Itō Calculus (An Abridged Overview)

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1 Itō Clculus (A Abrdged Overvew) Arturo Ferdez Uversty of Clfor, Berkeley Sttstcs 157: Topcs I Stochstc Processes Semr Aprl 14, Itroducto I my prevous set of otes, I troduced the cocept of Stochstc Itegrto through geerlzto of the Weer Process d some umercl exmples. I ths dscusso, I troduce oe formlzto of stochstc tegrto kow s Itō clculus. We beg by buldg up the geerl clss of stochstc processes tht c be tegrted. Flly, I preset Itō s formul d exmples where t pples. 2 Prelmres (Note: The followg troducto to Itō Clculus s bsed off the lecture otes referred to [1].) We beg wth some deftos cocerg the ture of probblty spces d mesures. Frst s the cocept of σ-lgebr, whch we wll eed defg collectos d fltrtos. Defto 1 (σ-lgebr). Let X be some set, d 2 X symbolclly represet ts power set. The subset S 2 X s clled σ-lgebr f t stsfes the followg three propertes: S s o-empty. There s t lest oe A X S. S s closed uder complemets. X\A. If A s S, the so s ts complemet, S s closed uder coutble uos. If A 1, A 2, A 3,... re S, the so s A = j=1 A j It follows by De Morg s Lws tht S s lso closed uder coutble tersectos. It s lso cler tht {X, } s σ-lgebr. Defto 2 (Probblty spce). A probblty spce, usully deoted s (Ω, F, P ), cossts of three m compoets: 1

2 Ferdez 2 The smple spce Ω s the set of ll possble outcomes. The σ-lgebr F s the collecto of ll evets we would lke to cosder. Evets re zero or more outcomes. The probblty mesure P s fucto tht returs the probblty of evet. It s mp P : F [, 1]. Defto 3 (Fltrto). I the theory of stochstc processes, fltrto, defed o mesurble spce (Ω, F), s cresg sequece of σ-lgebrs {F t } t wth the propertes: F t F t If t 1 t 2, the F t1 F t2 The dex whch t les vres by cotext. For exmple, t mght be dexed N, [, T ], [, + ). We wll cosder the fte tervl cse for the purpose of these otes. Defto 4 (Progressve Process). A cotuous-prmeter stochstc process X dpted to fltrto {G t } s progressvely mesurble or mesurble whe X(s, ω), s t, s lwys mesurble wth respect to B t G t, where B t s the Borel σ-lgebr 1 o [,t]. Adpted processes tht re left or rght cotuous re progressve. Cotuousprmeter processes wth cotuous smple pths re lso progressve. Defto 5 (No-Atcptg Fltrtos, Processes). Let W t be stdrd Weer process, {F t } the rght-cotuous completo of the turl fltrto 2 of W t, d G y σ-lgebr depedet of {F t }. The the o-tcptg fltrtos re those of the form σ(f t G), t <. 3 A stochstc process X s o-tcptg f t s dpted to some o-tcptg fltrto The defto of X s o-tcptg sys tht up to tme t t c deped o the hstory of W t up to the tme t d possbly other depedet rdom formto, but t s ot possble to predct the future behvor of W t (or X). Defto 6 (Elemetry Process). A progressve, o-tcptve process X s elemetry f there exsts cresg sequece of tmes t, coutble, s.t. X(t) = X(t ) whe t [t, t +1 ),.e. X s step-fucto of tme. Defto 7 (Me Squre[ Itegrble). A stochstc process s me-squretegrble (MSI) o [, b] f E ] b X2 (t) dt s fte. The clss of ll MSI processes s deoted s S 2 [, b]. 1 A borel set s y set tht c be formed from ope sets through coutble uos d complemets. The collecto of ll Borel sets o [, t] forms σ-lgebr kow s the Borel σ-lgebr o [, t] 2 Itutvely, the turl fltrto of W t cots the totl formto bout the behvour of W t up to tme t. 3 For F fmly of subsets of set X, we cll the uque, smllest σ-lgebr whch cots every set F the σ-lgebr geerted by F d deote t s σ (F ).

