A Change-of-Variable Formula with Local Time on Surfaces

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1 Sém. de Probab. L, Lecure Noes i Mah. Vol. 899, Spriger, 7, (69-96) Research Repor No. 437, 4, Dep. Theore. Sais. Aarhus (3 pp) A Chage-of-Variable Formula wih Local Time o Surfaces GORAN PESKIR 3 Le = ( ;... ; ) be a coiuous semimarigale ad le b : IR! IR be a coiuous fucio such ha he process b = b( ;... ; ) is a semimarigale. Seig C = f (x;... ; x ) IR j x < b(x;... ; x ) g ad D = f (x;... ; x ) IR j x > b(x;... ; x ) g suppose ha a coiuous fucio F : IR! IR is give such ha F is C i;...;i o C ad F is C i;...;i o D where each i k equals or depedig o wheher k is of bouded variaio or o. The he followig chage-of-variable formula holds: F ( ) = F () Z Z ( ;... ; s ) ( s ;... ; s ) F ( ;... s ) F j ( ;... ; s ) ( ;... ; s ) where `bs () is he local ime of o he surface b give by: `b s () = IP lim "# " Z s d i j ( s ;... ; s ) dh i ; j i s I( s = b s ) d`b s () I(" < r b r < ") dh b ; b i r ad d`bs () refers o iegraio wih respec o s 7! `bs (). The aalogous formula exeds o geeral semimarigales ad b as well. A versio of he same formula uder weaker codiios o F is derived for he semimarigale ((; ; S )) where ( ) is a Iô diffusio ad (S ) is is ruig maximum.. Iroducio Le ( ) be a coiuous semimarigale (see e.g. [3]) ad le b : IR! IR be a coiuous fucio of bouded variaio. Seig C = f (; x) IR IR j x < b() g ad D = f (; x) IR IR j x > b() g suppose ha a coiuous fucio F : IR IR! IR is give such ha F is C ; o C ad F is C ; o D. The he followig chage-of-variable formula is kow o be valid (cf. []): (.) F (; ) = F (; ) Z Z Z F (s; s )F (s; s ) F x (s; s )F x (s; s ) d s ds F xx (s; s ) I( s 6= b(s)) dh; i s 3 Nework i Mahemaical Physics ad Sochasics (fuded by he Daish Naioal Research Foudaio). Mahemaics Subjec Classificaio. Primary 6H5, 6J55, 6G44. Secodary 6J6, 6J65, 35R35. Key words ad phrases: Local ime-space calculus, Iô s formula, Taaka s formula, local ime, curve, surface, Browia moio, diffusio, semimarigale, weak covergece, siged measure, free-boudary problems, opimal soppig. gora@imf.au.dk

2 F x (s; s )F x (s; s ) I( s = b(s)) d`b s () where `bs () is he local ime of o he curve b give by: (.) `b s () = IP lim "# " Z s I(b(r)" < r < b(r)") dh; i r ad d`bs () refers o iegraio wih respec o he coiuous icreasig fucio s 7! `bs (). A versio of he same formula for a Iˆo diffusio derived uder weaker codiios o F has foud applicaios i free-boudary problems of opimal soppig (cf. []). The mai aim of he prese paper is o exed he chage-of-variable formula (.) o a mulidimesioal seig of coiuous fucios F which are smooh above ad below surfaces. Coiuous semimarigales are cosidered i Secio, ad semimarigales wih jumps are cosidered i Secio 3. A versio of he same formula uder weaker codiios o F is derived i Secio 4 for he coiuous semimarigale ((; ; S )) where ( ) is a Iô diffusio ad (S ) is is ruig maximum. This versio is useful i he sudy of free-boudary problems for opimal soppig of he maximum process whe he horizo is fiie (for he ifiie horizo case see [] wih refereces). The sudy of Secio 4 serves as a example of wha geerally eeds o be doe i order o relax he smoohess codiios o F from C ad D o C [ D. These relaxed versios of he formula are impora for applicaios. I is hus hoped ha he programme sared i Secio 3 of [] ad i Secio 4 of he prese paper will be coiued. For relaed resuls o he local ime-space calculus see [], [5], [3], [], [8]. Older refereces o he opic iclude [7], [4], [9], [5], [4].. Coiuous semimarigales Le = ( ;... ; ) be a coiuous semimarigale ad le b : IR! IR be a coiuous fucio such ha he process b = b( ;... ; ) is a semimarigale. [Noe ha he sufficie codiio b C is by o meas ecessary.] Seig: (.) C = f (x ;... ; x ) IR j x < b(x ;... ; x ) g (.) D = f (x ;... ; x ) IR j x > b(x ;... ; x ) g suppose ha a coiuous fucio F : IR! IR is give such ha: (.3) F is C i;...;i o C (.4) F is C i ;...;i o D where each i j equals or depedig o wheher j is of bouded variaio or o. More explicily, i meas ha F resriced o C coicides wih a fucio F which is C i;...;i o IR, ad F resriced o D coicides wih a fucio F which is C i;...;i o IR. [We recall ha a coiuous fucio F k : IR! IR is C i;...;i o IR if he parial derivaives k =@x j whe i j = as well F k =@x j whe i j ; i j = exis ad are coiuous as

