = Ex [ F] Ex [ F ] = x x= 0 u u Fiie mea codiio is ecessary o esure he exisece of he codiioal expecaio [3] Example of Coiuous Marigale: Sadard Browia Moio Le [0, ) B ) be a sadard Browia moio process defied o a filered probabiliy space Ω, F [0, ), P ) The, B [0, )) is a coiuous marigale wih respec o he filraio F ad he probabiliy measure P Proof [0, ) By defiiio, B [0, )) is a oaicipaig process ie F [0, ) - adaped process) wih he fiie mea E[ B ] = 0< for [0, ) For 0 u < : u Bu B db v = Usig he fac ha a Browia moio is a oaicipaig process, ie EB [ F ] = B: or i oher words: u E[ Bu B F] = E[ Bu F] E[ B F] = E[ B dbv F] B u EB [ u B F] = EB [ F] E[ dbv F] B EB [ B F ] = B 0 B= 0, u EB [ F u u ] = EB [ db F ] = EB [ F ] E [ db F ] = B 0 u v v EB [ F ] = B, u which is a marigale codiio 1

27 Le [0, ) B ) be a sadard Browia moio process defied o a filered probabiliy space Ω, F [0, ), P ) The, a Browia moio wih drif [0, ) ) µ σ B [0, ) ) is o a coiuous marigale wih respec o he filraio F [0, ) ad he probabiliy measure P Proof By defiiio, [0, )) is a oaicipaig process ie F [0, ) - adaped process) wih he fiie mea E [ ] = E[ µ σ B] = µ < for [0, ) ad µ R For 0 u< : u u = v d Usig he fac ha a Browia moio wih drif is a oaicipaig process, ie E [ F ] = : E [ F u u ] = E [ d F ] = E [ F ] E [ d F ] u v v E [ F ] = µ u ), u which violaes a marigale codiio Bu oe way o rasform omarigales io marigales is o make he process drifless I oher words, elimiaig he red of he process which is someimes called a deredig Cosider he followig example A dereded Browia moio wih drif defied as: µ ) µ σ B µ ) σb ), [0, ) [0, ) [0, ) is a coiuous marigale wih respec o he filraio F [0, ) ad he probabiliy measure P Proof For 0 u< : E [ µ u F ] = E [ µ ) u d µ u dv ) F ] u v

28 E [ u µ uf ] = E [ µ ) F ] E [ d dv v µ ) F ] E [ µ uf ] = µ µ u ) µ u ) u E [ µ uf ] = µ, u u u which saisfies a marigale codiio [33] Marigale Asse Pricig Mos of fiacial asse prices are o marigales because hey are o compleely upredicable ad mos fiacial ime series have reds Cosider a sock price process { S;0 T} o a filered probabiliy space Ω, F [0, T], P) ad le r be he risk-free ieres rae I a small ime ierval, risk-averse ivesors expec S o grow a some posiive rae This ca be wrie as uder acual probabiliy measure P : P E [ S F ] S > This meas ha a sock price S is o marigale uder P ad wih respec o F To be more precise, risk-averse ivesors expec S o grow a a rae greaer ha r because a sock is risky: P r E [ e S F ] S > The sock price discoued by he risk-free ieres rae P ad wih respec o F r e S is o marigale uder r How ca we cover a discoued sock price e S io a marigale? Firs approach is o elimiae he red The red i his case is he risk premium π which risk-averse ivesors demad for bearig exra amou of risk If we ca esimaeπ correcly, a r discoued sock price e S ca be covered io a marigale by deredig: P E [ e e S F] = E [ e S F = S π r P r π) ] Bu his approach ivolves he raher difficul job of esimaig he risk premium π ad is o used i quaiaive fiace Marigale asse pricig uses he secod approach o cover o-marigales io marigales by chagig he probabiliy measure We will ry o fid a equivale probabiliy measure Q called risk-eural measure) uder which a discoued sock price becomes marigale: Q r E [ e S F ] = S 3

