Part II CONTINUOUS TIME STOCHASTIC PROCESSES

Par II CONTINUOUS TIME STOCHASTIC PROCESSES 4

Chaper 4 For an advanced analyss of he properes of he Wener process, see: Revus D and Yor M: Connuous marngales and Brownan Moon Karazas I and Shreve S E: Brownan Moon and Sochasc Calculus Begnnng from hs lecure, we sudy connuous me processes A sochasc process X s defned, n he same way as n Lecure, as a famly of random varables X {X : T } bu now T [, or T [a, b] R Our man examples sll come from Mahemacal Fnance bu now we assume ha fnancal asses can be raded connuously We assume he same smple suaon as n Lecure, ha s we assume ha we have a smple fnancal marke conssng of bonds payng fxed neres, one sock and dervaves of hese wo asses We assume ha he neres on bonds s compounded connuously, hence he value of he bond a me s B B e r We denoe by S he me prce of a share and we assume now ha S can also change connuously In order o nroduce more precse models for S smlar o dscree me models lke exponenal random walk we need frs o defne a connuous me analogue o he random walk process whch s called Wener process or Brownan Moon 4 WIENER PROCESS: DEFINITION AND BASIC PROPERTIES We do no repea he defnons nroduced for dscree me processes n prevous lecures f hey are exacly he same excep for he dfferen me se Hence we assume ha he defnons of he flraon, marngale, are already known On he oher hand, some properes of he correspondng objecs derved n dscree me can no be aken for graned Defnon 4 A connuous sochasc process {W : } adaped o he flraon F s called an F -Wener process f W, for every s he random varable W W s s ndependen of F s, 3 for every s he random varable W W s s normally dsrbued wh mean zero and varance s In many problems we need o use he so called Wener process W x sared a x process defned by he equaon W x x + W The nex proposon gves he frs consequences of hs defnon I s a Proposon 4 Le W be an F -Wener process Then EW and EW s W mns, for all s, and n parcular EW The process W s Gaussan 4

3 The Wener process has ndependen ncremens, ha s, f 3 4 hen he random varables W W and W 4 W 3 are ndependen Proof We shall prove 3 frs To hs end noe ha by defnon of he Wener process he random varable W 4 W 3 s ndependen of F and W W s F -measurable Hence 3 follows from Proposon 3 f we pu F σ W 4 W 3 and G F Because W and W W W, he frs par of follows from he defnon of Wener process Le s Then CovW s, W EW s W EW s W W s + EWs EW s EW W s + EWs s mns, I s enough o show ha for any n and any choce of < < n he random vecor W W X s normally dsrbued To hs end we shall use Proposon 9 Noe frs ha by 3 he random varables W, W W,, W n W n are ndependen and herefore he random vecor W n W W W Y W n W n s normally dsrbued I s also easy o check ha X AY wh he marx A gven by he equaon A Hence follows from Proposon 9 Noe ha he properes of Wener process gven n Defnon 4 are smlar o he properes of a random walk process We shall show ha s no accdenal Le S h be he process defned n Exercse 8 Le us choose he sze of he space grd a h relaed o he me grd by he condon a h h 4 Then usng he resuls of Exercse 8 we fnd mmedaely ha ES h and mn, CovS h, S h h h Hence lm CovS h, S h mn, h For every > and h > he random varable S h s a sum of muually ndependen and dencally dsrbued random varables Moreover, f 4 holds hen all assumpon of he 4

Cenral Lm Theorem are sasfed and herefore S h has an approxmaely normal dsrbuon for small h and any mh wh m large More precsely, for any >, we fnd a sequence n kn n such ha n converges o and for a fxed n we choose h n n Then we can show ha y lm n h n y exp z dz n π I can be shown ha he lmng process S s dencal o he Wener process W nroduced n Defnon 4 Proposon 43 Le W be an F -Wener process Then W s an F -marngale, W s a Markov process: for s P W y F s P W y W s 4 Proof By defnon, he process W s adaped o he flraon F and E W < because has a normal dsrbuon The defnon of he Wener process yelds also for s whch s exacly he marngale propery We have EW W s F s EW W s P W y F s EI,y W F s EI,y W W s + W s F s and pung X W W s and Y W s we oban 4 from Proposon 4 The Markov propery mples ha for every and y R P W y W s x, W s x,, W sn x n P W y W sn x n for every collecon s s n and x,, x n R On he oher hand, he Gaussan propery of he Wener process yelds for s n < y P W y W sn x n π sn exp z x n y xn dz Φ s n sn Therefore, f we denoe by p, x exp x, π hen p s, y x s he condonal densy of W gven W s x for s < The densy p s, y z s called a ranson densy of he Wener process and s suffcen for deermnng all fne dmensonal dsrbuons of he Wener process To see hs le us consder he random varable W, W for < We shall fnd s jon densy For arbrary x and y, 6 yelds P W x, W y x x y P W y W z p, zdz p, y zp, zdz 43