3 fucios from IR o IR for all j; j where k equals or.] The he aural desire arisig i free-boudary problems of opimal soppig (ad oher problems where he hiig ime of D by he process plays a role) is o apply a chageof-variable formula o F ( ) so o accou for possible jumps of (=@x )(x;... ; x ) a x = b(x;... ; x ) beig measured by: (.5) `b s () = IP lim "# " Z s I(" < r b r < ") dh b ; b i r which represes he local ime of o he surface b for s [; ]. Noe ha he limi i (.5) exiss (as a limi i probabiliy) sice b is a coiuous semimarigale. I he special case whe he semimarigale equals (; ) i is evide ha he previous seig reduces o he seig leadig o he chage-of-variable formula (.) above. Furher paricular cases of he formula (.) are reviewed i []. The followig heorem provides a geeral formula of his kid for coiuous semimarigales (see also Secio 3 below). Theorem. Le = ( ;... ; ) be a coiuous semimarigale, le b : IR! IR be a coiuous fucio such ha he process b = b( ;... ; ) is a semimarigale, ad le F : IR! IR be a coiuous fucio saisfyig (.3) ad (.4) above. The he followig chage-of-variable formula holds: (.6) F ( ) = F () i ( s ;... ; s i ( s ;... ; s F ( ;... ; s ) ( s ;... ; s ) ( s ;... ; s ( s ;... ; s ) d i s dh i ; j i s I( s = b s ) d`b s () where `bs () is he local ime of o he surface b give i (.5) above, ad d`bs () refers o iegraio wih respec o he coiuous icreasig fucio s 7! `bs (). Proof.. Se Z = ^ b ad Z = _ b for > give ad fixed. Deoig ^ = ( ;... ; ; Z ), = ( ;... ; ; Z ) ad ~ = ( ;... ; ; b ), we see ha he followig ideiy holds: (.7) F ( ) = F( ^ ) F( ) F ( ~ ) where we use ha F (x) = F(x) = F(x) for x = (x;... ; x ; b(x;... ; x )). The processes (Z ) ad (Z ) are coiuous semimarigales admiig he followig represeaios: (.8) Z = (.9) Z = b b b b. 3

4 Recallig he Taaka formula: (.) j b j = j b j where sig() =, we fid ha: (.) dz = = (.) dz = = sig( s b s ) d( s b s ) `b () d( b ) sig( b ) d( b ) d`b sig( b ) d () sig( b ) db () d( b ) sig( b ) d( b ) d`b d`b () sig( b ) d sig( b ) db d`b (). I he sequel we se D i i ad D ij =@x j as well as D i =@x i.. Applyig he Iˆo formula o F( ^ ) ad usig (.) we ge: (.3) F( ^ ) = F( ^) = F( ^) D if( ^ s ) d ^ i s D ijf( ^ s ) dh ^ i ; ^ j i s D if( ^ s ) d i s sig( s bs ) D F( ^ s ) d s Z sig( s bs ) D F( ^ s ) db s D F( ^ s ) d`b s () D ijf( s ) I( s b s ) dh ~ i ; ~ j i s where i he las four iegrals we make use of he geeral fac: (.4) I(Y s = Y s ) dhy ; Y 3 i s = I(Y s = Y s ) dhy ; Y 3 i s wheever Y ; Y ad Y 3 are coiuous (oe-dimesioal) semimarigales. The ideiy (.4) ca easily be verified usig he Kuia-Waaabe iequaliy ad he occupaio imes formula (for more deails see he proof followig (3.) below). The righ-had side of (.3) ca furher be expressed i erms of ~ usig (.4) as follows: 4

5 (.5) F( ^ ) = F( ^) D if( s ) I( s b s ) d ~ i s D F( s ) I( s = b s ) d s D F( s ) I( s b s ) d ~ s Z D F( ~ s ) I( s = b s ) d ~ s D F( s ) I( s = b s ) d`b s () D ijf( s ) I( s b s ) dh ~ i ; ~ j i s. By groupig he correspodig erms i (.5) we obai: (.6) F( ^ ) = F( ^) D if( s ) I( s b s ) d ~ i s D if( ~ s ) I( s = b s ) d ~ i s D ijf( ~ s ) I( s > b s ) dh ~ i ; ~ j i s 4 D ijf( ~ s ) I( s = b s ) dh ~ i ; ~ j i s. 5

6 3. Applyig he Iˆo formula o F( ^ ) ad usig (.) we ge: (.7) F( ) = F( ) = F( ) D if( s ) d i s D ijf( s ) dh i ; j i s D if( s ) d i s sig( s bs ) D F( s ) d s Z sig( s bs ) D F( s ) db s D F( s ) d`b s Z () D ijf( s ) I( s > b s ) dh i ; j i s 4 D ijf( s ) I( s = b s ) dh i ; j i s 4 D ijf( ~ s ) I( s = b s ) dh ~ i ; ~ j i s D ijf( ~ s ) I( s b s ) d i s D if( s ) I( s = b s ) d i s D if( ~ s ) I( s b s ) d s D F( ~ s ) I( s b s ) dh i ; j i s 4 D ijf( s ) I( s = b s ) dh i ; j i s 4 D ijf( ~ s ) I( s = b s ) dh ~ i ; ~ j i s D ijf( ~ s ) I( s < b s ) dh ~ i ; ~ j i s. 6