29 [34] Submarigales ad Supermarigales Defiiio 8 Submarigale Cosider a filered probabiliy space Ω, F, P) A rcll sochasic process F ) [0, T ] ad uder he probabiliy measure 1) is oaicipaig ) E[ ] < for [0, T] Fiie mea codiio 3) E [ F ] for u> u is said o be a submarigale wih respec o he filraio P if i saisfies he followig codiios: [0, T] Iuiively, a submarigale is a sochasic process wih a posiive upward) red A submarigale gais or grows o average as ime progresses Defiiio 9 Supermarigale Cosider a filered probabiliy space Ω, F, P) A rcll sochasic process ) [0, T ] [0, T] is said o be a supermarigale wih respec o he filraio F ad uder he probabiliy measure P if i saisfies he followig codiios: 1) is oaicipaig ) E[ ] < for [0, T] Fiie mea codiio 3) E [ F ] for u> u Iuiively, a supermarigale is a sochasic process wih a egaive dowward) red A supermarigale loses or declies o average as ime progresses By defiiio, ay marigale is a submarigale ad a supermarigale Figure 11: Relaioship amog marigales, submarigales, ad supermarigales 4

30 [33] Markov Processes: Srucure o Codiioal Probabiliy This secio gives a brief iroducio o a class of sochasic processes called Markov processes which impose a resricio o he codiioal probabiliies This differs from marigales which impose a srucure o codiioal expecaios [331] Discree Time Markov Chais Defiiio 30 Discree ime Markov chai Cosider a discree ime sochasic process ) N ie = 0,1,, ) defied o a filered probabiliy space Ω, F, P N ) which akes values i a couable or a fiie se E called a sae space of he process A realizaio is said o be i sae i E a ime if = i A E -valued discree ime Markov chai is a sochasic process which saisfies for N ad i, j E: P = j,,,, = i) = P = j = i) This is called a Markov propery Markov propery meas ha he probabiliy of a radom variable 1 a ime 1 omorrow) beig i a sae j codiioal o he eire hisory of he sochasic process 0, 1,,, ) is equal o he probabiliy of a radom variable 1 a ime 1 omorrow) beig i a sae j codiioal oly o he value of a radom variable a ime oday) I oher words, he hisory sample pah) of he sochasic process,,,, ) is of o imporace i ha he way his 0 1 sochasic process evolved or he dyamics 1 0, 1,) does o mea a hig i erms of he codiioal probabiliy of he process The oly facor which iflueces he codiioal probabiliy of a radom variable 1 a ime 1 omorrow) is he sae of a radom variable a ime oday) The probabiliy P = 1 j = ) i which is a codiioal probabiliy of movig from a sae i a ime o a sae j a ime 1 is called a oe sep rasiio probabiliy I he geeral cases, rasiio probabiliies are depede o he saes ad ime such ha m N : P = j = i) P = j = i ) 1 m 1 m Whe rasiio probabiliies are idepede of ime, discree ime Markov chais are said o be ime homogeeous or saioary Defiiio 31 Time homogeeous saioary) discree ime Markov chai Cosider a discree ime sochasic process ) N ie = 0,1,, ) defied o a filered probabiliy space Ω, F, P N ) which akes values i a couable or a fiie se E called a sae space of he process A realizaio is said o be i sae i E a ime 5

31 if = i A E -valued ime homogeeous discree ime Markov chai is a sochasic process which saisfies for N ad i, j E: P = j,,,, = i) = P = j = i) = P = j = i) = P j i) 1 0 I oher words, rasiio probabiliies do o deped o ime ad oly deped o rasiio saes from i o j A marix of rasiio probabiliies P = P ji ) is called i, j E a rasiio probabiliy marix: P0 0) P1 0) P 0) P3 0) P0 1) P1 1) P 1) P3 1) P ji) = i, j E P0 ) P1 ) P ) P3 ) P0 i) P1 i) P i) P3 i) Trasiio probabiliies P j i) saisfy he followig codiios: 1) P ji) 0 for i, j E ) P ji) = 1 for i E j E Codiio ) guaraees he occurrece of a rasiio icludig a case i which he sae remais uchaged Proposiio 1 Defiig a discree ime Markov chai A E -valued geeral discree ime Markov chai ) N is compleely defied if i saisfies he followig codiios: 1) All rasiio probabiliies P 1 = i 1 = i) are kow ) The probabiliy disribuio of he iiial ie ime 0) sae of he Markov chai = i ) = P is kow P Proof Cosider obaiig he joi probabiliy disribuio of a E -valued geeral discree ime Markov chai ) From he defiiio of a codiioal probabiliy: N 6

