Computation of Greeks using Malliavin's calculus in jump type market models

(1)

HAL Id: inria-00070525

https://hal.inria.fr/inria-00070525

Submitted on 19 May 2006

HAL is a multi-disciplinary open access archive for the deposit and dissemination of sci- entific research documents, whether they are pub-

L’archive ouverte pluridisciplinaire HAL, est destinée au dépôt et à la diffusion de documents scientifiques de niveau recherche, publiés ou non,

Computation of Greeks using Malliavin’s calculus in jump type market models

Marie-Pierre Bavouzet, Marouen Messaoud

To cite this version:

Marie-Pierre Bavouzet, Marouen Messaoud. Computation of Greeks using Malliavin’s calculus in jump type market models. [Research Report] RR-5482, INRIA. 2005, pp.31. �inria-00070525�

(2)

ISRN INRIA/RR--5482--FR+ENG

a p p o r t

d e r e c h e r c h e

Thème NUM

INSTITUT NATIONAL DE RECHERCHE EN INFORMATIQUE ET EN AUTOMATIQUE

Computation of Greeks using Malliavin’s calculus in jump type market models

Marie-Pierre Bavouzet and Marouen Messaoud

N° 5482

February, 1 2005

(3)

(4)

-

$

-

$&.! /0 2134 56 7 $!

8:9<;&=?>A@&BC=D>E;&;&>GFH9JILKNMOP>PQ

∗ 92RST8:92;&KNM>JRT8U>EVWVX9YKLMS

†

Z[]\&^`_acbCdfe gJhJikjl\&^`_&inmPo]^p!qlrtsJou_&i

vwqlxzyk_jCd{wZ(|c}~

n]<xAqkj5]_%ql_&[u_&ql[u_mYwe }2_!]qlo]qkh A%eE]A_&i

!z] _¡o]ik_`jl[]_d¢t¢£r¤¥Jr£m¢t&o]¢to]iC¦DxAq%v§xr£ilikxAmG]qlxJ&_!ilil_!irtm3xAql]_!qj¨x

&xA^`]oujl_Hil_!m]ilr©j¨rt¥Jr©j¨r£_!iª¦DxAqC«¬ouqlxA<_mxAEj¨r£xm]iª®(r©j¨[3o]m]u_&ql¢©hJr£m]¦DxA¢t¢£xW®(r£mu)yko]^`

j¯hE<_%]rt°Lo]ikr£xAm0±

Z[]_²^rtm³<xArtmPj`rti¡jlx´ik_!jkj¨¢£_²m³r£mj¨_!Aq¨zj¨r£xm³h³Yqµj¨i¶¦DxAqk^Ho]¢¤³·Dilr£^`r£¢£qjlxGjl[]_

xAm]_r£mj¨[]_d¢£¢£r£¥Ertm)&¢£!o]¢£o]i¨¸¹¦DxAqH_&m]_!q¨¢0^o]¢tjlr£]rt^_!m]ilrtxAmY¢Yqlm]]xA^¥qlr£]¢£_

®([]r£[`[Yi¬m`uilxA¢touj¨_!¢thH!xAmj¨rtmJouxAo]i¬¢¤®³®(rtj¨[¡]r©°N_!ql_!mPjlr¤u¢£_5]_!m]ilr©j¯hA± _(Art¥_(m

_ºE]¢£rt&r©j_ºE]ql_!ilikr£xAmHx¦]j¨[]_¹]r©°N_!ql_&mjlr¤¢xA<_&qlj¨xqlir£m¥Ax¢t¥A_!Hr£m%j¨[ur£i ¦DxAql^o]¢¤(m]j¨[]rti

L_!ql^`rtj¨i»j¨x¡ilrt^Ho]¢¤zj¨_cjl[]_&^mu!xAm]ik_&sPo]_&mjl¢thj¨x¡qloum`d²xAmjl_¼5ql¢£x`¢tAxAqkrtj¨[u^²±

½²¾¿NÀkÁHÂ

zÃ§

d¢£¢tr¤¥Jr£m¢t&o]¢to]i&Jd²xmPjl_Ä,¼5ql¢tx¢£AxAqkrtjl[]^²«wo]¢£_!qwil[]_!^_E!xA^¶Ä

LxAoum]v§xArtililxm²]qkxE!_&iki&2il_!m]ilr©j¨rt¥Jr©j¨r£_!i&xA^`]ouj¨j¨rtxAm0±

∗ ÅDÆÇ ÅDÈÉ¤ÇÊËkÌ&ÍXÎµÏzËµÊÍXÐÒÑkÓWÎkÔÕ×ÖÙØtÚ]ÔÕ×ÐÛÖÜÎµÉ£ÝzÖÜÎkÐÒÐÛÎÞßzÕ¨àÊ!ÍXáµÎµÑ¯â§ÖÙÏXÐÛÖÙÕzÞãäÐ

† ÅDÆÇ ÅDÈÉ¤ÇÊËkÌ&ÍXÎµÏzËµÊÍXÐÒÑkÓWÎkÔÕ×ÖÙØtÚ]ÔÕ×ÐÛÊÍXÎkÏPÞÔåÎlæÒæÛÕ×Ê!Ízçâ§ÖÙÏXÐÛÖÙÕzÞãäÐ

(5)

- - -

76$(. %

-

(

m`ouj¨rt¢£rtil_5¢t_n&¢£!o]¢]]_cd¢t¢£r£W¥JrtmHLxo]q¬]_&i¹]qlxJ&_!iliko]iå]_nvxAr£ikilxAm)Ym

]_¶¢t&o]¢t_&q¢t_&iil_!m]ilrt]r£¢trtjlp&icÜxAEj¨r£xm]i_&o]qkxA<p&_&mum]_&i]xAmj¢£_HikxAo]i,Ätyl!_&mjilo]r©j%o]m]_

]rt°Lo]ilrtxAm¶i¨oEj¨i&±

_¬<xAr£mj]qkr£m]!r£Y¢_&ikj§p!j¨]¢£rtq0o]m]_¬¦DxAqk^Ho]¢£_wÜrtmPjlp&Aqlj¨rtxAm%YqYqµj¨rt_&i»·?ik_&^H]¢£]¢t_

&_!¢£¢t_]o &¢£!o]¢]_¶d¢£¢£r£¥Ertm,ikj¨m]Yql,¸n<xAo]qn]_&ic¥qkr¤u¢£_&iª¢tpjlxAr£qk_&i(^Ho]¢©j¨r©Ä