7 By groupig he correspodig erms i (.8) we obai: (.9) F( ) = F( ) D if( s ) I( s b s ) d ~ i s D if( ~ s ) I( s = b s ) d ~ i s D ijf( ~ s ) I( s > b s ) dh ~ i ; ~ j i s 4 D ijf( ~ s ) I( s = b s ) dh ~ i ; ~ j i s. 4. Combiig he righ-had sides of (.6) ad (.9) we coclude: (.) F ( ) = F( ^ ) F( ) F ( ~ ) = F () D i F ( s ;... ; s ) D if ( s ;... ; s ) d i s Z D ij F ( s ;... ; s ) D ijf ( s ;... ; s ) dh i ; j i s Z D F ( s ;... ; s ) D F ( s ;... ; s ) I( s = b s ) d`b s () R where he fial erm is give by: (.) R = F ( ) ~ D if( ~ s ) I( s > b s ) d ~ i s Z D ijf( ~ s ) I( s > b s ) dh ~ i ; ~ j i s 4 D ijf( ~ s ) I( s = b s ) dh ~ i ; ~ j i s D if( ~ s ) I( s = b s ) d ~ i s D if( ~ s ) I( s < b s ) d ~ i s 7

8 4 D ijf( ~ s ) I( s < b s ) dh ~ i ; ~ j i s D ijf( ~ s ) I( s = b s ) dh ~ i ; ~ j i s F ( ~ ). D if( ~ s ) I( s = b s ) d ~ i s Hece we see ha (.6) will be proved if we show ha R =. Noe ha if F = F he he ideiy R = reduces o he Iˆo formula applied o F ( ~ ). I he geeral case we may proceed as follows. 5. Sice F(x) = F(x) for x = (x;... ; x ; b(x;... ; x )), we see ha he wo semimarigales F( ) ~ ad F( ) ~ coicide, so ha: (.) (.3) I( s > b s ) d(f ( ~ s )) = I( s = b s ) d(f ( ~ s )) = I( s > b s ) d(f ( ~ s )) I( s = b s ) d(f ( ~ s )). Applyig he Iô formula o F( ~ s ) ad F( ~ s ) we see ha (.) ad (.3) become: (.4) (.5) = = D if( ~ s ) I(s > b s ) d ~ s i D if( ~ s ) I(s > b s ) d ~ s i D if( ~ s ) I(s = b s ) d ~ s i D if( ~ s ) I(s = b s ) d ~ s i D ijf( ~ s ) I( s > b s ) h ~ i ; ~ j i s D ijf( ~ s ) I( s > b s ) h ~ i ; ~ j i s D ijf( ~ s ) I( s = b s ) h ~ i ; ~ j i s D ijf( ~ s ) I( s = b s ) h ~ i ; ~ j i s. Makig use of (.4) ad (.5) we see ha F i he firs four iegrals o he righ-had side of (.) ca be replaced by F. This combied wih he remaiig erms shows ha he ideiy R = reduces o he Iô formula applied o F( ~ ). This complees he proof of he heorem. Remark. The chage-of-variable formula (.6) ca obviously be exeded o he case whe isead of oe fucio b we are give fiiely may fucios b; b;... ; b m which do o iersec. More precisely, le = ( ;... ; ) be a coiuous semimarigale ad le us assume ha he followig codiios are saisfied: (.6) b k : IR! IR is coiuous such ha b k; = b k ( ;... ; ) is a semimarigale for k m (.7) F k : IR! IR is C i;...;i for k m where each i j equals or depedig o wheher j is of bouded variaio or o 8

9 (.8) F (x) = F (x) if x b m (x ;... ; x ) where F : IR! IR is coiuous ad x = (x ;... ; x ) belogs o IR. The he chage-of-variable formula (.6) exeds as follows: (.9) F ( ) = F ( ) ;... ; s ) (s ;... ; s ) i F ;... ; s F (s ;... ; s ) j m ;... ; s ) ;... ; s ) k= d i s dh i ; j i s I(s = bk; s ) d`bk s () where `bk s () is he local ime of o he surface b k give i (.5) above, ad d`bk s () refers o iegraio wih respec o s 7! `bk s (). Noe i paricular ha a ope se C i IR (such as a ball) ca ofe be described i erms of fucios b ; b ;... ; b m so ha (.9) becomes applicable. Perhaps he mos ieresig example of a fucio F is obaied by lookig a D = iff > j D g ad seig F (x) = E x (G( D )) where G is a admissible fucio ad = x uder P x for x IR. Oe such example will be sudied i Secio 4 below. Remark.3 The chage-of-variable formula (.6) is expressed i erms of he symmeric local ime (.5). I is evide from he proof above ha oe could also use he oe-sided local imes defied by: (.3) `b s (.3) `b s () = IP lim "# () = IP lim "# " " Z s Z s I( r b r < ") dh b ; b i r I(" < r b r ) dh b ; b i r. The uder he same codiios as i Theorem. we fid ha he followig wo equivale formulaios of (.6) are valid: (.3) F ( ) = F ( ( s ;... ; i ( s ;... ; s 7) d i j ( s ;... ; s 7) dh i ; j i s ( s ;... ; s ) I( s = b s ) d`b6 s (). Clearly (.9) above ca also be expressed i erms of oe-sided local imes. Noe fially ha if b is a coiuous local marigale, he he hree defiiios (.5), (.3) ad (.3) coicide. 9