32 P = i, = i, = i,, = i ) = P = i = i, = i, = i,, = i ) P = i, = i, = i,, = i ) Sice ) N is a Markov chai: P = i 0 = i0, 1 = i1, = i,, 1 = i 1) = P = i 1 = i 1) i i 1 = P ) Joi probabiliy ca be calculaed as: P = i, = i, = i,, = i ) = P i i 1 P 0 = i0 1 = i1 = i 1 = i 1 ),,,, ) = P i i ) P = i = i, = i, = i,, = i ) P = i, = i, = i,, = i ) P i i 1 P i 1 i P 0 i0 1 i1 i = i = ) ) =, =, =,, ) = P i i 1) P i 1 i ) P i i1) P i1 i0) P 0 Cosider a rasiio probabiliy of a ime homogeeous discree ime Markov chai ) N from a sae i a ime k ie k = i ) o a sae j a ime k This is called a -sep rasiio probabiliy ad expressed as: P = = ) = P = = ) = P i) ) k j k i j 0 i j Proposiio sep rasiio probabiliy marix a special case of Chapma- Kolmogorov equaio) Cosider a ime homogeeous discree ime Markov chai ) N defied o a filered probabiliy space Ω, F N, P ) which akes values i a couable or a fiie se E called a sae space of he process The, is -sep rasiio probabiliy marix from a sae i a ime k ie k = i ) o a sae j a ime k is give by for k, N ad i, j E: P ) ) ) j i ) = P j i ) = P y v i ) P z j v ) = P y v i ) P z j v ) v E v E where y z = ad P 0) j i) is defied as: Proof P 0) 1 for i= j ji) = 0 for i j 7

33 Whe =1: P 1) j i) = P j i ) Whe = : P ) j i) = P v i) P j v) v E By iducio: 1) P j i) = P v i) P j v) v E Oe ieresig opic abou his sep rasiio probabiliy marix is is asympoic behavior as As becomes larger, he iiial sae i becomes less impora ad i he limi as, P j i) is idepede of i We recommed Karli ad Taylor 1975) for more deails [33] Markov Processes Defiiio 3 Markov Processes Coiuous ime Markov chais) Cosider a defied o a filered probabiliy space coiuous ime sochasic process [0, T] ) Ω, F, P) [0, T] which akes values i for simpliciy) called a sae space of he process is said o be a ime homogeeous Markov process if for h R ad i, j N : [0, T] ) N P j i) = P = j F ) = P = j = i) h h h Markov propery meas ha he probabiliy of a radom variable ha ime h omorrow) beig i a sae j codiioal o he eire hisory of he sochasic process F [0, ] [0, ] is equal o he probabiliy of a radom variable ha ime h omorrow) beig i a sae j codiioal oly o he value of a radom variable a ime oday) I oher words, he hisory sample pah) of he sochasic process is of o F[0, ] imporace i ha he way his sochasic process evolved or he dyamics does o mea a hig i erms of he codiioal probabiliy of he process We discuss Markov processes more i deail i secio 38 [4] Sample Pah Properies of Sochasic Processes 8