and herefore he jon densy of W, W s f, x, y p, y xp, x The same argumen can be repeaed for he jon dsrbuon of he random vecor W,, W n for any n The above resuls can be summarzed n he followng way If we know he nal poson of he Markov process a me and we know he ranson densy, hen we know all fne dmensonal dsrbuons of he process Ths means ha n prncple we can calculae any funconal of he process of he form E[F W,, W n ] In parcular, for n, E[F W, W ] x y F x, yp, y zp, zdz, for < The nex proposon shows he so called nvarance properes of he Wener process Proposon 44 Le W be an F -Wener process For fxed s, we defne a new process V W +s W s Then he process V s also a Wener process wh respec o he flraon G F +s For a fxed c > we defne a new process U cw /c Then he process U s also a Wener process wh respec o he flraon H F /c Proof Clearly V and he random varable V s F +s G-measurable For any h > we have V +h V W +h+s W +s and herefore V +h V s ndependen of F +h G wh Gaussan N, h dsrbuon Hence V s a Wener process The proof of s smlar and lef as an exercse We shall apply he above resuls o calculae some quanes relaed o smple ransformaons of he Wener process Example 4 Le T > be fxed and le B W T W T Then he process B s a Wener process on he me nerval [, T ] wh respec o he flraon G σw T W T s : s Proof Noe frs ha he process B s adaped o he flraon G We have also for s < T B B s W T s W T and because he Wener process W has ndependen ncremens, we can check easly ha B B s s ndependen of G s Obvously B B s has he Gaussan N, s dsrbuon and B Because B s a connuous process, all condons of Defnon 4 are sasfed and B s a Wener process Example 4 Le S m + σw The process S s called a Wener process wh drf m and varance σ Clearly ES m and E S m σ We shall deermne he jon densy of he random varables W, S Noe ha S s a Gaussan process and he random varable W, S W, m + σw s jonly Gaussan as well can you show ha? By he defnon of he Wener process we oban CovW, m + σw σew W σ mn, 44

Hence he covarance marx s C σ mn, σ mn, σ and W, S N m, σ mn, σ mn, σ Example 43 Le Y xe m+σw The process Y s called an exponenal Wener process and ubquous n mahemacal fnance Ths process s a connuous me analogue of he exponenal random walk Noe frs ha by Theorem, he process Y s adaped o he flraon F By 7 EY xe m+ σ We shall show now ha he process Y s an F -marngale f and only f Indeed, for s we oban and he clam follows m σ E xe m+σw F s E xe ms+m s+σw s e σw Ws F s Y s Ee m s+σw Ws Y s e m s Ee σw s Y s e m s+σ W W s In he nex example we shall need he followng: Lemma 45 Le X be a sochasc process such ha Then E T T E X s ds < X s ds T EX s ds Moreover, f he process X s adaped o he flraon F, hen he process Y X s ds, T s also adaped o F Gaussan Addonally, f X s a Gaussan process, hen he process Y s also 45

Proof We gve only a very bref dea of he proof By assumpon he process X s negrable on [, T ] and hence we can defne an approxmang sequence Y n kt k T X k n n n <kt/ n for he process Y I s no dffcul o see ha all of he properes saed n he lemma hold f we replace Y wh Y n By defnon of he negral lm Y n Y n for every > and remans o prove only ha he properes saed n he lemma are preserved n he lm Ths par of he proof s omed Example 44 Consder he process Y W s ds By Lemma 45 s an F -adaped Gaussan process We shall deermne he dsrbuon of he random varable Y for a fxed By Lemma 45 we have EY Now, for s, once more by Lemma 45 s s EY s Y E W u du W v dv EW u W v dudv because Hence s s s s 3 s3 + EW u W v dudv + mnu, vdudv + s s s s s EW u W v dudv sudu 3 s3 + ss s s EW u W v W s dudv + EW u W v W s dudv EY s Y 3 mns3, 3 + s mns, s s EW u W s dudv 4 WIENER PROCESS: PROPERTIES OF SAMPLE PATHS We dd no specfy he sample space of he Wener process ye In analogy wh he dscusson n Lecure we denfy he sample pon wh he whole Brownan pah ω rajecory W ω for or T, whch s a connuous funcon of me by defnon We shall show ha Brownan pahs are exremely rregular We sar wh some nroducory remarks Consder a funcon f : [, R, such ha f and T f d <, 46

