The minimal entropy martingale measures for exponential additive processes revisited Tsukasa Fujiwara

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1 Jornal of Math-for-Indstry, Vol22B-, pp5 25 Th minimal ntropy martingal masrs for xponntial additiv procsss rvisitd Tskasa Fjiwara Rcivd on Jn 2, 2 Abstract Th mimimal ntropy martingal masr for th stochastic procss dfind as th xponntial of an additiv procss with th strctr of smimartingal will b invstigatd Spcial attntion will b paid to th cas whn th ndrlying additiv procss has fixd tims of discontinity Th invstigation of this papr will stablish a nifid way that is applicabl both to th cas of Lévy procsss and that of th sms of indpndnt random variabls Kywords additiv procss, procss with indpndnt incrmnts, smimartingal, minimal ntropy martingal masr, xponntial momnt, Laplac cmlant, modifid Laplac cmlant Introdction Lt X t X t,, X d t t [,T ], T >, b an R d -vald additiv procss with th strctr of smimartingal, in othr words, a d-dimnsional smimartingal with indpndnt incrmnts According to [7], w will also call sch a stochastic procss as X t a d-dimnsional PII-smimartingal W always sppos that X Lt S t S t,, S d t t [,T ] b a stochastic procss dfind as th xponntial of X t : S i t : S i Xi t, i,, d whr w sppos that ach S i is a positiv constant W call sch a stochastic procss as S t a d-dimnsional xponntial additiv procss basd on th additiv procss X t or, for simplicity, an xponntial PII-smimartingal basd on X t Th prpos of this papr is to propos a condition ndr which th mimimal ntropy martingal masr MEMM for S t xists and to rprsnt th MEMM xplicitly by th charactristics of X t In a sris of prvios paprs, w hav discssd this problm in th cas whn X t is a Lévy procss in [5] and [2]; 2 X t is a stochastically continos PII-smimartingal in [3], rspctivly In this papr, w ar intrstd in th cas whn X t is a PII-smimartingal that is not ncssarily stochastically continos, in othr words, that may hav fixd tims of discontinity Hnc, w can say that th aim of this papr is to giv a final answr to th problm dscribd abov in th framwork of xponntial PIIsmimartingals In Sction 2, w will rviw svral proprtis of PIIsmimartingals In particlar, Thorm plays fndamntal rol in showing Corollary that nsrs th xistnc of xponntial momnts of intgrals of dtrministic procsss basd on X t and that givs th rprsntation of thm by th charactristics of X t In Sction 3, w will prcisly stat or main rslt of this papr, Thorm 2, whr th xistnc and th rprsntation of th MEMM for S t will b shown ndr a mild condition C Owing to rmoving th rstriction of stochastic continity of th ndrlying PII-smimartingal X t, w can also trat th cas whn it is dfind by a sm of indpndnt random variabls in a nifid framwork S Corollary 2 In Sction 4, w will giv a proof of Thorm 2 It follows on th stram proposd in th proof of Thorm 3 in [5] Howvr, w will s that sitabl modification of discssions and dpr considration will b ndd to ovrcom th difficltis arising from th xistnc of fixd tims of discontinity 2 Additiv procsss and xponntial additiv procsss additiv procss with th strctr of smimartingals Lt Ω, F, P b a probability spac qippd with a filtration F t that satisfis th sal conditions S [7] I2 p2 for th dfinition of th sal conditions Lt X t X t,, X d t t [,T ], T >, b an R d -vald additiv procss with th strctr of smimartingal dfind on th probability spac Ω, F, P with F t To b prcis, X t is an R d -vald adaptd càdlàg procss with X that has th following proprtis: 5