34 [41] Coiuous Sochasic Process I his secio we give he formal defiiio of he coiuiy of a sample pah of a sochasic process Noe ha he coiuiy of pah ad coiuiy of ime are differe subjecs I oher words, a coiuous ime sochasic process does o imply coiuous sochasic process For example, a Poisso process is a coiuous ime sochasic process, bu i has discoiuous sample pahs There are differe oios of coiuiy of sample pahs which use differe oios of covergece of radom variables we saw i secio 1 Defiiio 33 Coiuous i mea square A real valued sochasic process [0, T] ) o a filered probabiliy space Ω, F [0, T], P) is said o be coiuous i mea square if for [0, T] : lim E[ ] = 0 s s Coiuiy i mea square implies coiuiy i probabiliy followig Chebyshev s iequaliy Defiiio 34 Coiuous i probabiliy o a filered probabiliy space Ω, F, P) for [0, T] ad every ε R : [0, T] A real valued sochasic process ) [0, T] is said o be coiuous i probabiliy if s ε ) lim P > = 0, s or equivalely: s ε ) lim P = 1 s Iuiively speakig, coiuiy i probabiliy meas ha he probabiliy of closer o rises ad eveually coverges o 1) as s approaches s geig For example, a Browia moio process is coiuous i mea square ad coiuous i probabiliy alhough is proof is o ha easy Bu i urs ou ha he above defiiios of coiuiy are oo loose because a Poisso process ca be prove o be coiuous i mea square ad probabiliy cosul Karli ad Taylor 1975) for deails) Therefore, a more sric defiiio of coiuiy is used for he defiiio of a coiuiy of a sample pah of a sochasic process Defiiio 35 Coiuous sochasic process A real valued oaicipaig sochasic process [0, T] ) o a filered probabiliy space Ω, F [0, T], P) is said o be 9

35 almos surely) coiuous if a sample pah of he process [0, T] ω)) is almos surely a coiuous fucio for [0, T] I oher words, a sample pah of he process T ω)) saisfies for [0, T] : [0, ] 1) Righ limi of he process as s approaches from he above righ had side) exiss, ie lim s = Lef limi of he process as s approaches from he below s, s> lef had side) exiss, ie lim s = s, s< = = ) This meas ha a coiuous sochasic process is a righ coiuous ad lef coiuous sochasic process [4] Righ Coiuous wih Lef Limi RCLL) Sochasic Processes Defiiio 36 Righ coiuous wih lef limi rcll) sochasic processes A real valued oaicipaig sochasic process ) o a filered probabiliy space [0, T] Ω, F, P) is said o be a rcll sochasic process if for [0, T] : [0, T] 1) Righ limi of he process as s approaches from he above righ had side) exiss, ie lim s = Lef limi of he process as s approaches from he below s, s> lef had side) exiss, ie lim s = s, s< ) = I oher words, oly he righ coiuiy is eeded his allows jumps) Apparely, a coiuous sochasic process implies a rcll sochasic process bu he reverse is o rue) Wha we ecouer i fiace lieraures are all rcll sochasic processes for he modelig of sock price dyamics Rcll processes iclude jump discoiuous process such as Poisso processes ad ifiie aciviy Lévy processes Esseially discoiuous processes are useless i fiace because hey do have eiher or boh) of he lef limi or he righ limi 30

36 Figure 1: Relaioship bewee rcll, coiuous, ad jump discoiuous processes A) A coiuous sochasic process B) A jump discoiuous sochasic process Figure 13: Examples of rcll sochasic processes Suppose is a discoiuiy poi The jump of he sochasic process a is defied as: = A rcll oaicipaig sochasic process [0, T] ) ca have a fiie umber of large jumps ad couable umber possibly ifiie) of small jumps [43] Toal Variaio Defiiio 37 Toal variaio of a fucio Le f x) be a bouded fucio defied i he ierval[ ab], : f x):[ a, b] R The ierval ca be ifiie, ie[, ] Cosider pariioig he ierval [ ab], wih he pois: 31

37 a= x0 < x1 < x x 1 < x = b The, he oal variaio of a fucio f x) is defied by: i i 1, i= 1 T f) = sup f x ) f x ) where sup idicaes a supremum leas upper boud) Defiiio 38 Fucio of fiie variaio A fucio f x ) o he ierval [ ab, ] is said o be a fucio of fiie variaio, if is oal variaio o he ierval [ ab], is fiie: T f) = sup f x ) f x ) < i i 1 i= 1 Proposiio 3 Every bouded icreasig or decreasig fucio is of fiie variaio o he ierval[, ab] Proof Cosider a icreasig fucio f x ) o he ierval[ ab, ] By is defiiio, for i : f xi) f xi 1) 0 Ad: { } { 0 } { } T f) = sup f x ) f x ) f x ) f x ) f x ) f x ) f x ) f x0 ) T f) = sup f x ) f x ) T f) = sup f b) f a), which is fiie because f x) is bouded: < f a), f b) < Defiiio 39 Toal variaio of a sochasic process Cosider a real valued sochasic process [0, T] ) o a filered probabiliy space Ω, F [0, T], P) Pariio he ime ierval [0, T ] wih he pois: = < < < < = T 3