3 Ferdez 3 Note tht f X s bouded wth probblty 1 o [, b] the t s MSI o [, b]. There s lso the cocept of the S 2 orm but for the purposes of the presetto I wo t go to ts specfc propertes. I wll ote however tht the S 2 orm of X S 2 [, b] s defed s [ b ] 1/2 X S2 = E X 2 (t) dt (1) Defto 8 (Itō Itegrl for elemtry process). If X s elemetry, progressve, o-tcptg, d MSI o [, b], the ts Itō tegrl from to b s b X(t)dW X(t )[W (t +1 ) W (t )] (2) where the t re s the defto of elemetry process, ech bouded below by d bove by b. As we oted the Weer Process d Stochstc Itegrto Itro otes, ths s just the Rem-Steltjes Itegrl ppled to rdom fucto. Through seres proofs, t c be show [1] tht we c exted the oto of the Itō tegrl to more geerl stochstc process. I ll summrze ths process d gve cocludg result. Note: All of processes tht re beg pproxmted the followg flowchrt re progressve d o-tcptve d re pproxmted uder the S 2 [, b] lmt (.e. uder the S 2 orm). Elemetry Processes 1 Bouded, Cotuous Processes 2 Bouded Processes 3 Me Squre Itegrble Processes The pproxmtos 1 re of the form X (t) = X( 2 ) I [/2,(+1)/2 ](t) = d the result follows from the bouded covergece theorem. Those 2 re of the form X (t) = f (s t)x(s)ds where f (t) s rel-vlued fucto of fte support o ( 1/, ),.e. t s zero ywhere outsde of tht tervl, d the result lso follows from the bouded covergece theorem. Lstly, those 3 re of the form X (t) = ( X(t))

4 Ferdez 4 d the result follows from the domted covergece theorem. The ext lemm summrzes these results. Lemm 9 (Approxmto of MSI Processes by Bouded Processes). Let X be progressve, o-tcptg, d MSI o [, b], the there exsts sequece of elemetry processes {X } wth X s ther lmt S 2. Ths lemm s powerful becuse the Itō Isometry wll show tht covergece S 2 s bsclly the sme s covergece me or covergece L 2 whe tegrtg wth respect to the Weer process. Lemm 1 (Itō Isometry for Elemetry Processes). Suppose X s elemetry, progressve, o-tcptve, MSI o [, b] d bouded o [, b]. The [ ( b ) 2 ] [ b ] E X(t) dt = E X 2 (t) dt = X 2 S 2 (3) Proof. Let W = W (t +1 ) W (t ). Note tht Brow cremets re depedet d tht X s o-tcptve so tht W j s depedet of X(t )X(t j ) W for < j.usg equto (2) d the fct tht E [ ( W ) 2] = t +1 t, t follows tht [ ( b ) 2 ] E X(t) dt = E X(t )X(t j ) W W j,j =,j E [X(t )X(t j ) W W j ] sce X elemetry mples X 2 s elemetry. = E [ X 2 (t ) ] (t +1 t ) [ b ] = E X 2 (t) dt Lemms 9 d 1 the led to powerful sttemet bout the covergece of Itō tegrls of pproxmtg fuctos to the ctul Ito tegrl of MSI process (See [1], p. 137). Furthermore, t turs out tht Itō Isometry holds for ll X tht re Itō-tegrble. Defto 11 (Itō Itegrl). Let X be progressve, o-tcptve, d MSI o [, b]. Its Itō Itegrl s b X(t)dW lm b X (t) dw (4) uder the lmt L 2 d where the X re those referred to Lemm 9. We sy tht X s Itō-tegrble o [, b].