10 3. Semimarigales wih jumps I his secio we will exed he chage-of-variable formula (.6) firs o semimarigales wih jumps of bouded variaio (Theorem 3.) ad he o geeral semimarigales (Theorem 3.).. Le = ( ;... ; ) be a semimarigale (see e.g. []). Recall ha each sample pah 7! i is righ coiuous ad has lef limis for i. I Theorem 3. below we will assume ha each semimarigale i has jumps of bouded variaio i he sese ha: (3.) <s j i sj < where i s = i s i s for i. I his case each i ca be uiquely decomposed io: (3.) i = i i;c i;d where i;c = M i;c A i;c is a coiuous semimarigale ad i;d is a discree semimarigale (of bouded variaio) give by: (3.3) i;d = Moreover, if F : IR! IR <s i s. (3.4) F ( ) = F () F ( s ) F ( s ) <s = F () <s i is C he Iˆo s formula akes ay of he wo equivale i ( s ) d i i ( s ) d i;c F ( s ) F ( s F ( s ) i F j ( s ) d[ i;c ; j;c ] j ( s ) d[ i;c ; j;c ] s Boh of hese forms will be used freely below wihou furher meioig. Le b : IR! IR be a coiuous fucio such ha he process b = b( ;... ; ) is a semimarigale wih jumps of bouded variaio. The b is a semimarigale wih jumps of bouded variaio ad he local ime of o he surface b is well-defied as follows: (3.5) `b s () = IP lim "# " Z s I(" < r b r < ") d[ b ; b ] c r where [ b ; b ] c is he coiuous (pah by pah) compoe of [ b ; b ]. Recallig ha ;c ad b ;c are coiuous semimarigales associaed wih ad b as i (3.) above, we kow ha [ b ; b ] c = [ ;c b ;c ; ;c b ;c ]. The followig heorem exeds he chage-of-variable formula (.6) o semimarigales wih jumps of bouded variaio.

11 Theorem 3. Le = ( ;... ; ) be a semimarigale where each i has jumps of bouded variaio, le b : IR! IR be a coiuous fucio such ha he process b = b( ;... ; ) is a semimarigale wih jumps of bouded variaio, ad le F : IR! IR be a coiuous fucio saisfyig (.3) ad (.4) above. The he followig chage-of-variable formula holds: <s (3.6) F ( ) = F () ;... ; s ) (s ;... ; s) i F ;... ; ) F (s ;... ; s) j F ( s ) F ( s) ;... ; s ) ;... ; s) d i;c s d[ i;c ; j;c ] s I( s = b s ; s = b s ) d`b s () where `bs () is he local ime of o he surface b give i (3.5) above, ad d`bs () refers o iegraio wih respec o he coiuous icreasig fucio s 7! `bs (). Proof. The proof ca be carried ou similarly o he proof of Theorem. ad we will oly highligh a few ovel pois appearig due o he exisece of jumps. The remaiig deails are he same as i he proof of Theorem... We begi as i he proof of Theorem. by iroducig he processes Z ; Z ; ^; ; ~ ad observig ha (.7)-(.9) carries over uchaged. Sice ad b boh have jumps of bouded variaio, i is easily see ha so do Z ad Z as well. Thus he aalogue of (.) which is obaied by applyig he Taaka formula reads: (3.7) j b j = j b j where <s sig( sb s j s b s j j s b sj ) d( ;c s sig() =. Similarly o (.) ad (.) we fid ha: (3.8) dz ;c = (3.9) dz ;c = b ;c s ) `b () sig(b ) ;c d sig(b ) ;c db d`b () sig(b ) ;c d sig(b ) ;c db d`b ().. Applyig he Iˆo formula o F( ^ ) we ge: (3.) F( ^ ) = F( ^) <s D if( ^ s) d F( ^ s )F( ^ s). ^ i;c s D ijf( ^ s) d[ ^ i;c ; ^

12 Hece usig (3.8) ad proceedig i he same way as i (.3) ad (.5) we obai he aalogue of he ideiy (.6) where all i ad ~ i i he iegraors (icludig hose wih he agle brackes) are replaced by i;c ad ~ i;c (ow wrie as he square brackes). I may be oed (as i he proof of Theorem.) ha i he precedig derivaio (ad i he derivaio followig (3.) below) we eed o make use of he geeral fac: (3.) I(Y s = Y s ) d[y ;c ; Y 3;c ] s = I(Y s = Y s ) d[y ;c ; Y 3;c ] s wheever Y ; Y ad Y 3 are (oe-dimesioal) semimarigales. To verify (3.) oe ha he claim is equivale o he fac ha for wo (oe-dimesioal) semimarigales Y ad Y we have I(Y s = ) d[y ;c ; Y ;c ] =. To derive he laer we may ivoke he Kuia-Waaabe iequaliy (cf. [; p.6]) accordig o which i is eough o show ha I(Y s = ) d[y ;c ; Y ;c ] =. This ideiy however follows by he occupaio imes formula (cf. [; p. 68]) sice g = fg equals zero almos everywhere wih respec o Lebesgue measure o IR. This proves (3.) i he geeral case (recall also (.4) above). 3. Applyig he Iˆo formula o F( ) we ge: (3.) F( ) = F( ) <s D if( s ) d i;c s F( s )F( s ). D ijf( s ) d[ i;c ; Hece usig (3.9) ad proceedig i he same way as i (.7) ad (.8) we obai he aalogue of he ideiy (.9) where all i ad ~ i i he iegraors (icludig hose wih he agle brackes) are replaced by i;c ad ~ i;c (ow wrie as he square brackes). 4. Combiig he righ-had sides of he resulig ideiies we fid he aalogue of (.) o be: (3.3) F ( ) = F( ^ ) F( ) F ( ~ ) = F () D i F ( s ;... ; s ) D if ( s ;... ; s) <s where we use ha: (3.4) <s = <s d i;c s D ij F ( s ;... ; s ) D ijf ( s ;... ; s) d[ i;c ; j;c ] s D F ( s ;... ; s ) D F ( s ;... ; s) F ( s )F ( s ) R F( ^ s )F( ^ s ) F ( s )F( s ) <s F( s )F( s ) <s d`b s () F ( ~ s )F ( ~ s )