2 6 Jornal of Mathmatics for Indstry, Vol22B- X t is a procss with indpndnt incrmnts: for all s t, th incrmnt X t X s is indpndnt of F s ; 2 X t is a smimartingal with rspct to th filtration F t According to [7], w will also call sch a stochastic procss as X t a d-dimnsional PII-smimartingal W wold lik to mphasiz that throghot this papr w do not ncssarily assm that X t is stochastically continos Not that, in or schm, th stochastic continity is qivalnt to th proprty of having no fixd tim of discontinity and also to th qasi-lft continity S [6] Corollary 28 p38 and [7] Thorm II48 p7 Lt C t, ndtdx, B t b th charactristics of X t associatd with th trncation fnction hx : xi x x on R d In othr words, lt th canonical rprsntation of X t associatd with h b givn as follows [7] Thorm II234 p84: X t Xt c B t hx Ñddx whr : R d \ Hr, ȟx Nddx, Xt c is a continos local martingal with X c and C ij t X c,i, X c,j for i, j,, d t Nddx dnots th conting masr of th jmps of X t : N, t], A :, t]; X : X X A for A B, whr X : lim v X v and B is th Borl σ-fild on W dnot by Ñddx : Nddx nddx th compnsatd masr of Nddx, whr nddx is th compnsator of Nddx Also, w st ȟx : x hx Each componnt Bt i i,, d is a càdlàg fnction with finit variation on [, T ] [7] Dfinition II26 p76 As fndamntal proprtis of charactristics, th folllowing facts ar known: C t, ndtdx, B t ar dtrministic, sinc X t has indpndnt incrmnts [7] Thorm II45 p6 x 2 nddx <, whr α β : minα, β for α, β R, and n, [7] II23 p77 B hx n, dx [7] II24 p77 Also, not that th law of X t is charactrizd by th Lévy-Khinchin formla [7] Thorm II45 p6: E P [ ξ X t X s ] [ xp 2 ξ C t C s ξ ξ B t B s ξ x ξ hx s,t] ξ B ] I J c nddx [ ξ x n, dx ], whr a b dnots th innr prodct of a, b R d ; J dnots th st of all fixd tims of discontinity of X t, that is, J : t > ; nt, > -dimnsional PII-smimartingal and th xponntial momnt Lt Y t t [,T ], T >, b a -dimnsional PII-smimartingal and C Y t, n Y dtdy, B Y t th charactristics of Y t associatd with th trncation fnction h y : yi y y on R In [4], w hav shown th following rslt with an xplicit proof S Thorm thrin Thorm 2 Sppos that y n Y ddy < y> Thn, Y tk Y t t is a niformly intgrabl martingal with man, whr K Y t is th modifid Laplac cmlant of Y t at : 3 K Y t 2 CY t Bt Y y h y n Y ddy R log y n Y, dy R y n Y, dy R In particlar, 4 E[ Y t ] KY t intgral basd on th d-dimnsional PII-smimartingal X t : Lt θ θ,, θ d b an R d -vald Borl masrabl fnction Not that it is dtrministic W say that θ is

3 Tskasa Fjiwara 7 intgrabl with rspct to X t if th following conditions i iii ar satisfid: i ii iii d i θ dc θ : d i,j θ i dc ij θ j <, θ i dvarb i <, whr VarA t dnots th total variation of th fnction A on th intrval [, t], θ hx 2 nddx < W dnot by LX th st of all intgrabl fnctions with rspct to X t Not that an arbitrary bondd masrabl fnction blongs to LX Lt θ LX Thn, w can dfin an intgral θ dx of θ basd on X t by 5 θ dx : θ dx c θ db θ hx Ñddx θ ȟx Nddx Th following rslt is shown as Proposition in [4]: Proposition Lt θ LX Thn Y t : θ dx is a -dimnsional PII-smimartingal; th charactristics