]r£^`_&m]ikr£xAmum]_&¢t¢£_&iuilxA¢to]^_!mjH&xmPjlr£mPo]_!iHYqHql]<xAqkj ¢¤^_!ilo]qk_]__!L_!ilAou_

_!j]xmPj5¢¤]_!m]ilr©j¨pC_!ikj(]r©°Np!ql_!mPjlr¤u¢£_±aªxAoui5]xAm]muxAm]io]m]__×ºE]ql_!ililrtxAm_ºE]¢tr£&r©j¨_n]_

[YsPo]_xALp!q¨jl_&o]qnuYq¨r£ililmj(Ym]i(!_!jljl_¦Dxql^Ho]¢t_±¹¼å_!¢¤¶m]xAo]i(<_&qk^_jc¢£xAqki5]_

¢£_&i»ilrt^Ho]¢£_!q»_j5]xm]c]_cLxAoE¥AxArtq»^`_!jkj¨ql_ª_&m xJ_&ou¥Jqk_Co]m ¢tAxAqkrtj¨^`_]_%dxAmj¨_%¼5qk¢£x]±

Â X À ! % &¢£!o]¢P]_d¢£¢£r£¥Ertm0X¢tAxAqlr©j¨[]^`_ ]_5d²xAmj¨_×Äµ¼5qk¢£x]il[]_!^)ªÙ«wo]¢t_&q!

]qlxJ&_!ililouiª]_vxAr£ikilxAm&xA^`<xAilpY¢t&o]¢0u_&i(il_!m]ilrt]r£¢trtj¨p!i&±

(6)

> ?

¡ §`7

-

!

~¯mj¨[u_c¢¤ikj¹hA_qli&J¦DxA¢£¢txW®(r£m]j¨[u_ª]rtxAm]_!_&qlrtm]]L_!qli

[^@ ], [ ] ¢£xj»x¦0®åxq/A¡&xAmu&_&qkmEÄ r£m]%j¨[]_nmPo]^`_&qlrt¢2]]¢£rtjlr£xAmuiwx¦0j¨[u_ªiµj¨xJ[Yikjlr£(¥qlr£j¨rtxAmY¢E¢t&o]¢to]iC·Òd¢t¢£r¤¥Jr£m

¢t&o]¢to]i¸ª[]iL_!_&mG]xAm]_±HZ[]rti^)rtm]¢th!xAm]&_!qlm]iu]¢£rtjlr£xAm]iªr£m3^j¨[u_&^j¨rt¢

YmYm]!_B !xA^`]oujzj¨r£xm]iwx¦L&xAm]urtj¨rtxAmY¢E_ºE<_&!j¨j¨rtxAm]ic·D®([]rt[)]L_&q¬r£m¶j¨[u_ª^_!qlr©Ä

mxAuj¨rtxAmuqlr£!r£m]uX¦Dxq§_×º]^]¢t_W¸m]x¦Yil_!m]ilr©j¨r©¥Er©j¨rt_&iå·Dj¨[u_»ikxC¢t¢£_!DCql_!_<APi¸×± Z[]_

^xJ]_&¢tiCjc[Ym]Gqk_Ho]ikoY¢£¢©h¢£xAÄ m]xAqk^)¢0j¯hJ<_¶]rt°Lo]ikr£xAm]iCm] j¨[u_&mxAm]_H^houil_

j¨[]_%ikj¨m]Yqld¢£¢tr¤¥Jr£m&¢£!o]¢£oui&±FEåoujnm]xW®åYhJiª<_&x]¢£_qk_^xAqk_%m]^xql_r£mJÄ

j¨_&qk_&iµj¨_&`r£m(yko]^Hj¯hJ<_]rt°Lo]ikr£xAm]i·?il_!_

[ ]^¸ muHj¨[]_!m`xm]_[Yijlxo]il_jl[]_ikjlxE[Yikj¨rt

¥qlr¤zj¨r£xmY¢ &¢£!o]¢£o]i%!xAqlqk_&il<xAm]ur£m]jlx vxAr£ikilxAm´LxArtmj]qkxE!_&ikil_&i!±3gEou[ ¢£&oJÄ

¢£o]iH[]i¶¢tql_uh+<_&_!m ]_!¥_&¢£xL_!³rtm

[ ] !xAm]&_!qlm]rtm] j¨[u_ m]xAr£ik_&x^rtm]¦Dqlx^ jl[]_

^u¢£rtjlo]]_%x¦¬j¨[u_nyko]^`]iCmu3r£m

[ ] !xAm]!_&qlmur£m]jl[]_nyko]^`j¨rt^_!i&±%dxAql_Hqk_&!_&mj¨¢©h r£m [ ] xm]_HAr©¥A_&i%o]m]r]_&G]]qkx[3o]ikr£m]jl[]_H¢£m]AoYA_x¦wjl[]_HGcrtqlr£[u¢£_!jª¦DxAql^`i&±

{cm]xjl[]_&q»Lxr£mj5x¦N¥Ert_!®%]]il_! xAm[Yxi5]_&!xA^`LxAikrtjlr£xAm`^)h`<_ª¦DxAo]mu r£m

[^I ] [ ] [^X ] [ ] [^A ]^±

Z[u_ rt^ x¦Cxo]q]L_!qrti`jlx´rt¥A_²´!xAm]!ql_!jl_ u]¢£rtjlr£xAm x¦ªj¨[]_ d¢t¢£r¤¥Jr£m

¢t&o]¢to]i ]uqlx[ ¦DxAq ik_&m]ikrtjlrt¥Jrtjlr£_&i&xA^`]ouj¨j¨rtxAm ·Cql_!_<APi¸¦Dxqyko]^ urt°Lo]ilrtxAm

^xJ]_&¢ti&± _Ar©¥A_Cj¯®»x_×º]^]¢t_&iJBwr£mj¨[u_ ]qlikj(xm]_]®»_%o]il_j¨[u_d¢£¢tr¤¥Jr£m¢t&o]¢to]i

®(rtj¨[ql_&ikL_!!jCjlx)j¨[u_nyko]^`3^`]¢£r©j¨o]]_!i&±n~¯m j¨[]_il_!&xAm]xAmu_L®»_¡] KEåqlxW®(m]r£m