n whch case f f sds Le us calculae how fas are he oscllaons of he funcon f on a fxed nerval [, T ] end for every n we dvde [, T ] no a sequence of subnervals To hs P n, n, P n n, n,, P n n, T The whole dvson wll be denoed by P n Wh every dvson P n of [, T ] we assocae s sze defned by he equaon d P n max n k k n k, n where n and n T For a gven dvson P n we defne he correspondng measure of he oscllaon of he funcon f V n f f n k f n k Then I follows ha V n f n k kn f ydy n k V n f T n k n k f y dy f y dy < and hs bound s ndependen of he choce of he dvson P n hen he lm sup V n f < n V f lm V n f dp n If T f y dy exss and s called a varaon of he funcon f A funcon wh hs propery s called a funcon of bounded varaon For a gven dvson P n we shall calculae now he so-called quadrac varaon V n f f k f k of he funcon f Fnally V n f Usng he same argumens as above we oban by Schwarz nequaly k n k n k f y dy max n k <k n k n k n k k n k n k n k n k f y dy f y T dy max n k n k n k n k f y dy V n f d P n T f y dy 43 47

and herefore lm V dp n n f We say ha he funcon f wh he square negrable dervave has zero quadrac varaon We shall show ha he behavor of brownan pahs s very dfferen from he behavor of dfferenable funcons descrbed above Defnon 46 A sochasc process {X : } s of fne quadrac varaon f here exss a process X such ha for every lm E V dp n n X X, where P n denoes a dvson of he nerval [, ] Theorem 47 A Wener process s of fne quadrac varaon and W for every Proof For a fxed > we need o consder Noe frs ha and herefore EV n W V n W W n W n E W W n n n n E V n W kn E W n W n n n Because ncremens of he Wener process are ndependen Fnally, we oban E V n W kn E W n W n n n E V n W d P n n n sup n kn n n n n and herefore W as desred Corollary 48 Brownan pahs are of nfne varaon on any nerval: P V W Proof Le [, ] be such an nerval ha for a ceran sequence P n of dvson lm n W n W n 48

Then W n W n sup W n W n W n W n Now he lef hand sde of hs nequaly ends o and connuy of he Wener process yelds lm sup W n n W n Therefore necessarly lm n W n W n Theorem 49 Brownan pahs are nowhere dfferenable Consder he process M W whch s obvously adaped We have also E M F s E W F s E W W s + W s W W s F s E W W s F s + Ws E W F s W s s + W s W s M s and herefore he process M s an F -marngale We can rephrase hs resul by sayng ha he process M W W s a marngale, he propery whch wll be mporan n he fuure 43 EXERCISES a Le {Y : } be a Gaussan sochasc process and le f, g : [, R be wo funcons Show ha he process X f + gy s Gaussan Deduce ha he process X x + a + σw, where W s a Wener process s Gaussan for any choce of a R and σ > b Show ha X s a Markov process Fnd EX and CovX s, X c Fnd he jon dsrbuon of X, X Compue E e X X for d Fnd he dsrbuon of he random varable e Compue X Z X s ds f Show ha he process X W s also a Wener process 49