C Y, n Y, B Y associatd with h on R ar givn by Ct Y n Y, t], A Bt Y θ dc θ ; R I A θ x nddx, A BR ; θ db h θ x θ hx nddx xponntial momnt of θ dx : Th following rslt is shown as Corollary in [4]: Corollary 6 Lt θ LX and sppos that θ x> θ x nddx < Thn, θ dx K X θ t t is a niformly intgrabl martingal with man, whr K X θ t is th modifid Laplac cmlant of X t at θ : 7 K X θ t 2 θ dc θ log θ db θ x θ hx nddx θ x n, dx In particlar, 8 E[ θ dx ] KX θ t θ x n, dx Rmark If θ is bondd, th condition 6 can b rplacd by th on θ x nddx < x > Rmark 2 Th sm of th first thr trms in th right hand sid of 7 is calld th Laplac cmlant of X t at θ and dnotd by K X θ t : 9 K X θ t 2 θ dc θ θ db θ x θ hx nddx Th modifid Laplac cmlant and th Laplac cmlant ar rlatd to ach othr throgh th following rlation: KX θ t E K X θ t S [8] and [7] for fndamntal proprtis of th modifid Laplac cmlant in th framwork of th thory of smimartingals xponntial additiv procss Lt S t S t,, S d t b an R d -vald stochastic procss dfind by S i t : S i Xi t, i,, d, whr w will asm that all of S i ar positiv constants W call S t th xponntial additiv procss basd on th additiv procss X t For simplicity, w will also call it th xponntial PII-smimartingal basd on X t Thn, it follows from Itô s formla that St i S i Sd i X, i whr Xi t : X i t 2 Xi,c t X i X i

4 8 Jornal of Mathmatics for Indstry, Vol22B- Combining this dfinition with, w s that th canonical rprsntation of X t associatd with h is givn as follows: Proposition 2 2 X t Xt c B t 2 C t hex I hx nddx hex I Ñddx ȟex I Nddx, whr w st Ex : x,, xd for x x,, x d, I :,, R d and d 3 C i t : C ii t It is also immdiat from Proposition 2 that th following proposition holds: Proposition 3 Th charactristics Ĉ, n, B of X t associatd with h ar givn by n, t], G B t B t 2 C t Ĉ t C t ; I G Ex I nddx, G B; hex I hx nddx 3 Th minimal ntropy martingal masrs for xponntial additiv procsss W will s th sam notation as in Sction 2 In particlar, rcall that X t t [,T ] is a d-dimnsional PII-smimartingal, dfind on a filtrd probability spac Ω, F, F t, P satisfying th sal conditions, with charactristics C t, ndtdx, B t associatd with th trncation fnction hx Morovr, S t dnots th xponntial PII-smimartingal dfind by In this sction, w will prcisly stat or main rslt Thorm 2 blow To this nd, w prpar som notion minimal ntropy martingal masr For a probability masr Q on th masrabl spac Ω, G, whr G is a sb-σ-fild of F, th rlativ ntropy of Q on G with rspct to P is dfind as follows: dq E Q [log ], if Q is absoltly continos with rspct dp G H G Q P : to P on G, othrwis, whr dq dp G is th Radon-Nikodym drivativ of Q with rspct to P on G Nxt, w dnot by ALMMP th st of all absoltly continos probability masrs on Ω, F T with rspct to P sch that S t is an F t -local martingal ndr Q An lmnt of ALMMP is calld an absoltly continos local martingal masr for S t For a class D of probability masrs on Ω, F T, w call an qivalnt martingal masr th qivalnt minimal ntropy martingal masr MEMM for S t in D if it minimizs th vals of th fnction: Q D H FT Q P main rslt W ar now in a position to stat or main rslt Or main objctiv is to show th xistnc and th rprsntation of th MEMM for S t in ALMMP To stablish this, w propos th following condition C for S t, which is dscribd by th charactristics C t, ndtdx, B t of th ndrlying PII-smimartingal X t Condition C: Thr xists an R d -vald bondd Borl masrabl fnction θ, [, T ], that satisfis th following i and ii: i for ach i,, d, 7 xi θ ExI nddx < ; x > ii for all t [, T ], Bt c 8 2 C t 9 dc θ Ex I θ ExI hx I J c nddx ; Ex I θ t ExI nt, dx Hr, B c t dnots th continos part of th fnction with finit variation B t : B c t : B t B Thorm 2 Sppos that th condition C holds Thn w hav th following I III I [ xp θ d X K Xθ ] t t [,T ]