YqkjHm] ]r©°N_!ql_!mPjlr¤jl_`®(r©j¨[ qk_&il<_&jHj¨x3<xjl[+j¨[]_%yko]^ ^u¢£rtjlo]]_!imu+j¨xjl[]_

r£_&mu_&qr£mu&ql_!^_!mj¨i&±

Z[u_cd²¢£¢tr¤¥Jr£mrtmj¨_&q¨jlr£xAmHh¶]qkjli ¦DxAql^Hou¢¤co]ik_&`r£mHjl[]r£iwYL_!qwr£iwm¡_!¢£_&^`_&mJÄ

jqµhxAm]_±¹aªxjlr£&_jl[Yjnmh mPo]^`_&qkr£¢0il[]_!^_o]ik_&rtmdxAmj¨_¼5ql¢tx¡¢£Axqlrtjl[]^

]<_qkii¦Do]m]!jlr£xAmGx¦5Ymurtj¨_¡mPo]^H<_&qx¦åq¨m]uxA^ ¥qlr£]¢£_!i

H1, . . . , Hn

±~ j

j¨o]qkm]i%xAouj%j¨[Yzjrt¦¹j¨[]_`¢¤®x¦åjl[]_&ik_qlm]]x^ ¥qlr£]¢£_!iCr£iuilxA¢touj¨_!¢th !xAmj¨rtmJouxAo]i

®(rtj¨[ql_!il<_&!j§j¨xnj¨[]_ _!L_!ilAou_å^`_iko]ql_åm][]i nil^`xExjl[u_&m]ikrtj¯hAj¨[]_!mj¨[]_¹ikjlq¨jµÄ

_&h u_!¥A_!¢£xAuL_! r£m j¨[u_Hd¢£¢£r£¥Ertm¢£&ou¢£o]i(^WhL_%r£^`]¢t_&^`_&mj¨_!¦Dxq

H1, . . . , Hn

m]´m+rtmPjl_&Aqlj¨rtxAm3hGYqkjli¦DxAql^Hou¢¤ r£i%xAuj¨r£mu_&0±Z[]r£i%r£i%m]xjikL_!&r Y`¦Dxq%jl[]_

EåqlxW®(m]r£m^xjlr£xAm xq5¦DxAq(vxAr£ikilxAm<xAr£mj]qkxE!_&ikiª]oEjªqk_&]qk_&ik_&mj¨icm²_!¢£_!^_!mPj¨qkh

]ikjlq¨jn¢t&o]¢to]i&±

Z[u_Y<_&qr£ixAqlm]r0L!_& i¦DxA¢£¢txW®(i&±~¯m gE_!!j¨rtxAmU`®å_uql_&ik_&mj¶j¨[u_)]qkxA]¢£_!^MB

®å_Art¥_jl[]_iµj¨xJ[Yikjlr£_&sPoYjlr£xAm¥A_&qkrY_! hj¨[]_o]m]]_!ql¢£rtm]ilik_!jC®([]rt[ r£in)xm]_

]r£^`_&m]ikr£xAm]¢Pyko]^`]rt°Lo]ilrtxAmG·?xm]_^h!xAm]ilrt]_&qn^Ho]¢©j¨räÄ¯]rt^_!m]ilrtxAmY¢Y]r©°Nouilr£xm]ini

®å_!¢£¢2]ouj¬®»_A_!_&rtm`jl[]_nxAm]_n]r£^`_&m]ikr£xAm]¢]il_jljlr£m]5yko]ikj¹j¨x¥AxArt`jl_&[]m]rt¢2]rON`&o]¢äÄ

(7)

j¨r£_!i¸×±Z[]_!m)®å_n]qk_&&rtil_([]xW® m`r£mjl_&Aqlj¨rtxAm¶h¡Yqkjli¹¦Dxql^Ho]¢£C^)h¶<_(o]il_!)rtm¡jl[]_

d²xAmj¨_¼5qk¢£x¡¢tAxAqkrtj¨[u^rtm xAqlu_&q5j¨x¡!xA^uouj¨_Cik_&m]ikrtjlrt¥Jrtjlr£_&i!±w~¯m²gE_&j¨r£xmH®»_%Art¥_

j¨[]_%r£mjl_&Aqlj¨rtxAm hYqkjli¦DxAql^Hou¢¤u±w~¯mgE_&j¨rtxAm¶®»_o]ik_j¨[]_d¢t¢£r¤¥Jr£mÜi»¢t&o]¢to]i

®(rtj¨[ qk_&ikL_!!jnj¨xHj¨[u_^]¢trtjlo]]_&i»x¦j¨[]_»yko]^`]irtm xAql]_!q5j¨x¡&x^]oEj¨_il_!m]ilr©j¨r©¥Er©j¨rt_&i&±

~¯mik_&!jlr£xAm

4®»_(&xAm]ikr£]_!qåd²_!qkjlxAmi»^`xEu_&¢]rtm¡®([ur£[)iµj¨xJ[Yiµj¨r£(rtmPjl_&Aql¢£i]qlr©¥A_&m

Ph² EåqlxW®(m]r£m^`xj¨rtxAm²m]h!xA^<xAo]mu v§xr£ilikxAm ]qkxE!_&iki]<_q!± _[Y¥A_

j¨[]_%[]xArt&_%x¦o]ilrtm]¶j¨[]_d¢t¢£r£W¥JrtmÜiå¢t&o]¢to]i5®(rtjl[²qk_&il<_&jªjlx¶j¨[]_ EåqlxW®(m]r£m^xzÄ

j¨r£xm0Aj¨xj¨[u_c^`]¢trtj¨ou]_&i x¦Nj¨[u_ ykou^]i¬xAqwjlx<xjl[x¦Lj¨[]_!^²±

_n<_&qk¦Dxql^ jl[]_j¨[]qk_&_

¢£Axqlrtjl[]^`iCm]3®»_!xA^`Yql_Hjl[]_¡_&^`]r£qkr£&¢§¥qkr¤m]!_¶x¦»_[´x¦»jl[]_&^±¶~ j%j¨o]qkm]i

xAouj%j¨[Yzjjl[]_`^xAiµjL_!qk¦DxAqk^)mj¢tAxAqkrtj¨[u^ ·D®(r©j¨[j¨[]_¡ik^)¢t¢£_!ikj¥qlr£m]!_W¸Cr£ij¨[]j