13 ad he fial erm i (3.3) is give by: (3.5) R = F ( ) ~ 4 D if( ~ s) I(s > b s ) d ~ i;c s D ij F( ~ s) I( s > b s ) d[ ~ i;c ; ~ D if( ~ s) I(s = b s ) d ~ i;c s 4 D ij F( ~ s) I( s = b s ) d[ ~ i;c ; ~ D if( ~ s) I(s < b s ) d ~ i;c s D ij F( ~ s) I( s < b s ) d[ ~ i;c ; ~ D if( ~ s) I(s = b s ) d ~ i;c s D ij F( ~ s) I( s = b s ) d[ ~ i;c ; ~ F ( ~ ) c where F ( ) ~ c is he coiuous semimarigale par of F ( ) ~. From (3.3) ad (3.5) we see ha (3.6) will be proved if we show ha R =. 5. The same argumes as hose give i (.)-(.5) show agai ha F i he firs four iegrals o he righ-had side of (3.5) ca be replaced by F. This combied wih he remaiig erms shows ha he ideiy R = reduces o applyig he Iˆo formula o F( ~ ) ad ideifyig he coiuous par of he resulig semimarigale. This complees he proof of he heorem.. The codiio (3.) applied o he semimarigale b is he bes kow sufficie codiio for he local ime of o he surface b o be give by meas of he explici expressio (3.5) above. I he case of geeral semimarigales ad b, however, he local ime of o he surface b (i.e. he local ime of he semimarigale b a zero) ca sill be defied by meas of he Taaka formula (3.7) reaiig is role as he occupaio desiy relaive o he radom clock [ b ; b ] c (see [; p.68]) bu we do o have he explici represeaio (3.5) aymore ad he use of he local ime is somewha less raspare. If = ( ;... ; ) is a geeral semimarigale (o ecessarily saisfyig (3.) above) he each i ca sill be decomposed io (3.) wih i;c = M i;c A i;c ad i;d = M i;d A i;d where M i;c is a coiuous local marigale, A i;c is a coiuous process of bouded variaio, M i;d is a purely discoiuous local marigale, ad A i;d is a pure jump process of bouded variaio. Sice he codiio (3.) may fail (due o he exisece of may small jumps) we kow ha Iô s formula akes oly he firs form i (3.4) above. I is well-kow (ad easily verified 3

14 by localizaio usig Taylor s heorem) ha he firs series over < s i (3.4) is absoluely coverge (eve if (3.) fails o hold). The followig heorem exeds he chage-of-variable formula (.6) o geeral semimarigales. Noe ha (.), (.6) ad (3.6) above are special cases of he geeral formula (3.6) below. Theorem 3. Le = ( ;... ; ) be a semimarigale, le b : IR! IR be a coiuous fucio such ha he process b = b( ;... ; ) is a semimarigale, ad le F : IR! IR be a coiuous fucio saisfyig (.3) ad (.4) above. The he followig chage-of-variable formula holds: (3.6) F ( ) = F () ;... ; s ) (s ;... ; s) i F ;... ; ) F (s ;... ; s) j <s F ( s )F ( ( s ;... ; s d i s d[ i;c ; j;c ] s ;... ; s ) (s ;... ; s) i ) ;... ; s) i s! I( s = b s ; s = b s ) d`b s () where `bs () is he local ime of o he surface b give by meas of (3.7) below ad d`bs () refers o iegraio wih respec o he coiuous icreasig fucio s 7! `bs (). Proof. The proof ca be carried ou similarly o he proof of Theorem. ad Theorem 3. ad we will oly highligh a few ovel pois appearig due o he absece of he codiio (3.). The remaiig deails are he same as i he proof of Theorem. ad Theorem 3... We begi as i he proof of Theorem. by iroducig he processes Z ; Z ; ^; ; ~ ad observig ha (.7)-(.9) carries over uchaged. The aalogue of (.) which is obaied by applyig he Taaka formula ow reads: (3.7) j b j = j b j sig(sb s ) d( s b s ) `b () j s b s j j sb sj sig( sb s ) ( b ) s where <s sig() =. Similarly o (.) ad (.) we ow fid ha: (3.8) dz = sig( b ) d sig( b ) db d`b () dj () (3.9) dz = sig( b ) d sig( b ) db d`b () dj () where we deoe: (3.) J () = <s j s bs j j s bs j sig( s bs ) ( b ) s. 4

15 . Applyig he Iˆo formula o F( ^ ) we ge: (3.) F( ^ ) = F( ^) <s D if( ^ s) d ^ i s F( ^ s )F( ^ s) D i F( ^ s) i s D ijf( ^ s) d[ ^ i;c ; ^ Hece usig (3.8) ad proceedig i he same way as i (.3) ad (.5), makig use of he geeral fac (3.), we obai he aalogue of he ideiy (.6) where all i ad ~ i i he iegraors wih he agle brackes are replaced by i;c ad ~ i;c ow wrie as he square brackes, ad he righ-had side of he ideiy coais a ew erm give by: (3.) D F( ^ s) dj s () due o he appearace of dj () i (3.8). 3. Applyig he Iˆo formula o F( ) we ge: (3.3) F( ) = F( ) <s D if( s) d i s F( s )F( s) D i F( s) i s. D ijf( s) d[ i;c ; Hece usig (3.9) ad proceedig i he same way as i (.7) ad (.8), makig use of he geeral fac (3.), we obai he aalogue of he ideiy (.9) where all i ad ~ i i he iegraors wih he agle brackes are replaced by i;c ad ~ i;c ow wrie as he square brackes, ad he righ-had side of he ideiy coais a ew erm give by: (3.4) D F( s) dj s () due o he appearace of dj () i (3.9). 4. Combiig he righ-had sides of he resulig ideiies we fid he aalogue of (.) o be: (3.5) F ( ) = F( ^ ) F( ) F ( ~ ) = F () D i F (s ;... ; s ) D if (s ;... ; s) where he fial erm is give by:. d i s D ij F (s ;... ; s ) D ijf (s ;... ; s) d[ i;c ; j;c ] s D F ( s ;... ; s ) D F ( s ;... ; s) d`b s () R 5