5 Tskasa Fjiwara 9 is a tr maringal ndr th probability P, whr X t is th stochastic procss of and K Xθ t is th modifid Laplac cmlant of X t at θ : 2 K Xθ t θ dc θ 2 θ dc θ db 2 θ ExI θ hx nddx log θ ExI n, dx θ ExI n, dx In particlar, 2 E P [ θ d X ] K X θ t Thrfor, a probability masr P on F T is consistntly dtrmind by 22 dp dp : Ft E P [ θ d X θ d X ] II Undr th probability masr P of 22, th stochastic procss X t of 2 is an additiv procss; th charactristics C t, n dtdx, B t associatd with th trncation fnction hx : xi x x ar givn by C t C t ; θ t ExI n dtdx K Xθ ndtdx; t Bt B t dc θ θ ExI hx R d K Xθ nddx, whr K Xθ t is th Laplac cmlant of X t at θ : 26 K Xθ t 2 θ dc θ θ db θ dc 2 θ ExI θ hx nddx and 27 K Xθ t θ t ExI nt, dx Frthrmor, P is an qivalnt martingal masr for S t of III Th probability masr P of 22 attains th minimal ntropy in th class ALMMP : 28 min H F T Q P H FT P P K Xθ T Q ALMMP discrt cas Lt ξ, ξ 2, b a sqnc of R d -vald random variabls dfind on a probability spac Ω, F, P with a filtration G k of sb-σ-filds of F; sppos that ξ k is adaptd to G k for ach k N and that, for all j < k, ξ k is indpndnt of G j Lt 29 X t : ξ k, [t] k whr [t] dnots th gratst intgr that dos not xcd th ral nmbr t Thn X t of 29 can b rgardd as a PII-smimartingal with rspct to th filtration F t : G [t] Th canonical rprsntation of X t associatd with h is givn as follows: X t B t hx Ñddx ȟx Nddx, whr B t [t] k E[hξ k]; N, t], A : k N, t]; ξ k A for A B; th compnsator of Nddx is givn by nk, A P [ξ k A] Not that J N and that I J cnddx In th stting abov, th condition C is rdcd to th following on: Condition C d : Thr xisits an R d -vald fnction θk, k,, [T ], that satisfis th following: for ach k,, [T ] and i,, d, i ii x > xi θ k ExI nk, dx E[ ξi k θ k Eξ ki ; ξ k > ] < ; xi θ k ExI nk, dx E[ ξi k θ k Eξ ki ] Thn it is asy to obtain th following rslt from Thorm 2

6 2 Jornal of Mathmatics for Indstry, Vol22B- Corollary 2 Sppos that th condition C d holds Thn th probability masr P dfind by dp [t] k θ Eξ ki : dp Ft [t], t [, T ] k k EξkI ] E[θ is th mimimal ntropy martingal masr for S t : S [t] k ξ k in th class ALMMP ; min H F T Q P H FT P P Q ALMMP [t] log E[ θ k EξkI ] k 4 Proof of Thorm 2 In this sction, w will giv a proof of Thorm 2 Roghly spaking, it follows on th stram proposd in th proof of Thorm 3 in [5]; howvr, w nd to considr mor finly to ovrcom th difficltis arising from th xistnc of fixd tims of discontinity Proof of I W will apply Corollary to th stochastic intgral θ d X Rcall that, as w hav shown in Proposition 3, th charactristics Ĉ, n, B of X t associatd with h ar givn by 4, 5 and 6 Thn, it follows from th condition C-i that <, x > θ x nddx ExI > x >/α θ ExI nddx θ ExI nddx bcas Ex I > x > /α for som α > Thrfor, w s from Corollary and Rmark that θ d X K X θ t t is a