Yil_!²xm<xjl[jl[]_ EåqlxW®(m]r£m^xj¨r£xm m] j¨[u_^]¢trtjlo]]_&iåx¦j¨[]_åyko]^`]i&±¹gJx¡jl[]_

mJou^_!qlr£&¢_ºE<_&qkr£^`_&mj¨i5·Dj¨[u_&ql_¹r£im]x(j¨[]_!xAqlrt¢Pqk_&ilou¢tj×¸0rtm]]r£&j¨_wjl[Yj§xAm]_¹[Yij¨xcouil_

j¨[]_(r£mjl_&Aqlj¨rtxAmHh¶Yqµj¨iw¦DxAqk^Ho]¢¤n®(rtj¨[¡qk_&il<_&j»jlx¢£¢Jj¨[]_(m]xArtil_5®([]rt[rtiw¥r£¢£]¢£_

r£mjl[]_H^`xE]_!¢Û±CZ[u_HmPo]^_!qlrt¢ql_!ilo]¢©j¨i%qk_&xmPj¨r£mu_&3r£mil_&j¨rtxAm

5^± _H&x^Yql_

j¨[]_²d¢t¢£r£W¥Jrtm ^_j¨[]xJ ®(rtjl[ jl[]_ Ym]r©j¨_ ]r©°N_!ql_&mu&_^_j¨[]xJ0± }YxAq¡«wo]qlxL_&m &¢£¢

xAuj¨rtxAm]i!Ej¨[u_Cql_!ilo]¢©j¨i(qk_Cq¨jl[]_&qåilr£^`r£¢£q¬]ouj¦DxAq»]r£rtj¢LxAEj¨r£xm]i&Pj¨[]_qk_&iko]¢tjliArt¥_&m

Ph3jl[]_)d¢£¢tr¤¥Jr£m ^`_!jl[]xJ+qk_¡^Ho][+<_!jkj¨_!q%jl[YmGj¨[]_¡xAm]!_&iArt¥_&mGPh3jl[]_ Ym]rtjl_

]rt°L_&qk_&m]!_^`_!jl[]xJ0± Z[]rtin&mL__×ºE]¢¤rtm]_&hjl[]_¦?!jjl[Yjj¨[u_]Whx° ¦Do]mu!j¨rtxAm

r£i^Ho][^`xAql_rtqlql_!Ao]¢£q&±

§ $(

_»&xA^`]oujl_»j¨[u_åil_!m]ilr©j¨r©¥Er©j¨rt_&ix¦<mxAuj¨rtxAm®(r©j¨[YhAx°

φ®([u_&ql_»j¨[]_åilik_!j

(St)t≥0

¦DxA¢£¢txX®(i(¶xAmu_Ä¯ur£^`_&m]ikr£xAmY¢Jyko]^`urt°Lo]ilrtxAm]qlxJ&_!ilin]qlr©¥A_&m²h`!xA^<xAo]mu²vxAr£i,Ä

ilxAm]qlxJ&_!ili!± _jªouinuql_&!r£ik_xo]q(]qlxAu¢£_&^±

_j

N <_!xA^<xAo]mu²vxAr£ikilxAm²]qlxJ&_!ili!± _]_!m]xj¨_h

(Ti)i∈N

jl[]_ykou^j¨rt^_!i

x¦

t → Nt

¹m] h

(Jt)t≥0

jl[]_!xAqlqk_&il<xAm]ur£m]ikj¨m]Yql v§xr£ilikxAm ]qlxJ&_!iliH®(r©j¨[

Yq¨^_j¨_&q

λ > 0jl[Yjr£i

Jt

!xAo]mj¨i%j¨[]_mPo]^H<_&qx¦yko]^`]io]´jlxjm]G¦DxAq¢£¢

n≥1 Tn−Tn−1

[Yi5_×ºELxAmu_&mj¨r£¢N]rtikjlqlr£uouj¨rtxAm`®(r©j¨[^`_m

λ±¬}YxAq»¢t¢

i∈N^∗^E®»_

]_ Ymu_

∆i =NTi−NTi−1

Eilx

∆i

qk_&]qk_&ik_&mj¨i5jl[]_¬yko]^`^]¢trtjlo]]_(j»j¨r£^`_

Ti

±wZ[]_

q¨m]uxA^ ¥qlr£]¢t_&i

(∆i)i∈N∗

qk_ªrtm]]_!L_!m]]_&mjåm]r£]_!mPjlr£&¢£¢©h]rtikjlqlr£uouj¨_!0± _ci,Ä

ilo]^`_ªjl[Yjåjl[]_c¢£® x¦

∆i

r£i»]ikxA¢£oEj¨_&¢©h¡&xAmjlr£mPo]xAo]i»®(rtj¨[ql_!il<_&!jjlxjl[]_ _&<_&ikAo]_

^_&ilo]qk_¡m]3®»_¶]_&m]xj¨_¡h

ρ rtjli]_&m]ikrtj¯h0j¨[Yj%rti

∆i ∼ ρ(a)daN¦DxAq¢£¢

i ∈ N^∗^±

_®(r£¢t¢0ilou]Lxil_jl[Yj

ρ r£i(]rt°L_&qk_&mj¨r£]¢t_m]¥m]r£ik[]_&ini

x→ ±∞ ·Dil_!_ik_&!jlr£xAm

(8)

4^¸±

_]_¢å®(rtjl[ j¯®»xG^xJ]_&¢ti&±´~¯m³jl[]_ Yqliµj¡xAm]_¬j¨[u_²ilil_j

St

rti¡uo]ql_yko]^`

]rt°Lo]ilrtxAm ilx¢£oujlr£xAm x¦j¨[u_gGC«

St =β+

Jt

X

i=1

c(Ti,∆i, S_T− i ) +

Z t

0

b(u, Su)du . ^·,X¸

~¯mjl[]_%il_&!xAm]xAm]_]®å_]²HEåqlxW®(m]r£mYqkjB

St =β+

Jt

X

i=1

c(Ti,∆i, S_T⁻

i ) + Z t

0

b(u, Su)du+ Z t

0

σ(u, Su)dWu. ^·ÒA¸

ouq»r£^ rtiwj¨x&xA^`]oujl_

∂βE(φ(ST))Jjl[Yj¹r£i jl[]_Gª_&¢tj¨x¦ m«wo]qlxA<_m)xAujlr£xAm¡x¦

YhAx°

φ±w~¯m il_¥A_!q¨¢0]L_!qli%·Dilo][iÜÒ

@

ÒäW¤¸ExAm]_cjlql_zjåj¨[ur£i5]qkxA]¢t_&^ o]ilrtm]