16 (3.6) R = F ( ) ~ 4 4 D if( ~ s) I( s > b s ) d ~ i s D ijf( ~ s) I( s > b s ) d[ ~ i;c ; ~ D if( ~ s) I( s = b s ) d ~ i s <s <s D ijf( ~ s) I( s = b s ) d[ ~ i;c ; ~ D if( ~ s) I( s < b s ) d ~ i s D ijf( ~ s) I( s < b s ) d[ ~ i;c ; ~ D if( ~ s) I( s = b s ) d ~ i s D ijf( ~ s) I( s = b s ) d[ ~ i;c ; ~ F ( ~ ) F( ^ s ) F( ^ s) F( s ) F( s) D F( s) dj s () D i F( ^ s) ^ i s D i F( s) i s D F( ^ s) dj s (). 5. The same argumes as hose give i (.) ad (.3) ow lead o he followig aalogues of (.4) ad (.5) respecively: (3.7) (3.8) = D if( ~ s) I(s > b s ) d ~ i s I(s > b s ) <s F( ~ s ) F( ~ s) D if( ~ s) I(s > b s ) d ~ i s I(s > b s ) <s D ijf( ~ s) I( s > b s ) [ ~ i;c ; ~ F( ~ s ) F( ~ s) D if( ~ s) I( s = b s ) d ~ i s D i F( ~ s) ~ i s D ijf( ~ s) I( s > b s ) [ ~ i;c ; ~ D i F( ~ s) ~ i s D ijf( ~ s) I( s = b s ) [ ~ i;c ; ~ 6

17 = <s I( s = b s ) F( ~ s ) F( ~ s) D if( ~ s) I(s = b s ) d ~ i s I(s = b s ) <s F( ~ s ) F( ~ s) D i F( ~ s) ~ i s D ijf( ~ s) I( s = b s ) [ ~ i;c ; ~ D i F( ~ s) ~ i s Makig use of (3.7) ad (3.8) we see ha F i he firs four iegrals i (3.6) ca be replaced by F upo akig io accou he four series over < s appearig i (3.7) ad (3.8). Addig ad subracig he same series over < s we see ha he firs ie erms o he righ-had side of (3.6), ogeher wih he series added, assemble exacly he expressio obaied by applyig he Iˆo formula o F( ~ ). Sice F ( ~ ) = F( ~ ) hece we see ha he firs e erms obaied o he righ-had side of (3.6) equals he eleveh erm which is he series subraced. Recallig also he four series from (3.7) ad (3.8) his shows ha: (3.9) R = I(s > b s ) F( ~ s ) F( ~ s) <s I(s > b s ) F( ~ s ) F( ~ s) <s <s <s <s <s <s I( s = b s ) I( s = b s ) F( ~ s ) F( ~ s) D i F( ~ s) ~ i s F( ~ s ) F( ~ s) F( ~ s ) F( ~ s) F( ^ s ) F( ^ s) F( s ) F( s) D F( s) dj s () D i F( ~ s) ~ i s D i F( ^ s) ^ i s D i F( s) i s. D i F( ~ s) ~ i s D i F( ~ s) ~ i s D i F( ~ s) ~ i s D F( ^ s) dj s (). From (3.5) we hus see ha he proof of (3.6) reduces o verify he followig ideiy: (3.3) R = F ( s ) F ( s) I(s b s ) D if( s) i s. i s 7

18 To his ed i is helpful o oe ha: (3.3) Z D F ( s) dj s () D F ( ^ s) dj s () = <s <s <s <s I( s > b s ; s = b s ) D F ( ~ s) D F ( ~ s) I( s > b s ; s b s ) D F ( s) D F ( ~ s) ( s b s ) ( s b s ) ( s b s ) I( s < b s ; s = b s D ) F ( ~ s) D F ( ~ s) ( s b s ). A leghy bu sraighforward verificaio shows ha he wo sides i (3.3) coicide i.e. ha he righ-had side of (3.9) equals he righ-had side of (3.3). This ca be doe by recallig ha each series over < s i (3.9) ad (3.3) is absoluely coverge so ha all eleve of hem appearig o he righ-had side of (3.9) ca be combied io a sigle series of he fiie sum of he eleve idividual erms. Muliplyig he sum by each of he idicaors I(s > b s ; s = b s ), I(s = b s ; s = b s ), I( s b s ), I( s b s ), I(s > b s ; s < b s ), I( s = b s ; s < b s ), I( s < b s ; s < b s ) ad comparig he resul wih he correspodig expressio o he righ-had side of (3.3) i is see ha all eigh of hem coicide. This esablishes he ideiy (3.3) ad complees he proof of he heorem. Remark 3.3 I is evide ha he coes of Remark. ad Remark.3 carry over o he seig of Theorem 3. (or Theorem 3.) wihou major chage. By addig he correspodig jump erms o (.9) ad (.3) oe obais heir exesio o geeral semimarigales (or semimarigales wih jumps of bouded variaio). We will omi he explici expressios of hese formulas. 4. The ime-space maximum process I his secio we firs apply he chage-of variable formula (.6) o a hree-dimesioal coiuous semimarigale ad he derive a versio of he same formula uder weaker codiios o he fucio. This versio is useful i he sudy of free-boudary problems.. Le be a diffusio process solvig: (4.) d = (; ) d (; ) db i Iˆo s sese. The laer more precisely meas ha saisfies: (4.) = (r; r ) dr (r; r ) db r for all where ad are locally bouded (coiuous) fucios for which he iegrals 8