38 The, he oal variaio of a sochasic process ) o he ime ierval [0, T ] is defied by: [0, T] i i 1, i= 1 T ) = sup ) ) where sup idicaes a supremum leas upper boud) Defiiio 40 Sochasic process of fiie variaio A real valued sochasic process [0, T] ) o a filered probabiliy space Ω, F [0, T], P) o he ierval [0, T ] is said o be a sochasic process of fiie variaio, if he oal variaio o he ierval [0, T ] of a sample pah of he process is fiie wih probabiliy 1: P T ) = sup ) ) < ) = 1 i= 1 i i 1 33

39 [3] Lévy Processes I his chaper some heorems ad proposiios are preseed wihou proofs This is obviously because i is o he goal of his sequel o prove heorems ad some proofs are beyod wha we eed while cosumig oo may pages Bu for hose iquisiive readers, we provide he iformaio abou where o look for more deails of he subjecs ad heir proofs Our goal is o prese he foudaios of he mahemaics of Lévy processes as simple as possible [31] Defiiio of Lévy Processes Defiiio 31 Lévy processes A real valued sochasic process ) o a filered probabiliy space Ω, F, P) is said o be a Lévy process o R if i saisfies he followig codiios: [0, ) [0, ) 1) Is icremes are idepede I oher words, for 0 1 < < < < : P ) = P ) ) ) ) P 0 1 P 0 P 1 1 ) Is icremes are saioary ime homogeeous): ie for h 0, h has he same disribuio as h I oher words, he disribuio of icremes does o deped o 3) P = 0) = 1 The process sars from 0 almos surely wih probabiliy 1) 0 4) The process is sochasically coiuous: ε > 0, lim P ε ) = 0 5) Is sample pah rajecory) is rcll almos surely Nex, we describe wha each codiio implies Cosider a icreasig sequece of ime 0 < 1 < < < < < u< where is he prese As a resul of idepede icremes codiio: h 0 h P,,, ) u P u 1 0,,, ) 1 = P,,, ) P u ) P 1 0,,, ) 1 = P,,, ) = P ), u which meas ha here is o correlaio probabilisic depedece srucure) o he icremes amog he pas, he prese, ad he fuure 34

40 For example, idepede icremes codiio meas ha whe modelig a log sock price l S as a idepede icreme process, he probabiliy disribuio of a log sock price i year is idepede of he way he log sock price icreme has evolved over he years ie sock price dyamics), ie i does maer if his sock crushes or soars i year ): P l S l S,l S l S,l S l S,l S l S ) = Pl S l S ) Usig he simple relaioship ) for a icreasig sequece of ime 0 < 1 < < < < < u< : u u P,,,,, ) = P ),,,,, ) u 0 1 u 0 1 = P ), u which holds because a icreme u ) is idepede of by defiiio ad he value of depeds o is realizaio ω ) This is a srog probabilisic srucure imposed o a sochasic process because his meas ha he codiioal probabiliy of he fuure value depeds oly o he previous realizaio ω ) ad o o he eire u pas hisory of realizaios, 0,,,, 1 ie called Markov propery which is discussed soo) Alhough his codiio seems oo srog, i imposes a very racable propery o he process Because if wo variables ad Y are idepede: E[ Y ] = E[ ] E[ Y ], Var[ Y ] = Var[ ] Var[ Y ], Cov[, Y ] = 0 ie Corr[, Y ] = 0) Saioary icremes codiio ) meas ha he disribuios of icremes h do o deped o he ime, bu hey deped o he ime disace h of wo observaios ie ierval of ime) I oher words, he probabiliy desiy fucio of icremes does o chage over ime For example, if you model a log sock price l S as a process wih saioary icremes, he disribuio of a log sock price icreme i is he same as ha i : l S l S dl S l S Processes saisfyig he codiios 1) ad ) are called processes wih idepede ad saioary icremes Idepede icremes codiio is a resricio o he 35