d²¢£¢tr¤¥Jr£m&¢£!o]¢£o]i(]]qlxA[0± _®(rt¢£¢L¦Dx¢£¢£xW® ¶ilrt^rt¢¤q¹ikjlq¨jl_&h rtmxAo]q¦Dq¨^`_

o]ilrtm]d¢£¢tr¤¥Jr£m&¢£!o]¢£oui5¦DxAq§yko]^`j¯hJ<_%]rt°Lo]ilrtxAm]i!±

_®(qlr©j¨_

∂

∂βE(φ(ST)) = E

φ⁰(ST)∂ST

∂β

= E

φ⁰(ST)∂ST

∂β 1JT=0

+

∞

X

n=1

E

φ⁰(ST)∂ST

∂β |σ(Ti, i∈N)

1J_T=n

.

m

{JT = 0}jl[]_&qk_Cr£i»m]xnyko]^`)xm

]0, T]^Pj¨[Po]i ST

m]

∂ST

∂β

ilx¢t¥A_C]_!jl_&qk^rtm]r£iµj¨r£

r£mj¨_!Aq¨¢Y_!sPoYj¨rtxAm0± ~¯mj¨[]_n_ºu^`]¢£_!i¹jl[Yj¹®å_ª&xAmuilr£u_&qårtm`jl[]r£i¹Y<_&q!Jjl[]_nilxA¢touj¨rtxAm

x¦§jl[]_&ik__!sPoYj¨rtxAm]i(qk_%_ºE]¢£rt&r©juikx`jl[Yjjl[]r£i5jl_&ql^r£i_ºE]¢tr£!rtj¨¢©h:APm]xW®(m0±

{ªiliko]^_nm]xW® jl[Yj

JT =n 6= 0±Z[]_!m0Eouilr£mum`r£mjl_&Aqlj¨rtxAm¶h¡Yqkjliw¦DxAql^Hou¢¤

xAm {JT =n}]®»_®(qlrtjl_HB

E

φ⁰(ST)∂ST

1JT=n

= E

φ(ST)Hn(ST,∂ST

∂β )|σ(Ti, i∈N)

1JT=n,

(9)

®([]_&qk_

Hn

rti¬®»_&rtA[j¹r£m¥AxA¢©¥Jr£m] µd¢£¢tr¤¥Jr£m]_&qkrt¥j¨r©¥A_!i x¦

ST

m]`x¦

∂S_T

∂β

± Z[]_!il_

]rt°L_&qk_&mj¨r£¢¬xAL_!q¨jlxAqliqk_)ilrt^rt¢¤qCjlx²jl[]xAil_)r£m

[ ] ]oujj¨[]_`¦Dq¨^`_`[]_&qk_r£i^Ho][

^xAqk_5ilrt^u¢£_ilr£mu&_j¨[]_!ql_(qk_(m]x&!o]^Ho]¢£j¨rtxAmHx¦Lil^¢£¢Wyko]^`]i&± gEx®å_®(r£¢t¢u]_!qlr©¥A_

j¨[]rtir£mj¨_!Aq¨jlr£xAm h ]qkjli(¦Dxql^Ho]¢£Ho]ilrtm]¡_&¢t_&^`_&mjqµh qkAo]^`_&mj¨i!± _%xAuj¨r£m

∞

X

n=1

E

φ⁰(ST)∂ST

1JT=n

= E

φ(ST)HJT(ST,∂ST

∂β )1JT≥1

.

~¯m xAql]_!qj¨x_&^`]¢£xWhj¨[]rti¦DxAqk^Ho]¢¤r£m d²xAmj¨_×Äµ¼5qk¢£x¢tAxAqlr©j¨[]^ ®å_ uqlxJ&_&_!³i

¦DxA¢£¢txX®(i!± _ilrt^Ho]¢£j¨_ j¨[u_¡yko]^`³jlr£^`_&i

(T_nⁱ)n∈N

i = 1, . . . , M ^m] jl[]_Hyko]^

^u¢£rtjlo]]_!i

(∆ⁱ_n)n∈N

i= 1, . . . , M<m] ®»_!xA^uouj¨_Cjl[]_%&xAqkql_&ikLxm]]r£mu

J_Tⁱ S_Tⁱ

m]

H_Jⁱi T

±wZ[]_!m²®»_®(qlrtjl_

E

φ(ST)HJT(ST,∂ST

∂β )1JT≥1

' 1

M

X

i=1

φ(S_Tⁱ)H_Jⁱi

T 1J_Tⁱ≥1.

~¯mj¨[]_ik_Hx¦¹d_&qkjlxAmiC^`xE]_!¢ ®å_HuqlxJ&_&_!3in¦DxA¢£¢txX®(i!±n}r£qkikjªx¦¬¢t¢®»_ikr£^Hou¢¤jl_

j¨[]_yko]^²j¨r£^`_&i

Ti, i ∈N^± m]!_j¨[]_h ql_uºE_! ®å_H!xAm]iµj¨qlou!j%m«wo]¢t_&qªil[]_!^_

r£m²®([ur£[jl[]_jlr£^`_&i

Ti, i∈ N ql_r£m]!¢£o]]_!rtm²j¨[u_]rtil&qk_!jlr0Lzj¨r£xmAqkr£0±(Z[]_!m ®å_

ilr£ô]¢¤jl_`jl[]_^`]¢£r©jö]u_&ix¦(j¨[]_ykou^]iHm] j¨[u_4EåqlxW®(m]r£m r£mu&ql_!^_!mjï¡m] ®»_

L_!qk¦DxAqk^6j¨[u_dxAmj¨_Ä,¼5ql¢tx`¢£Axqlrtjl[]^±

# $$&(')!

-

$

-

$!.z% ! $!

-

%$&

m5]qkxAYur£¢£r©j¯hnilY!_

(Ω,F, P)®»_¬&xAmuilr£u_&qil_!sJou_&m]!_»x¦Er£mu]_&<_&m]u_&mjq¨m]uxA^

¥qlr¤]¢£_!i

(Vn, n∈N^∗)^± _%ilo]uLxAik_jl[Yj

¿wÂ ¾ Ò

* 35

n∈N^∗

6 'O

Vn

0 '(# '(5 -6 )!6 1 !