19 i (4.) are well-defied (he secod beig Iˆo s) so ha iself is a coiuous semimarigale (he process B is a sadard Browia moio). To esure ha is o-degeerae we will assume ha >. Associaed wih we cosider he maximum process S defied by: (4.3) S = max r r _ S. The ((; ; S )) is a coiuous semimarigale akig values i IR E where we se E = f (x; s) IR j x sg.. Le b : IR IR! IR be a coiuous fucio such ha he process b defied by b = b(; S ) is a semimarigale. Seig: (4.4) C = f (; x; s) IR E j x > b(; s) g (4.5) D = f (; x; s) IR E j x < b(; s) g suppose ha a coiuous fucio F : IR E! IR is give such ha: (4.6) F is C ;; o C (4.7) F is C ;; o D i he sese explaied followig (.3) ad (.4) above. [A sligh oaioal chage i he defiiio of he process ((; ; S )) ad he ses C ad D i compariso wih hose give i Secio above is made o mee he oaio used i [] ad relaed papers.] Moreover, sice > i follows ha: = for r h; ] (4.8) P r = b r so ha uder (4.6) ad (4.7) he chage-of-variable formula (.6) akes he simpler form: (4.9) F (; ; S ) = F (; ; S ) F (r; r ; S r ) I( r 6= b r ) dr F x (r; r ; S r ) I( r 6= b r ) d r F xx (r; r ; S r ) I( r 6= b r ) dh; i r F x (r; r ; S r ) F x (r; r ; S r ) d`b r () where `br () is he local ime of o he surface b give by: (4.) `b r () = IP lim "# " Z r I(" < u b u < ") dhb ; b i u F s (r; r ; S r ) I( r 6= b r ) ds r ad d`br () i (4.9) refers o iegraio wih respec o he coiuous icreasig fucio r 7! `br (). [The appearace of i d`br () is moivaed by he fac ha S is a fucioal 9

20 of.] Noe also ha usig (4.) he formula (4.9) ca be rewrie as (4.) below. 3. I urs ou, however, ha similarly o he case sudied i Secio 3 of [] he codiios (4.6) ad (4.7) are o always readily verified. The mai example we have i mid (arisig from he free-boudary problems meioed above) is: (4.) F (; x; s) = E ;x;s G( D ; D ; S D ) where ( ; S ) = (x; s) uder P ;x;s, a admissible fucio G is give ad fixed, ad: (4.) D = if f r > j (r; r ; S r ) D g. The oe direcly obais he ierior codiio (4.3) by sadard meas while he closure codiio (4.6) is harder o verify a b sice (uless we kow a priori ha r 7! b(r; s) is Lipschiz coiuous or eve differeiable) boh F ad F xx may i priciple diverge whe b is approached from he ierior of C. Moivaed by applicaios i free-boudary problems we will ow prese a versio of he formula (4.9) where (4.6) ad (4.7) are replaced by he codiios: (4.3) F is C ;; o C (4.4) F is C ;; o D. The raioale behid his versio is he same as i []. Give ha oe has some basic corol over F x a b (i free-boudary problems meioed above such a corol is provided by he priciple of smooh fi) eve if F is formally o diverge whe he boudary b is approached from he ierior of C, his deficiecy is couerbalaced by a similar behaviour of F xx hrough he ifiiesimal geeraor of, ad cosequely he firs iegral i (4.) below is sill well-defied ad fiie. 4. Give a subse A of IR E ad a fucio f : A! IR we say ha f is locally bouded o A ( i IR E ) if for each a i A here is a ope se U i IR E coaiig a such ha f resriced o A \ U is bouded. Noe ha f is locally bouded o A if ad oly if for each compac se K i IR E he resricio of f o A\K 6= ; is bouded. Give a fucio g : [; ]! IR of bouded variaio we le V (g)() deoe he oal variaio of g o [; ]. To grasp he meaig of he codiio (4.9) below i he case of F from (4.) above, leig ( =@x deoe he ifiiesimal geeraor of, recall ha he ifiiesimal geeraor IL of ((; ; S )) ca formally be described as follows (cf. []): (4.5) IL = IL i x < = a x = s. Deoig C s = f (; x) j (; x; s) C g ad D s = f (; x) j (; x; s) D g hece we see ha: (4.7) ILF = i C s (4.8) ILF = ILG i D s. This shows ha ILF is locally bouded o C s [ D s as soo as ILG is so o D s. The laer