! Le u, x Ef x + W for a ceran bounded funcon f Use he defnon of expeced value o show ha u, x fx + y exp y dy π Usng change of varables show ha he funcon u s wce dfferenable n x and dfferenable n > Fnally, show ha u, x u, x x 3! Le S be an exponenal Wener process sarng from S x > a Fnd he densy of he random varable S for > b Fnd he mean and varance of S c Le for x > u, x EfxS for a bounded funcon f dfferenably properes as n and u, x σ x u x, x + m + σ x u, x x 4 Le W be a Wener process and le F denoe s naural flraon a Compue he covarance marx C for W s, W Show ha u has he same b Use he covarance marx C o wre down he jon densy fx, y for W s, W c Change he varable n he densy f o compue he jon densy g for he par of random varables W s, W W s Noe hs densy facors no a densy for W s and a densy for W W s, whch shows ha hese wo random varables are ndependen d Use he densy g n c o compue Verfy ha φu, v E exp uw s + vw φu, v exp u, vc u v 5 Le {X : } be a connuous and a Gaussan process, such ha EX and E X s X mns, Show ha he process X s a Wener process wh respec o he flraon F X 6 Brownan brdge Le W be a Wener process Le x, y be arbrary real numbers The Brownan brdge beween x and y s defned by he formula V x,y x + W T T W T y, T a Show ha Deduce ha for all T P W a W T y P V,y a, T P W a where f T denoes he densy of W T P V,y a f T y, 5

b For any T and any n T show ha P W a,, W n a n W T y P V,y a,, V,y n a n for any real a,, a n c Compue EV x,y and Cov V x,y, V x,y for, T and arbrary x, y d Show ha he process { V x,y T : T } s also a Brownan brdge process 7 Le W be an F -Wener process and le X W a Fnd P X x and evaluae he densy of he random varable X b Wre down he formula for P X s x, X y as a double negral from a ceran funcon and derve he jon densy of X s, X Consder all possble values of s, and x, y c For s < fnd P X y X s x n he negral form and compue he condonal densy of X gven X s x d Is X a Markov process? 5

Chaper 5 STRONG MARKOV PROPERTY AND REFLECTION PRINCIPLE A good reference for hs chaper s: Karazas I and Shreve S E: Sochasc Calculus and Brownan Moon Le W be a real-valued F -Wener process For b > we defne a random varable I s clear ha τ b mn { : W b} 5 { } {τ b } max W b s Ths deny shows ha he even {τ b } s compleely deermned by he pas of he Wener process up o me and herefore for every {τ b } F ha s, τ b s a soppng me We shall compue now s probably dsrbuon Noe frs ha P τ b < P τ b <, W > b + P τ b <, W < b P W > b + P τ b <, W < b Usng n a heursc way he deny we oban P τ b <, W < b P τ b <, W > b P W > b P τ b < P W > b b/ π e x / dx b Φ 5 Hence for every b > he soppng me τ b has he densy f b b exp b, 53 π 3 Le us recall he resul Proposon 43 from Lecure 4 whch says ha he Wener process sars afresh a any momen of me s n he sense ha he process B W s+ W s s also a Wener process The nuve argumen appled o derve 5 s based on he assumpon ha he Wener process sars afresh f he fxed momen of me s s replaced wh a soppng me τ b, ha s, s mplcly assumed ha he process W τ b + W τ b s also a Wener process The rgorous argumen s based on he followng srong Markov propery: P W τ+ x F τ P W τ+ x W τ 54 for all x R In hs defnon we assume ha τ s a fne soppng me, ha s P τ < In he same way as n he case of he Markov propery 54 yelds E f W τ+ F τ E f W τ+ W τ 5

Theorem 5 The Wener process W enjoys he srong Markov propery Moreover, f τ s any soppng me hen he process B W τ+ W τ s a Wener process wh respec o he flraon G F τ+ and he process B s ndependen of G Corollary 5 Le W be an F -Wener process and for b > le τ b be gven by 5 he densy of τ b s gven by 53 I follows from 5 ha b P τ b < lm P τ b lm Φ Then I means ha he Wener process canno say forever below any fxed level b Theorem 5 allows us o calculae many mporan probables relaed o he Wener process Noe frs ha for he random varable W max W s s we oban for x x P W x P τ x > Φ Hence he random varables W P W x and W have he same dsrbuons Proposon 53 For > he random varable W, W has he densy { y x fx, y π exp y x f x y, y, oherwse Proof Noe frs ha by he symmery of Wener process P b + W s a P W s b a P b + W s b a Hence denong B W τ y+ W τ y and usng he srong Markov propery of he Wener process we oban Hence and because P W x, W y P W x τ y P τ y P y + W W τ y x τ y P τ y P W x, W he proposon follows easly P y + B τ y x τ y P τ y P y + B τ y y x τ y P τ y P y + W W τ y y x τ y P τ y P W y x W P W y x, W y P W y x π y x e z / dz y P W y y P W x P W x, W y fx, y x y P W x, W y 53