niformly intgrabl martingal with man and hnc E[ θ d X ] K X θ t, whr K Xθ t is th modifid Laplac cmlant of X t at θ : K Xθ t 2 θ dĉ θ log θ d B θ x θ hx nddx θ x n, dx θ x n, dx Frthrmor, it is asy to s from Proposition 3 that 2 holds Proof of II W will first show that 26 and 27 hold By th dfinition of th Laplac cmlant, K Xθ t θ dĉ θ θ d 2 B θ x θ hx nddx Hnc, as in th proof of 2, it follows from Proposition 3 that 26 holds Morovr, sinc C t is continos and B t hx nt, dx, w s that K Xθ t θt B t θ t ExI θ t hx nt, dx θ t ExI nt, dx Nxt, w will show th following proposition: Proposition 4 Lt Dt b th dnsity procss of P against P, that is, Dt dp ˇM t, whr w st dp Ft 3 ˇM t : θ d X K Xθ t Thn, 3 D t ˇM t EM t, whr 32 Mt : θ dx c θ ExI R d K Xθ Ñddx Bfor giving a proof of this proposition, w not th following rslt, which nsrs that th scond trm in th right hand sid of 32 maks sns as a martingal

7 Tskasa Fjiwara 2 Lmma Lt V t : K Xθ t Thn /V t t is niformly bondd Proof For ach n N, lt inf, T ]; > n, V /2 n : if, T ]; > n, V /2 ; T if, T ]; > n, V /2 W will show that thr xists n N sch that n T To s this, sppos that for all n N, n < T Thn, V n /2, and hnc K Xθ n /2 Thrfor, for all n, < K Xθ n /2 /2, which implis that K Xθ n /2 Hnc w hav K Xθ K Xθ n 2 n n Howvr, this rslt contradicts to th fact that K Xθ t is a fnction with finit variation on [, T ] Hnc th hypothsis that n < T for all n N is rjctd Ths, it holds that thr xists n N sch that n T This mans that th nmbr of tims t, T ] sch that V t /2 is finit Also, sinc V t > for ach t, w s that sp t /V t < Proof of Proposition 4 By Proposition 2 and th rprsntation 2, w s that 33 ˇM t θ d X K Xθ t θ dx c θ Ex I Ñddx θ dc θ 2 θ ExI θ Ex I nddx log K Xθ K Xθ, whr w hav sd 27 to rprsnt th last trm in th right hand sid Also, not that Hnc, it follows from Itô s formla [7]p57 that 35 ˇM t ˇM θ dx c ˇM θ Ex I Ñddx ˇM θ 2 dc θ ˇM θ ExI 2 ˇM θ Ex I nddx log K Xθ K Xθ ˇM θ dc θ ˇM ˇM ˇM ˇM θ dx c ˇM θ Ex I Ñddx ˇM θ ExI θ Ex I nddx ˇM θ ExI R d K Xθ θ Ex I Nddx ˇM K Xθ ˇM θ dx c ˇM θ ExI K Xθ Ñddx θ ExI ˇM K Xθ K Xθ nddx ˇM K Xθ R d K Xθ Nddx ˇM K Xθ Not that 36 K Xθ for J c 34 ˇM θ E X I log K Xθ Thrfor, w s that th sm of th last thr trms in

8 22 Jornal of Mathmatics for Indstry, Vol22B- th right hand sid is qal to 37 ˇM θ ExI ˇM K Xθ K Xθ I J nddx ˇM K Xθ ˇM K Xθ K Xθ I J Nddx θ ExI K Xθ K Xθ n, dx K Xθ K Xθ ˇM ˇM K Xθ ˇM K Xθ K Xθ K Xθ Combining 35 and 37, w obtain ˇM t ˇM θ dx c ˇM K Xθ K Xθ K Xθ θ ExI K Xθ Ñddx Thrfor, if w tak Mt of 32, w hav ˇM t ˇM dm, that is to say, w obtain 3 Ths, w hav compltd th proof of Proposition 4 Proposition 5 Undr th probability P, th canonical rprsntation of X t associatd with h is givn as follows: 38 X t X,c t Bt hx Ñ ddx ȟx Nddx, whr 39 4 X,c t : Xt c dc θ, Ñ ddx : Nddx n ddx, and B t is th dtrministic procss dfind by 25 Proof Rcall that, ndr P, Xt c is a continos local martingal with X c,i, X c,j t Cij t Hnc, it follows from th rprsntation 3 of th dnsity procss Dt that D d X c, D d X c, D dc θ, ] D θ dx c Thrfor, by Thorm 49 in [], w s that X,c t of 39 is a continos local martingal ndr P It is also clar that X,c,i, X,c,j t Cij t Nxt, w will show that th compnsator N ddx of Nddx ndr P is qal to n ddx of 24 Lt A b any st in BR d \ satisfying n, T ], A < and st A ε : A ε < x < /ε for an arbitrary ε, Also, lt N t : I Aε x Ñddx Thn, it follows from th rprsntation 3 of th dnsity procss D t that N, D t and hnc I Aε x D θ ExI K Xθ nddx I Aε x n, dx D θ ExI K Xθ n, dx, D d N, D ExI I Aε x θ R d K Xθ nddx I Aε x n, dx K Xθ K Xθ Hr again, w hav sd th rlation 27 to gt th last trm Thrfor, w s that