21 codiio (i free-boudary problems) is easily verified sice G is give explicily. The mai resul of he prese secio may ow be saed as follows (see also Remark 4. below for furher sufficie codiios). Theorem 4. Le be a diffusio process solvig (4.) i Iô s sese, le b : IR IR! IR be a coiuous fucio such ha he process b defied by b = b(; S ) is a semimarigale, ad le F : IR E! IR be a coiuous fucio saisfyig (4.3) ad (4.4) above. If he followig codiios are saisfied: (4.9) F F x ( =)F xx ( ; ; s) is locally bouded o C s [ D s (4.) F x ( ; b( ; s)6"; s)! F x ( ; b( ; s)6; s) uiformly o [; ] as " # (4.) sup V (F ( ; b( ; s)6"; s))() < for some > <"< for each s give ad fixed, he he followig chage-of-variable formula holds: (4.) F (; ; S ) = F (; ; S ) F F x ( =)F xx (r; r ; S r ) I( r 6= b r ) dr (F x )(r; r ; S r ) I( r 6= b r ) db r F s (r; r ; S r ) I( r 6= b r ; r = S r ) ds r Z F x (r; r ; S r )F x (r; r ; S r ) I( r = b r ) d`b r() where `br() is he local ime of a he surface b give by (4.) above, ad d`br() refers o iegraio wih respec o he coiuous icreasig fucio r 7! `br(). Proof. The key observaio is ha off he diagoal x = s i E he process (; ; S ) ca be ideified wih a process (; ) ad he surface process b(; S ) ca be ideified wih a curve b(). By slighly exedig he wo-map argume give i Remark 4. of [6] he previous observaio ca be embedded rigorously i a well-defied mahemaical seig. I his seig he problem becomes equivale o he problem reaed i Theorem 3. of []. Applyig he same mehod of proof, upo makig use of (.6) ad (.9) above, ad relyig upo he properies of he local ime ad Helly s selecio heorem, i is see ha he codiios (3.6)-(3.8) i Theorem 3. of [] become he codiios (4.9)-(4.) above. As his verificaio is leghy, bu i priciple he same, furher deails will be omied (for more deails see []). Remark 4. I is evide ha all of he umber of sufficie codiios discussed i [], which are eiher o imply (4.9)-(4.) or could be used isead, ca easily be raslaed io he prese seig. We will sae explicily oly oe se of hese codiios. Assume ha F saisfies (4.3) ad (4.4)

22 above. If (4.9) is saisfied ad for each s give ad fixed we have: (4.3) x 7! F (r; x; s) is covex or cocave o [b(r; s); b(r; s)] ad covex or cocave o [b(r; s); b(r; s)] for each r [; ] wih some > (4.4) r 7! F x (r; b(r; s)6; s) is coiuous o [; ] wih values i IR he boh (4.) ad (4.) hold. This shows ha (4.3) ad (4.4) imply (4.) whe (4.9) holds. The codiio (4.3) ca furher be relaxed o he form where: (4.5) F xx ( ; ; s) = G ( ; ; s) G ( ; ; s) o C s [ D s where G ( ; ; s) is o-egaive (o-posiive) ad G ( ; ; s) is coiuous o C s ad D s for each s give ad fixed. Thus, if (4.4) ad (4.5) hold, he boh (4.) ad (4.) hold implyig also (4.) whe (4.9) holds. REFERENCES [] EISENBAUM, N. (). Iegraio wih respec o local ime. Poeial Aal. 3 (33-38). [] EISENBAUM, N. (4). Local ime-space sochasic calculus for Lévy processes. Prepri. [3] ELWORTHY, K. D., TRUMAN, A. ad ZHAO, H. Z. (3). A geeralized Iˆo formula ad asympoics of hea equaios wih causics, i oe-dimesio. Prepri. [4] FÖLLMER, H., PROTTER, P. ad SHIRYAYEV, A. N. (995). Quadraic covariaio ad a exesio of Iô s formula. Beroulli (49-69). [5] GHOMRASNI. R. ad PESKIR, G. (3). Local ime-space calculus ad exesios of Iˆo s formula. Proc. High Dim. Probab. III (Sadbjerg ), Progr. Probab. Vol. 55, Birkhäuser Basel (77-9). [6] GRAVERSEN, S. E. ad PESKIR, G. (998). Opimal soppig ad maximal iequaliies for geomeric Browia moio. J. Appl. Probab. 35 (856-87). [7] ITÔ, K. (944). Sochasic iegral. Proc. Imp. Acad. Tokyo (59-54). [8] KYPRIANOU, A. E. ad SURYA, B. A. (4). A chage-of-variable formula wih local ime o curves for Lévy processes of bouded variaio. Prepri. [9] MEYER, P. A. (976). U cours sur les iégrales sochasiques. Sém. Probab., Lecure Noes i Mah. 5 (45-4). [] PESKIR, G. (998). Opimal soppig of he maximum process: The maximaliy priciple. A. Probab. 6 (64-64). [] PESKIR, G. (5). A chage-of-variable formula wih local ime o curves. J. Theore. Probab. 8 ( ). [] PROTTER, P. (4). Sochasic Iegraio ad Differeial Equaios. Spriger-Verlag, Berli.

23 [3] REVUZ, D. ad YOR, M. (999). Coiuous Marigales ad Browia Moio. Spriger- Verlag, Berli. [4] TANAKA, H. (963). Noe o coiuous addiive fucioals of he -dimesioal Browia pah. Z. Wahrscheilichkeisheorie ud Verw. Gebiee (5-57). [5] WANG, A. T. (977). Geeralized Iô s formula ad addiive fucioals of Browia moio. Z. Wahrscheilichkeisheorie ud Verw. Gebiee 4 (53-59). Gora Peskir Deparme of Mahemaical Scieces Uiversiy of Aarhus, Demark Ny Mukegade, DK-8 Aarhus home.imf.au.dk/gora gora@imf.au.dk 3

STK4080/9080 Survival and event history analysis

STK4080/9080 Survival and event history analysis STK48/98 Survival ad eve hisory aalysis Marigales i discree ime Cosider a sochasic process The process M is a marigale if Lecure 3: Marigales ad oher sochasic processes i discree ime (recap) where (formally