Noe ha by he nvarance propery of he Wener process P max W s b P max W s b P mn W s b s s s Proposon 53 allows us o deermne he dsrbuon of he random varable W W We sar wh smple observaon ha W W max s W s W max s W s W Snce he process B s W W s s a Wener process on he me nerval [, ] we fnd ha P W W x P max B s x P B x s Noe ha we have proved ha he random varables W, W, and W W have he same densy 53 We end up wh one more example of calculaons relaed o he dsrbuon of he maxmum of he Wener process Le X x + W wh x > We shall calculae P X y mn s > s For y hs probably s equal o zero Le y > By he defnon of he process X P X y mn X s > P W y x mn W s > x s s P W x y mn W s < x s P W x y max W s < x s P B x y max B s < x s P B x y, max s B s < x P max s B s < x where B W s a new Wener process Now, we have P B x y, max B s < x P B x y P B x y, B x s and by Proposon 53 Noe ha P B x y, B x y P B x y, B x x y x x x v π exp v u π 3 v u exp dvdu v x y v x y exp π 3 v x y exp dv π x + y dv 54

Therefore he condonal densy of X gven ha mn s X x > s f y mn X s > X s y P y mn X s > s P τ x y P B x y y P B x y, B x x y exp x + y exp P τ x π π We end up hs lecure wh some properes of he dsrbuon of he soppng me τ b If W s a Wener process hen he process W b b W b s also a Wener process We defne a soppng me { } τ b mn : W b Proposon 54 We have τ b b τ b and consequenly τ b has he same densy as he random varable b τ b b Moreover, he random varables τ b, W and b W have he same denses Proof By defnon τ b mn { : b W } { } mn b : W b { } b mn : W b b τ b In order o prove he remanng properes s enough o pu b and show ha τ has he same densy as Noe frs ha W W has he same densy as W Therefore and hs concludes he proof P τ x P W x P xw P W x 5 EXERCISES Le X x + σw, where W s a Wener process and σ, x > are fxed Le τ mn { : X } a Fnd he condonal densy of X gven X s x for s b Fnd he densy of he soppng me τ c Compue P X y, τ > d Le Y { X f, f > Fnd P Y y Y s x for s and he condonal densy of Y gven X s x Le τ b be he soppng me 5 Show ha Eτ b 55

3 For a fxed > we defne he barrer b { a f, b f >, where < a < b Le W be a Wener process and τ mn { : W b} Compue P τ for 4 Le τ b be defned by 5 Usng he Srong Markov Propery show ha for a < b τ b τ a nf { : W τ a+ W τ a b a} Derve ha he random varable τ b τ a s ndependen of he σ-algebra Fτ W a Fnally, show ha he sochasc process {τ a : a } has ndependen ncremens 5 Le W be a Wener process Show ha he condonal densy of he par W +s, W+s gven W a and W b s fx, y a, b y x a πs 3 exp y x a s 6 Fnd he jon dsrbuon of he random varable W s, W for s 7 Le W and B be wo ndependen Wener processes a Show ha he random varables B W and B W have he same Cauchy densy fx π + x b Le τ b be he soppng me defned by 5 for he process W Apply Proposon 54 o show ha he random varable B τ b has he same dsrbuon as he random b varable B W and deduce from a he densy of he random varable B τ b 56

Chaper 6 CESS MULTIDIMENSIONAL WIENER PRO- Reference: Karazas I and Shreve S E: Sochasc Calculus and Brownan Moon Le W, W,, W d be a famly of d ndependen Wener processes adaped o he same flraon F An R d -valued process s called a d-dmensonal Wener process W W W d For hs process he followng properes hold Proposon 6 Le W be a d-dmensonal Wener process adaped o he flraon F Then EW and EW s W T The process W s Gaussan mns, I for all s, and n parcular EW W T I 3 The Wener process W has ndependen ncremens Proposon 6 Le W be a d-dmensonal Wener process adaped o he flraon F Then W s an F -marngale W s a Markov process: for s and real numbers y,, y d P W y,, W d y d F s P W y,, W d y d W s 6 Le X and Y be wo real-valued connuous sochasc processes For any dvson P n of he nerval [, ] we defne V n X, Y l X n l X n l Y j n l Y j 6 n l Defnon 63 Le X and Y be wo R-valued connuous processes We say ha X and Y have jon quadrac varaon process X, Y f for every here exss random varables X, Y such ha lm E V dp n n X, Y X, Y If X Y hen we wre X nsead of X, X and call quadrac varaon process of X If X and Y are wo R d -valued connuous sochasc processes, ha s X T X,, X d and Y T Y,, Y d such ha X, Y j exss for all, j,, d hen he marx-valued process X, Y X, Y j,j d s called a jon quadrac varaon of he processes X and Y Hence for vecor-valued processes quadrac varaon s a marx valued sochasc process whch can be defned usng marx noaon X, Y lm dp n l where P n, n, s a sequence of dvsons of [, ] Xl X l Yl Y l T, 63 57