5 Ferdez 5 3 Exmples of Stochstc Itegrls 3.1 dw We beg by otcg tht the tegrl b dw s fct oe of elemetry process. Let t be such tht 1 = X(t ) for t [t, t +1 ). For exmple, It follows from equto (2) tht t =, X(t ) = 1 t 1 = + b 2, X(t 1) = 1 t 2 = b, X(t 2 ) = 1 b dw = (1)[W (t +1 ) W (t )] = W (b) W () Notce tht we could hve doe wthout the extr mdpot brekpot. More mporttly, ddg more brekpots to elemetry fucto wll ot chge ts tegrl. 3.2 W dw Clerly, W s mesurble d o-tcptve but s t MSI? Yes, by Fub s Theorem, whch llows us to exchge tegrls, [ ] E W 2 (s) ds = E [ W 2 (s) ] ds = whch s fte for bouded tervls [, t]. Ths exmple wll led us to very peculr property of Itō tegrls becuse lthough oe mght expect W s dw to equl Wt 2 /2, oe would be correct ths cse. We ll beg by usg elemetry fuctos to pproxmte W. Let t = t 2, 2 1. Defe φ (t) = 2 1 = s ds W (t )I [t,t +1 ) (5) It c be show ([1], p.139) tht φ (t) W t. We wll focus our tteto o the lmt of the tegrls of the pproxmtg fuctos sce t should be the sme.

6 Ferdez 6 Uder the ssumptos tht the tegrls wll coverge, let W s dw = lm φ (t) dw 2 1 = lm W (t )[W (t +1 ) W (t )] = 2 1 = lm = W (t ) W (t ) Defe W 2 (t ) W 2 (t +1 ) W 2 (t ). Notce tht sce W () = = W 2 (), W (t) = W (t ) (6) W 2 (t) = W 2 (t ) (7) sce the sums re just telescopg seres. Although t ws metoed tht W 2 /2 by tself ws ot the exct the soluto, t s fct prt of the soluto. We ow wll try to mpulte these formuls to try to get W 2 to show up d wll del wth other terms s they come up. We beg by tryg to get W W term from W 2 (t ). W 2 (t ) = W 2 (t +1 ) W 2 (t ) = [W (t +1 ) W (t )] [W (t +1 ) + W (t )] = W (t ) [W (t +1 ) W (t ) + W (t ) + W (t )] = ( W (t )) 2 + 2W (t ) W (t ) Pluggg ths result to Equto 7, W 2 (t) = [ ( W (t )) 2 + 2W (t ) W (t ) ] (8) Alertvely wrtte s W (t ) W (t ) = 1 2 W 2 (t) 1 ( W (t )) 2 (9) 2 We ow show tht lm 2 1 = ( W (t )) 2 = t L 2. Proof. We beg by showg covergece me. [ 2 ] 1 E ( W (t )) 2 = E [ (W (t 1 ) W (t )) (W (t 2 ) W (t 2 1)) 2] = Sce these cremets re..d = E [ 2 (W (t +1 ) W (t ))) 2] = 2 E [ (W (t/2 )) 2] = 2 = t t 2

7 Ferdez 7 We cotue by showg tht the vrce wll coverge to. By the equlty dstrbutos tht we oted bove, t follows tht so tht Vr[( W (t )) 2 ] = 2 2 Vr(W 2 (t)) = 2 2 (< ) Vr( ( W (t )) 2 ) = C = 2 C whch wll pproch zero s. Flly, we hve tht W s dw = lm 2 1 = = 1 2 W 2 (t) 1 2 lm W (t ) W (t ) ( W (t )) = = 1 2 W 2 (t) t 2 Our result suggests tht the chge of vrbles method tht we re ccustomed to clculus wll ot pply exctly d tht we should try to vod clcultg Itō tegrls by the defto. 4 Itō s formul 4.1 Geerlzto Ths secto wll gve the bsc de behd Itō s formul wthout proof d provde exmples where t pples. Defto 12 (Itō Process). If A s o-tcptve, mesurble process, B s Itō-tegrble, d X s L 2 rdom vrble depedet of W, the X(t) = X + A(s)ds + s Itō process. Equvletly ths s commoly wrtte s B(s)dW s (1) dx = Adt + BdW (11) It c be show tht X s o-tcptg but rther we wll cotue o to preset Itō s formul.