9 Tskasa Fjiwara 23 N t D d N, D N, t], A ε θ ExI I Aε x K Xθ nddx I Aε x K Xθ K Xθ nddx I Aε x n, dx K Xθ K Xθ Sinc th sm of th last two trms in th right hand sid is qal to, w obtain N t D d N, D θ ExI N, t], A ε I Aε x R d K Xθ nddx N, t], A ε I Aε x n ddx On th othr hand, it follows from Thorm 49 in [] that N t D d N, D is a local martingal ndr P Thrfor, w s that N, t], A ε I R d Aε x n ddx is a local martingal ndr P for any ε, Ltting ε, w can concld that N, t], A I R d A x n ddx is a local martingal ndr P s p82 in [3] for a prcis argmnt, which implis that th compnsator N ddx of Nddx ndr P is qal to n ddx of 24 By th discssion abov, th compnsatd masr Ñ ddx of Nddx ndr P is givn by 4 Hnc, th canonical rprsntation of X t associatd with h ndr P is givn as follows: X t Xt c B t hx Ñddx X,c t B t ȟx Nddx dc θ θ ExI hx R d K Xθ nddx hx Ñ ddx ȟx Nddx Thrfor, th third componnt B t of th charactristics of X t ndr P is givn by 25 Ths, w hav compltd th proof of Proposition 5 By Proposition 5, it is clar that th charactristics C t, n dtdx, B t associatd with h of X t ndr P ar givn by 23, 24 and 25 Also, sinc thy ar dtrministic and Bt is a procss with finit variation on [, T ], X t is a PII-smimartingal ndr P In th rmaindr of this part Proof of II, w will show that P is an qivalnt martingal masr for S t of For ach i,, d, lt θ i :,,,,,, R d, whr is on th i-th componnt, and Y i t : θi dx K X θ i t Not that w hav rgardd X t as a smimartingal with rspct to P and hav dnotd by K X θ t th modifid Laplac cmlant of X t at θ Hnc, 4 K X θ i t θ i dc θ i 2 log θ i db θ i x θ i hx n ddx θ i x n, dx θ i x n, dx By Lmma and th condition C-i, w can chck that θi x n ddx < x > Thrfor, it follows from Corollary that Y i t t is a niformly intgrabl martingal ndr P Nxt, w will show that 42 K X θ i t Sinc th charactristics C t, n dtdx, B t associatd with h of X t ndr P ar givn by 23, 24 and 25, it follows from 4 that 43 K X θ i t Bt i 2 Ci t log dc θ i θ ExI xi K Xθ h i x nddx x i n, dx x i n, dx Hr, not that it follows from 9 that θ ExI 44 Ex I K Xθ n, dx,

10 24 Jornal of Mathmatics for Indstry, Vol22B- which implis that th last trm in th right hand sid of 43 is qal to Frthrmor, w can s that th condition C-ii implis that th sm of th first for trms in th right hand sid of 43 is also qal to as follows: Lmma 2 Th condition C-ii implis that 45 B t 2 C t dc θ Ex θ ExI I R d K Xθ hx nddx Proof By th fact that B hx n, dx and 44, w hav Ex θ ExI B I R d K Xθ hx n, dx Hnc, w obtain 46 Ex θ ExI B I R d K Xθ hx I J nddx On th othr hand, d to th proprty 36, w s from 8 that Bt c 47 2 C t dc θ Ex θ ExI I K Xθ hx I J c nddx Ths, combining 46 and 47, w s that 45 holds Ths, w hav shown that 42 holds and hnc w s that Xi t t is a tr martingal ndr P, in othr words, P is an qivalnt martingal masr for S t Proof of III Lt Q b an arbitrary absoltly continos martingal masr for S t satisfying H FT Q P < Thn, Xi t is a local martingal with rspct to Q S S i ds