Lemma 64 If X, Y, Z are real valued processes of fne quadrac varaon hen X, Y Y, X 4 X + Y 4 X Y 64 and ax + by, Z a X, Z + b Y, Z 65 Proof The symmery of jon quadrac varaon follows mmedaely from he defnon To show he second equaly n 63 s enough o noce ha for any real numbers a, b ab 4 a + b a b 4 and o apply hs deny o 6 Equaon 64 follows easly from 6 We shall fnd he quadrac varaon of he d-dmensonal Wener process W usng 6 and he resul for one dmensonal Wener process W we fnd ha W, W If j hen We shall show ha for j W, W j 66 Noe frs ha EV n W, W j because W and W j are ndependen By defnon V n W, W j l m W n l W W n l W n m W j n m n l W j W j n W j l n m n m Then ndependence of he ncremens and ndependence of Wener processes W and W j yeld E V n W, W j and he las expresson ends o zero l l E W n l W n l E W j n l n l n l d P n Hence we proved ha W j n l W I, 67 where I denoes he deny marx Consder an R m -valued process X BW, where B s an arbrary m d marx and W s an R d -valued Wener process In fuure we shall develop oher ools o nvesgae more general processes, bu properes of hs process can be obaned by mehods already known Frs noe ha X s a lnear ransformaon of a Gaussan process and hence s Gaussan self We can easly check ha EX BEW and The process X s also an F -marngale CovX s, X EX s X T mn s, BB T 68 Indeed, for s E X X s F s BE W W s F s 58

We wll fnd quadrac varaon of he process X usng 6: V n X, X Therefore usng 66 we oban B l Xl X l l Wl W l BV n W, W B T Xl X l T Wl W l T B T X BB T 69 In parcular, f d and X bw hen X b Noe ha follows from 67 and 68 ha wo coordnaes of he process X, X and X j say, are ndependen f and only f X, X j Example 6 Le W and W be wo ndependen Wener processes and le and B aw + bw B cw + dw We wll fnd he jon quadrac varaon B, B Lemma 64 yelds B, B aw + bw, cw + dw where he las equaly follows from 66 ac W, W + ad W, W + bc W, W + bd W, W ac + bd, 6 EXERCISES Connuaon of Example 6 Assume ha a + b Show ha n hs case he process B s also a Wener process Fnd W, B Assume ha W s a d-dmensonal Wener process and le U be a d d unary marx, ha s U T U Show ha he process X UW s also a Wener process 3 Le W be a d-dmensonal Wener process a For any a R d show ha he process a T W s a marngale a T W a, where a s he lengh of he vecor a b Show ha he process M W d s a marngale c Le X BW, where B s an m d marx Show ha he process M X r BB T 59

s a marngale, where for any m m marx C rc e T Ce + + e T mce m and e,, e m are bass vecors n R m, ha s { f j, e T e j f j Show also ha he process M,j X X j X, X j s a marngale for any choce of, j m X X T X s a marngale Deduce ha he marx-valued process 4 Le W be a d-dmensonal Wener process and le X x x + W, where x s any sarng pon n R d The process X x s called a Bessel process sarng from x Usng he defnon of χ dsrbuon or oherwse wre down he densy of he random varable X Usng properes of he Wener process show ha he random varables X x and X x have he same dsrbuon 6