8 Ferdez 8 Theorem 13 (Itō s Formul Oe Dmeso). Suppose X s Itō process (dx = Adt + bdw ). Let f(t, x) : R + R R be fucto such tht f t d 2 f x 2 re cotuous. The F (t) f(t, X(t)) s Itō process, d Equvletly, F (t) F () = df = f f (t, X(t)) + t x (t, X(t))dX B2 (t) 2 f (t, X(t))dt (12) x2 [ f + (s, X(s)) + A(s) f t B(s) f x (s, X(s))dW s x (s, X(s))dX B2 (s) 2 f x ] (s, X(s)) 2 The proof of Itō s formul s qute terestg d c be foud t [1], p We ow pply Itō s formul to two exmples. 4.2 Exmple 1: W dw The bsc de s to strt wth Itō process d the use Itō s formul to smplfy our results. I ths cse, we wt Itō process wth W (t) term d pproprte f tht wll smply uder Itō s formul. We beg by lettg X =, A =, d B = 1 Equto (1) so tht X(t) = W (t). We ow eed f tht wll smplfy the followg verso of Itō s formul: df = f f dt + t x dw f dt (13) 2 x2 To get W dw term out of equto (13) t looks lke settg f = x 2 /2 or f = x 3 /3 mght help, but usg the cubc fucto would leve us wth strge lookg W 2 dw tht we hve t ecoutered yet. We let f(x, t) = x 2 /2. The So tht the secod form of Itō s formul becomes F (t) = X 2 (t)/2 = W 2 (t)/2 (14) f t = (15) f x = x (16) 2 f x 2 = 1 (17) 1 2 W 2 (t) 1 2 W 2 () = 1 (1) ds + 2 ds (18) W s dw s (19) W s dw s = 1 2 W 2 (t) 1 2 t (2) whch s the exct sme result tht we rrved to through the defto of the Itō tegrl.

9 Ferdez Exmple 2: Geometrc Brow Moto (Note: Ths exmple s bsed off the lecture otes referred to [2].) I ths cse, we beg wth the stochstc process, F (t, W t ) = e (µ 1 2 σ2 )t+σw t, kow s geometrc brow moto. Accordgly, f(t, x) = e (µ 1 2 σ2 )t+σx so tht f t = (µ 1 2 σ2 )f(t, x) (21) f = σf(t, x) x (22) 2 f x 2 = σ2 f(t, x) (23) Smlr to the prevous exmple, we let our Itō process X(t) = W (t) by lettg A =, B = 1. It follows tht [ F (t) F () = (µ 1 2 σ2 )F (s) ] 2 (1)σ2 F (s) ds + (1)σF (s) dw s F t F = µ F s ds + σ (1)F s dw s We coclude tht stochstc dfferetl equto (SDE) ssocted wth geometrc brow moto s df = µf dt + σf dw (25) 5 Remrks From the Frst Retur Probblty of Rdom Wlk to Brow Moto d Itō Clculus, I would lke to thk our Sttstcs 157 Professor Jm Ptm for the mzg opportuty to eroll ths clss d ler so my terestg thgs. To the reder: If you ever desre to ler bout somethg, I ecourge seres of browser tbs wth Wkped rtcles, thorough google serch for lecture otes, d publshed reserch ppers d results tht re sure to chllege you d push you further. Also, lwys red crefully d tke your tme to let the mterl relly sk. (24)

10 Ferdez 1 Refereces [1] Cosm Shlz, Stochstc Itegrls d Stochstc Dfferetl Equtos [PDF Documet] , Advced Probblty II or Almost Noe of the Theory of Stochstc Processes. Retreved from Lecture Notes Ole Web ste: [2] Atoell Bsso, Mrt Nrdo, THE STOCHASTIC INTEGRAL [PDF Documet]. Accessed t: StochstcItegrl27.pdf