6- K 6 /"6 :/ ; D 6 ;# ! 5

pn

#- 0 -6 !661 '(5

; 6 ))<'0D

R ^{;8 6-M} ∀k∈N lim

y→±∞|y|^kp(y) = 0^! ^{'=K #:}

∂yln(p(y)) = ∂yp(y) p(y)

: 96'(5, )' "" #

(10)

I

_(r£mjlqlxJ]o]&_(ikxA^_muxjjlr£xAmN± }YxAq

k≥ 1 _(]_&m]xj¨_nPh

C_↑^k(Rⁿ) jl[]_(ilY&_nx¦Ljl[]_

¦Do]m]!jlr£xAmui

f : Rⁿ →R ®([]rt[ql_A)j¨rt^_!i5]r©°N_!ql_&mjlr¤]¢t_%m]ilo][j¨[Yzj¦wm]r©j¨i

]_&qkrt¥j¨r©¥A_!iªo]²j¨x)xql]_!qA [Y¥A_Hjc^`xAiµjc<xA¢thJm]x^r£¢0AqkxX®jl[0±n}YxAqc`^Ho]¢©j¨r©Ä r£m]u_º

α = (α1, . . . , αk) ®»_²]_!m]xjl_

∂_α^kf = ∂^k

∂α1. . . ∂α_k

f±³~¯m³jl[]_ik_

k = 0 C_↑⁰(Rⁿ)

]_&ikr£Am]i¬j¨[u_cil_jåx¦0j¨[]_(¦Do]m]j¨rtxAm]i

f : Rⁿ→R®([]rt[qk_ª!xAmj¨rtmJouxAo]i»®(r©j¨[ql_&ikL_!!j j¨x¡_[²qkAo]^`_&mjªm][Y¥A_j(^`xAikj(<xA¢thJm]x^r£¢LqlxW®j¨[0±

_%]_ ]m]_muxX® j¨[]_%ikr£^`]¢£_c¦Doum]!jlr£xAmY¢£imujl[]_%ilr£^`]¢t_C]qlxJ&_!ilik_&i&±

{ qlm]]x^¥qlr£]¢£_

F r£i¢£¢£_!`ikr£^`]¢£_c¦Doum]!jlr£xAmY¢ rt¦j¨[]_!ql_%_ºEr£iµj¨inilxA^`_

n ∈N^∗

m]ilxA^`_%^_&ilouq¨]¢t_C¦Do]m]j¨r£xm

f : Rⁿ→R iko][²jl[Yj

F =f(V₁, . . . , Vn).

_]_!m]xjl_h

S(n,k)

j¨[u_ikY!_x¦jl[]_%ilrt^]¢t_ª¦Do]mu!j¨rtxAmY¢ti5ilou[²j¨[]j

f ∈ C_↑^k(Rⁿ)^±

{ ikr£^`]¢£_CuqlxJ&_&iki(x¦¢t_&m]j¨[

n r£i(¡il_&sPo]_!m]&_x¦§q¨m]uxA^6¥qkr¤]¢t_&i

U = (Ui)i≤n

iko][²jl[Yj

Ui =ui(V1(ω), . . . , Vn(ω)).

_]_&m]xj¨_h

P_(n,k) jl[]_)ikY&_)x¦j¨[]_ilrt^]¢t_¡]qlxJ&_!ilil_!i¶x¦¢t_&m]jl[

n ^iko][ ^j¨[]j ui ∈ C_↑^k(Rⁿ), i= 1, . . . , n±¹aªxjl_Cj¨[Yjnr©¦

U ∈P_(n,k) jl[]_&m

Ui ∈S_(n,k) i∈N^∗

±

_)u_ Ym]_ muxX® jl[]_]r©°N_!ql_!mPjlr¤¢¬xA<_&qlj¨xqli%®([]r£[ uL_&qrtm´j¨[u_ d¢t¢£r£W¥JrtmÜi

¢t&o]¢to]i&±

Z[]_Hd²¢£¢tr¤¥Jr£m)u_&qlr©¥j¨r©¥A_

D:S_(n,1) →P_(n,0)^B

rt¦

F =f(V1, . . . , Vn)^]j¨[]_!m

DiF := ∂if(V₁(ω), . . . , Vn(ω)) = (∂f

∂xi

)(V₁(ω), . . . , Vn(ω)), DF := (DiF)i≤n∈P(n,0).

Z[]_Hg#AxAqkxA[]xJrtmj¨_&q¨¢

δ :P_(n,1) →S_(n,0)^B δ(U) := −

n

X

i=1

(DiUi+θi(Vi)Ui)

= −

n

X

i=1

∂ui

∂xi

(V1, . . . , Vn) +θi(Vi)ui(V1, . . . , Vn)

,

(11)

®(rtj¨[

θi(y) := ∂yln(pi)(y) = (^p_p⁰ⁱ

i)(y) ^rt¦ pi(y)>0,

:= 0 ^rt¦ pi(y) = 0.

¾ +] * 35

F ∈S_(n,1) ^; ^{* 5} p∈N ^, E

n

X

j=1

|D_iF|²

!p

<∞ ^; ^{* 35} U ∈P(n,1)

E (|δ(U)|^p)<∞

#0 '!'0" 3 6,J#- 9

E|Vi|^p <∞ ^7'!' ^!,- lim

x→±∞|y|^kpi(y) = 0

; 3 6J#- . 9

ui

; ∂pi

pi

i = 1, . . . , n ^{*: 7:} ^6'(5 ^, ^)'

"" #

Z[]_%xAL_!q¨jlxAq

δ r£i5jl[]_zykxArtmjx¦

D ^·j¨[]rtir£ij¨[u_ql_ilxAmx¦ <_&rtm]¶x¦

θi

¸ B

Â wÂ

D

Â

9

F ∈S_(n,1) ^; U ∈P_(n,1) ^#6

E(< DF, U >) =E(F δ(U)), ^·?¸

6

< ., . > ^{0 #6%-/'O} ^<;6- ^! Rⁿ Â0Â m]_®(qkrtj¨_!i

E(< DF, U >) =E

n

X

i=1

DiF ×Ui

!