Chaper 7 STOCHASTIC INTEGRAL Reference: Karazas I and Shreve S E: Sochasc Calculus and Brownan Moon Le be fxed and le P n be any dvson of [, ] We defne Noe frs ha by smple algebra M n W n W n W n Hence W kn W n W n W n W n + M n W W n W n W n W n W n bu he second erm on he rgh hand sde s known o converge o he quadrac varaon of he Wener process, and fnally lm E dp n M n W Therefore we are able o deermne he lm of he negral sums M n whch can jusfably be called he negral W s dw s W In general he argumen s more complcaed bu repeas he same dea sochasc process X we defne an negral sum I n deermned by a dvson P n of he nerval [, ] X n W n W n Theorem 7 Assume ha X s a process adaped o he flraon F and such ha Then here exss an F -measurable random varable For an arbrary X s ds < 7 such ha for every ε > I X s dw s lm P I n I > ε n 6

If moreover, hen EX s ds < 7 lm E I n I n If 7 holds for every T hen he above heorem allows us o defne an adaped sochasc process M Theorem 7 If X s an adaped process and T X s dw s X s ds < hen he sochasc negral enjoys he followng properes M ; for all T X s dw s + X s dw s X s dw s ; 3 f Y s anoher adaped process such ha hen for all T T Y s ds < 4 f moreover, hen he process ax s + by s dw s a E T M X s dw s + b X s ds < X s dw s Y s dw s defned for T s a connuous marngale wh respec o he flraon F and EM EX s ds I wll be mporan o know wha s he quadrac varaon of he marngale defned by a sochasc negral The answer s provded by he nex heorem 6

Theorem 73 If hen M M X s dw s X s ds Corollary 74 Le M X s dw s and N where Y s, X s are wo F -adaped processes such ha X s + Ys ds <, and W, W are wo F -adaped Wener processes M, N Then X s Y s d W, W s Y s dw s, Proof We prove he corollary for W W The general case s lef as an exercse Invokng he resul from chaper 5 we fnd ha M, N 4 M + N, M + N M N, M N X s + Y s ds X s Y s ds 4 X s Y s ds and he corollary follows Havng defned a sochasc negral we can nroduce a large class of processes called semmarngales In hese noes a semmarngale s a sochasc process X adaped o a gven flraon F such ha X X + a s ds + b s dw s, 73 where X s a random varable measurable wh respec o he σ-algebra F and a, b are wo adaped processes wh he propery T We ofen wre hs process n a dfferenal form as + b s ds < dx a d + b dw Equaon 73 s called a decomposon of he semmarngale X no he fne varaon par A a s ds 63

and he marngale par M b s dw s For connuous semmarngales hs decomposon s unque If T Y s b sds < and T Y s a s ds < hen we can defne an negral Y s dx s of one semmarngale wh respec o anoher: Y s dx s Y s a s ds + Y s b s dw s, T, and he resul s sll a semmarngale Because processes of fne varaon have zero quadrac varaon, he quadrac varaon of a semmarngales X s he same as he quadrac varaon of s marngale par: X M I follows ha, for any semmarngale X, f X for every, hen s marngale par s zero and X X + b sds a s ds If he process X adaped o he flraon F s a semmarngale, hen s decomposon 73 no a marngale and a process of bounded varaon s called he semmarngale represenaon of X We consder now d semmarngales adaped o he same flraon F X X + A + M X + where,, d and W are F -Wener processes a sds + b sdw s, 74 Theorem 75 Io s formula Le F : R d R be a funcon wh wo connuous dervaves and le he semmarngales X,, X d be gven by 74 Then F X,, X d F X,, X d + d + d F x X s,, X d s dm s + F X x s,, Xs d da s In a more explc form he Io s formula can be wren as F X,, X d F X,, X d + d + d F x X s,, X d s b s dw s + d,j F X x s,, Xs d a s ds d,j F X x x s,, Xs d d X, X j s j F X x x s,, Xs d b s b j sd W, W j s j 64