41 probabilisic depedece srucure of icremes amog he pas, prese, ad fuure Saioary icremes codiio is a resricio o he shape of he disribuio of icremes amog he pas, prese, ad fuure The codiio 4) which is implied by he codiios ), 3), ad 5)) does o imply he coiuous sample pahs of he process I meas ha if we are a ime, he probabiliy of a jump a ime is zero because here is o uceraiy abou he prese Jumps occur a radom imes This propery is called sochasic coiuiy or coiuiy i probabiliy which we saw i secio 4 Rcll codiio 5) does o eed o be imposed This is because a real valued Lévy process i law which is a process saisfyig codiios 1), ), 3), ad 4) is modified o a Lévy process which saisfies he codiios 1), ), 3), 4), ad 5) heorem 115 of Sao 1999)) I oher words, he codiio 5) resuls from he codiios 1), ), 3), ad 4) hrough a heorem Defiiio 3 Righ coiuous wih lef limi rcll) sochasic process A real valued oaicipaig sochasic process ) o a filered probabiliy space [0, T] Ω, F, P) is said o be a rcll sochasic process if for [0, T] : [0, T] 1) Righ limi of he process as s approaches from he above righ had side) exiss, ie lim s = Lef limi of he process as s approaches from he below s, s> lef had side) exiss, ie lim s = s, s< ) = As you ca see, he fac ha lef coiuiy is o eeded allows he process o have jumps A coiuous sochasic process implies a rcll sochasic process bu he reverse is o rue All sochasic processes used i fiace lieraures for he modelig of asse price dyamics are rcll sochasic processes Rcll processes iclude jump discoiuous process such as Poisso processes ad ifiie aciviy Lévy processes Esseially discoiuous processes are useless i fiace because hey do have eiher or boh) of he lef limi or he righ limi 36

42 Figure 31: Relaioship bewee rcll, coiuous, ad jump discoiuous processes A) A coiuous sochasic process B) A jump discoiuous sochasic process Figure 3: Examples of rcll sochasic processes Suppose is a discoiuiy poi The jump of he sochasic process a is defied as: = A rcll oaicipaig sochasic process [0, T] ) ca have a fiie umber of large jumps ad couable umber possibly ifiie) of small jumps We saw he defiiio of a Lévy process Nex we discuss ifiie divisibiliy of a disribuio I urs ou ha we cao separae Lévy processes from ifiiely divisible disribuios because Lévy processes are geeraed by ifiiely divisible disribuios [3] Ifiiely Divisible Radom Variable ad Disribuio Defiiio 33 Ifiiely divisible radom variable ad disribuio A real valued radom variable wih he probabiliy desiy fucio P x) is said o be ifiiely divisible if for N here exis iid radom variables 1,,, saisfyig: 37

43 d1 P x) is said o be a ifiiely divisible disribuio Defiiio 34 Characerisic fucio Le be a radom variable wih is probabiliy desiy fucio P x) A characerisic fucio φ ω ) wih ω R is defied as he Fourier rasform of he probabiliy desiy fucio P x) usig Fourier rasform parameers ab, ) = 1,1): ) [ )] i ω x iωx φω F P x e P x) dx E[ e ] 31) I erms of a characerisic fucio, ifiie divisibiliy is defied as follows Proposiio 31 Ifiiely divisible radom variable ad disribuio A real valued radom variable wih he probabiliy desiy fucio P x) ad he characerisic fucio φ ω ) is said o be ifiiely divisible if for N here exis iid radom variables 1,,, each wih a characerisic fucio φ ω ) such ha: i φ ω) = φ ω)) or i φ ω = φ ω 1/ )) ) i P x) is said o be a ifiiely divisible disribuio Proof Cosul Applebaum 004) secio 1 Examples of ifiiely divisible disribuios iclude ormal disribuios o R, gamma disribuios o R, α -sable disribuios o R, Poisso disribuios o R, compoud Poisso disribuios o R, geomeric disribuios o R, egaive biomial disribuios o R, expoeial disribuios o R From Sao 1999), probabiliy measures wih bouded suppors eg uiform ad biomial disribuios) are o ifiiely divisible Suppose ha a radom variable Y is draw from a ormal disribuio wih he mea µ ad he variaceσ : 1 1 y µ ) P y) = exp πσ σ 38