=

n

X

i=1

E

ui

∂f

∂xi

(V1, . . . , Vn)

=

n

X

i=1

Z

Rn

ui

∂f

∂xi

(a1, . . . , an)p₁(a1). . . pn(an)da₁. . . dan

=−

n

X

i=1

Z

Rⁿ

f(a₁, . . . , an)

∂iui+ui

p⁰_i pi

pi(ai) Y

j6=i

pj(aj)da₁. . . dan,

j¨[]_¡¢¤ikj%_&sPoY¢£rtj¯hL_!r£m]xAEjrtm]_&Gh3r£mj¨_!Aq¨zj¨r£xm3Ph]qkjli&± _`_&^`]¢£xWh[]_&qk_¡jl[]_

¦?!j%j¨[]j¦Dxq¢£¢

k ∈ N

lim

y→±∞|y|^kpi(y) = 0 p⁰_i pi

∈ C_↑⁰(Rⁿ) ^mu´¦, ui ∈ C_↑¹(Rⁿ)^±

(12)

@

¼åxA^rtm]¶YAj¨x`_ºEL_!!j¨j¨rtxAm]i®å_%xAEjrtm+·?A¸×±

_¡r£mj¨qkxEuo]&_¶m]xW® j¨[]_¡ikx&¢£¢t_& qlmuikj¨_!r£mGbª[u¢£_&mP<_&A3xA<_&qlj¨xq

L = δ(D) : S_(n,2) →S_(n,0) :

LiF := DiDiF +θiDiF , ^¦DxAq i= 1, . . . , n

LF := −

n

X

i=1

LiF

= −

n

X

i=1

(∂i(∂if)(V1, . . . , Vn) +θi(Vi) (∂if)(V1, . . . , Vn)) . ^·DP¸

¾ +] !,-

θi

f ∈ C²_↑(Rⁿ) ^, E|LF|^p <∞ ^'!'

gEo]]<xAil_¶jl[Yj

F, G∈S(n,2)

j¨[]_!m0i%mr£^`^_!]r¤zj¨_!xAm]ik_&sPo]_&mu&_)x¦¹jl[]_¡]oY¢trtj¯h

ql_&¢£j¨rtxAm·?¸åxAm]_%xAuj¨r£mui

E(F LG) =E(< DF, DG >) =E(G LF). ^·ÛA¸

d²xAql_!xW¥A_&q!Ejl[]_(ikjm]Yqk)]r©°N_!ql_&mjlr¤¢Y&¢£!o]¢£o]i¬qko]¢£_!i¹rt¥A_jl[]_n¦Dx¢£¢£xW®(rtm][Yrtmqko]¢£_±

¹¾ +

9

φ ∈ C_b¹(R^d) ^;4'0 F = (F¹, . . . , F^d), Fⁱ ∈ S(n,1). ⁶ φ(F)∈S_(n,1) ^;

Dφ(F) =

d

X

k=1

∂kφ(F)DF^k, ^· ^¸

!

9

F, G∈S(n,2)

#6

L(F G) = F LG+G LF −2 < DF, DG > . ^· ^I ^¸

}r£mY¢£¢thArt¥_&m

F = (F¹, . . . , F^d), Fⁱ ∈ S_(n,1)®»_r£mj¨qkxEuo]&_¡j¨[]_d¢t¢£r¤¥Jr£m!xÄ

¥qlr¤m]&_^j¨qkr©º

σ_F^ij=< DFⁱ, DF^j >=

n

X

p=1

∂pfⁱ∂pf^j

(V1, . . . , Vn),

®([]_&qk_

Fⁱ =fⁱ(V1, . . . , Vn)^±

_qk_m]xW®T]¢t_Cj¨x`ikj¨j¨_Cjl[]_%r£mj¨_!Aq¨jlr£xAm hYqkj¨i5¦Dxql^Ho]¢£u±

(13)

¾JÂ ¾ 9

F = (F¹, . . . , F^d)∈S_(n,2)^d G∈S(n,1)

:% 6

8 3

σF

0 !6* 3 )<'0 ;K;# ,

γF :=σ⁻¹_F

'=D 16

E (detγF)⁴

<∞. ^·?¸

6 * 35H : #%3#,-

φ:R^d →R

* 5

i= 1, . . . , d

E(∂iφ(F)G) =E(φ(F)Hⁱ(F, G)), ^· ^@ ^¸

!

Hⁱ(F, G) =

d

X

j=1

E G γ^ji_F LF^j −γ_F^ji < DF^j, DG > −G < DF^j, Dγ_F^ji >

.

·µ¸

Â0Â ±¹bnilr£mu¡jl[]_%[Yrtmqko]¢£_C®»_xAujr£m¦DxAq_&[

j = 1, . . . , d

< Dφ(F), DF^j > =

n

X

r=1

< Drφ(F), DrF^j >

=

n

X

r=1 d

X

i=1

∂iφ(F) < DrFⁱ, DrF^j >

=

d

X

i=1

∂iφ(F)σ^ij_F ,

ilx¡j¨[Yj

∂iφ(F) =

d

X

j=1

< Dφ(F), DF^j > γ_F^ji±»d²xql_&xW¥_&q&2o]ikr£m]²·

I ¸

< Dφ(F), DF^j >= 1

2 −L(φ(F)F^j) +φ(F)LF^j +F^jL(φ(F)) .

(14)

Z[]_&mj¨[]_%]o]¢£r©j¯hql_!¢¤jlr£xAm)Art¥_&i

E(∂iφ(F)G)

=1 2E

G

d

X

j=1

−L(φ(F)F^j) +φ(F)LF^j+F^jL(φ(F)

!!

γ_F^ji

!

=

d

X

j=1

1

2E φ(F) −F^jL(γ_F^jiG) +G γ_F^jiLF^j +L(G F^jγ_F^ji)

=

d

X

j=1

E φ(F) G γ_F^jiLF^j−< DF^j, D(G γ_F^ji)>

,

m] j¨[]_%]qkxEx¦rti&xA^`]¢£_j¨_!0±

?