Corollary 76 Le X X + a sds + b sdw s and le F : [, ] R R have wo connuous dervaves Then F, X F, X + + F s s, X s a s ds + F x s, X s b s dw s + F x s, X s a s ds F x s, X s b sds Proof I s enough o pu n Theorem 75 d, X and X X I s ofen convenen o wre he Io s formula n s nfnesmal form F df, X, X F + a x, X + F b x, X F d + b x, X dw Example 7 Le X W n Then X and herefore nws n dw s + n n n Ws ds n Ws n dw s n W n n For n we recover n a smple way he fac already known ha Le and X X + Y Y + W s dw s a s ds + u s ds + Ws n n n dw s + W b s dw s v s dw s Ws n ds W n s ds 75 be wo semmarngales wh possbly dependen Wener processes W and W We wll fnd he semmarngale represenaon of he process Z X Y The Io formula appled o he funcon F x, y xy yelds mmedaely X Y X Y + + X s u s + Y s a s ds X s v s dw s + We can wre hs formula n a compac form X s dy s X Y X Y Y s b s dw s + whch can be called a sochasc negraon by pars formula b s v s d W, W s Y s dx s X, Y For M X s dw s, 65

consder he process N exp M M We wll apply he Io formula o he process F X, X, where X M X s ds, X M X s dw s and F x, x e x x Then and herefore he Io formula yelds N + N s Xs ds + F x, x x F x, x, F x, x F x x x, x F x, x N s X s dw s + N s X s ds + N s X s dw s As a by-produc we fnd ha N N s X s ds The process N wll be mporan n many problems Below are he frs applcaons Example 7 Le S be an exponenal Wener process Clearly S S exp m + σw S S e m+ σ e σw σ S e m+ σ N In hs case, usng he example from Lecure 5 we fnd ha T σ ENs ds < and herefore he process N s a marngale Hence we oban he semmarngale represenaon of he exponenal Wener process S S + m + σ S s ds + σs s dw s and f m σ hen, as we already know, S s a marngale wh he represenaon S S + σs s dw s In fnancal applcaons he coeffcen m s usually wren n he form m r σ 66

and hen S S + rs s ds + σs s dw s Le us consder a smple bu mporan case of a sochasc negral wh he deermnsc negrand f such ha T I follows from Theorem 7 ha he process M f sds < fsdw s T, s a connuous marngale bu n hs case we can say more Proposon 77 Le M X s dw s The process M s Gaussan f and only f s quadrac varaon process M X s ds s deermnsc or, equvalenly X s s a deermnsc funcon Theorem 78 Levy Theorem Le M be an R d -valued connuous F -marngale sarng from zero wh he quadrac varaon process M M j Then M s a Wener process f and only f { M j f j, 76 f j Proof If W s a Wener process hen s quadrac varaon s gven by 76 Assume now ha he process M sasfes he assumpons of he heorem We need o show ha M s a Wener process We shall apply he heorem from chaper Le, for every a R d, F a, x e at x be a funcon defned on R d Then F x j a, x a j F a, x, F x j x k a, x a j a k F a, x By he Io formula F a, M e at M s + Hence d a j e at M u dm u j s d,j a j s e at M u du e at M M s + d a j e at M u M s dm u j s d,j a j s e at M u M s du 77 67

By he properes of sochasc negrals E e at M u dm u F s s Le A F s If we denoe Then E e at M M s I A EI A a E f E e at M M s I A s e at M u M s I A du hen we oban an equaon or f P A a s f u du f a f wh he nal condon f s P A I s easy o show ha he unque soluon o hs dfferenal equaon s gven by he formula f P A exp a s Fnally, akng no accoun he defnon of condonal expecaon we proved ha E e at M M s I A E I A E e at M M s F s E I A exp a s and herefore E e at M M s F s e a s/ and herefore he random varable M M s s ndependen of F s and has he normal N, si dsrbuon and he proof s fnshed We wll apply hs heorem o quckly show he followng: Corollary 79 Le W be an R d -valued Wener process and le he marx B be unary: B T B Then he process X BW s also an F -Wener process Proof We already know ha X s a connuous marngale wh quadrac varaon X BB T Now, by assumpon X I and he Levy Theorem concludes he proof 68

7 EXERCISES Fnd he semmarngale represenaon of he process where a, b are arbrary consans X e a x + e a e as b dw s, Show ha X x + a X s ds + bw Use he Io s Formula o wre a semmarngale represenaon of he process Y cos W Nex, apply he properes of sochasc negrals o show ha he funcon m EY sasfes he equaon m msds Argue ha m m and m Show ha m e / 3 Le where Show ha X s a Wener process b X b s dw s, { f W, f W < 4 Le W be an F -Wener process and le F : [, ] R R sasfes he assumpons of he Io s Lemma Fnd condons on he funcon F under whch he process Y F, W s a marngale 5 Prove he general verson of Corollary 74 69