i Thorm III29 in [9] p28 Hnc, θ d X is a local martingal with rspct to Q S Thorm IV29 in [9] p7 Thrfor, thr xists an incrasing sqnc τ n n of stopping tims sch that τ n as n and that,t τ n ] θ d X is a martingal with rspct to Q for ach n N In particlar,,t τ n ] θ d X is intgrabl with rspct to Q; dp log dp FT τ n,t τ n ] θ d X K Xθ T τn Not that K Xθ T τn is niformly bondd with rspct to n and ω, sinc K Xθ t is actally a fnction with dp finit variation on [, T ] Hnc, log is intgrabl with rspct to Q Thrfor, w s from Lmma dp FT τn 2 in [5] that H FT Q P H Q P FT τ n dp E Q [log ] dp FT τ n E Q [ θ d X ] E Q [K Xθ T τn ],T τ n ] E Q [K Xθ T τn ] On th othr hand, sinc τ n as n, it follows from th bondd convrgnc thorm that lim n EQ [K Xθ T τn ] E Q [K Xθ T ] K Xθ T Ths, w hav shown that Nxt, w will show that H FT Q P K Xθ T H FT P P K Xθ T To this nd, w not th following fact: Proposition 6 48 Xt X,c t Ex I Ñ ddx Proof As w hav shown in th stp 2 that, ndr P, X t is a PII-smimartingal with charactristics C t, n dtdx, B t, w s from Proposition 2 that th corrsponding canonical rprsntation is givn by X t X,c t Bt 2 C t hex I hx n ddx hex I Ñ ddx ȟex I Nddx

11 Tskasa Fjiwara 25 Hnc, w hav X t X,c t Bt 2 C t hex I hx n ddx ȟex I n ddx Ex I Ñ ddx Not that th sm of th scond trm throgh th fifth trm in th right hand sid is qal to Bt 2 C t Ex I hx n ddx B t 2 C t dc θ θ ExI Ex I R d K Xθ hx nddx, whr w hav sd Lmma 2 to gt th last qality Ths, w hav obtaind 48 By Proposition 6, w s that E P [ θ d X ], and hnc H FT P P E P [ K Xθ T θ d X K Xθ T ] Ths, w hav th conclsion of th stp 3: for any absoltly continos martingal masr Q for S t satisfying H FT Q P <, Rfrncs [] Dllachri, C, Myr, P A: Probabilitis and Potntial B Thory of Martingals, North-Holland, 982 [2] Fjiwara, T: From th minimal ntropy martingal masrs to th optimal stratgis for th xponntial tility maximization: th cas of gomtric Lévy procsss, Asia-Pacific Financial Markts 26, [3] Fjiwara, T: Th minimal ntropy martingal masrs for xponntial additiv procsss, Asia-Pacific Financial Markts 6 29, [4] Fjiwara, T: On th xponntial momnts of additiv procsss with th strctr of smimartingals, Jornal of Math-for-Indstry 22A-2 2, 3 2 [5] Fjiwara, T, Miyahara,Y: Th minimal ntropy martingal masrs for gomtric Lévy procsss, Financ and Stochastics 7 23, [6] H, SW, Wang, JG, Yan, JA: Smimartingal Thory and Stochastic Calcls, Scinc Prss/CRC Prss, 992 [7] Jacod, J, Shiryav, A N: Limit Thorms for Stochastic Procsss, Scond dition, Springr, 23 [8] Kallsn, J, Shiryav, A N: Th cmlant procss and Esschr s chang of masr, Financ and Stochastics 6 22, [9] Prottr, P: Stochastic Intgration and Diffrntial Eqations A Nw Approach, Appl Math 2 Scond dition, Springr, 24 Tskasa Fjiwara Dpartmnt of Mathmatics, Hyogo Univrsity of Tachr Edcation, Kato, Hyogo , Japan tskasaathyogo-acjp H FT Q P K Xθ T H FT P P W hav at last compltd or proof of Thorm 2 Acknowldgmnts Th athor wold lik to thank th anonymos rfr for giving svral commnts and many corrctions on this papr Th contnts of this papr ar basd on th athor s prsntation at th workshop: Mathmatical financ and th rlatd filds hld on Janary 2 and 22, 2 at Nagoya Univrsity Th athor wold lik to xprss his gratitd to th organizr Profssor Koichiro Takaoka for giving him a chanc of prsntation