¡"%% ¡1 §×

§( 3

- 4

T/

$!×7(

~¯m j¨[ur£i`il_&j¨rtxAm ®»_ o]ik_²j¨[u_²rtmPjl_&Aqlj¨rtxAm³Ph³Yqkj¨i¡¦DxAqk^Ho]¢¤³·

@

¸¶®(r©j¨[ ql_&ikL_!!jj¨x

∆i ∼ ρ(a)da i ∈ N^∗

n®([]r£[ ql_ jl[]_+^]¢trtjlo]]_&i)x¦jl[]_)yko]^`]ix¦

(Nt)_t≥0

ilxA¢touj¨rtxAm x¦j¨[]_g GC«

St =β+

Jt

X

i=1

c(Ti,∆i, ST_i−) + Z t

0

b(r, Sr)dr . ^·,AX¸

_®»xAq/Aoum]]_&qjl[]_¦DxA¢£¢txX®(rtm]H[hJLxj¨[]_!ilr£i!±

¿wÂ ¾ Ò

ρ 0-6 )!66 '(5&; 6 ))<'0 ρ⁰ ρ

: 6'(5, )'

"" #

R

;7 '!'

k∈N

y→±∞lim |y|^kρ(y) = 0

¿wÂ

¾

Ò

63,- )

(a, x) → c(t, a, x) ^; x → b(t, x) ^-

-6 !66 '(5K; 6 )<'0 ;: #6 H 0 DM- 6

K > 0 ^; α ∈N ^6-

(15)

i)|c(t, a, x)| ≤K(1 +|a|)^α(1 +|x|)

|b(t, x)| ≤K(1 +|x|) ii)|∂xc(t, a, x)|+

∂_x²c(t, a, x)

+|∂ac(t, a, x)| ≤K(1 +|a|)^α

|∂xb(t, x)|+

∂_x²b(t, x) ≤K

¿wÂ ¾ Ò

|∂ac(t, a, x)| ≥η >0.

!"#$

_%r£mjlqlxJ]o]&_Cjl[]_¦DxA¢£¢txW®(r£m]Hu_!j¨_!ql^`r£m]rtikjlr£C_!sJo]j¨rtxAm0±

_uºikxA^_

0< u₁ < . . . < un < T2m]®å_]_!m]xj¨_

u= (u₁, . . . , un)^± ^_¢£ikx

uº

a= (a1, . . . , an)∈Rⁿ^±¬Z§x u ^m] a]®»_ilikxE!r¤jl_Cj¨[]_%_!sJo]j¨rtxAm

st =β+

Jt(u)

X

i=1

c(ui, ai, s_u− i ) +

Z t

0

b(r, sr)dr, 0≤t≤T , ^·µXA¸

®([]_&qk_

_¹o]il_¹j¨[]rti§u_!j¨_!ql^`r£m]rtikjlr£w_&sPoYzj¨r£xmr£mxAqk]_&qj¨xª_ºE]ql_!ili

St

inilrt^]¢t_ ¦Do]m]j¨r£xmY¢B

xAm {Jt =n} xAm]_%[Yi

St = st(T1, . . . , Tn,∆1, . . . ,∆n),

∂∆iSt = ∂aist(T1, . . . , Tn,∆1, . . . ,∆n),

∂²_∆_j_,∆_iSt = ∂_a²_j_,a_ist(T1, . . . , Tn,∆1, . . . ,∆n),

∂βSt = ∂βst(T₁, . . . , Tn,∆₁, . . . ,∆n).

St

<®»_[]W¥_jlx)&x^]oEj¨_%j¨[]xAik_x¦

st

±

bªm]]_!q[hE<xjl[]_&ikr£i]±ÙEN®»_¡^)h²L_!qk¦DxAqk^ jl[]_&ik_¶&xA^`]ouj¨j¨rtxAm]iCh urt°L_&ql_!mj¨r¤zj¨r£mu

®(rtj¨[qk_&il<_&jªjlx

aj

j = 1, . . . , n rtm´·µXA¸×± _%xAuj¨r£m

(16)

}YxAq

i≤Jt(u), ∂a_ist

ilj¨rti Y_!injl[]__&sPoYjlr£xAm

∂aist =∂ac(ui, ai, s_u⁻

i ) +

Jt(u)

X

j=i+1

∂xc(uj, aj, s_u⁻

j )∂ais_u⁻

j

+ Z t

ui

∂xb(r, sr)∂aisrdr . ^·µ¸

∂_a²_ist =∂²_ac(ui, ai, s_u⁻

i ) +

Jt(u)

X

j=i+1

∂_x²c(uj, aj, s_u⁻

j )

∂ais_u⁻

j

2

+

Jt(u)

X

j=i+1

∂xc(uj, aj, s_u−

j )∂_a²_is_u−

j +

Z t

ui

∂xb(r, sr)∂_a²_isrdr

+ Z t

u_i

∂_x²b(r, sr) (∂aisr)² dr . ·µ&P¸

}YxAq

i < j

∂_a²_j_,a_ist =∂_a,x² c(uj, aj, s_u− j ) +

Jt(u)

X

k=j+1

∂_x²c(uk, ak, s_u−

k)∂ais_u−

k ∂ajs_u− k

+

Jt(u)

X

k=j+1

∂xc(uk, ak, s_u⁻

k)∂_a²_j_,a_is_u⁻

k +

Z t

uj

∂xb(r, sr)∂_a²_j_,a_isrdr

+ Z t

uk

∂²_xb(r, sr)∂aisr∂ajsrdr , ^·µWA¸

m] ¦DxAq

i > ju®»_xAujr£m

∂_a²_j_,a_ist

h iµhE^`^`_!j¨qµhA±

∂βst

r£i&x^]oEj¨_&h

∂βst = 1 +

Jt(u)

X

i=1

∂xc(ui, ai, s_u−

i )∂βs_u−

i +

Z t

0

∂xb(r, sr)∂βsrdr . ^·, ^¸

(17)

¹¾ + 663#,-

(a1, . . . , an) → st(u1, . . . , un, a1, . . . , an) ∈ C_↑²(Rⁿ)

;

(a1, . . . , an) → ∂βst(u1, . . . , un, a₁, . . . , an) ∈ C_↑¹(Rⁿ)

!

|∂anst|² ≥η²e^{−T K} >0

Â0Â ·?r¸§v¬qkxE!_&_!)h¡ql_!&o]qkql_&mu&_couilr£mu[hJ<xj¨[]_!ilrti»]±Ùr£m²·µX¸L·µA¸L·,P¸<·,XA¸×

m]r£m´·µ ¸±

·?r£r¤¸»bnilr£mu·,W¸×

∂ansT =∂ac(tn, an, s_t− n) +

Z T

tn

∂xb(r, sr)∂ansrdr^uilx

|∂ansT|=

∂ac(tn, an, s_t⁻

n) exp

Z T

tn

∂xb(r, sr)dr

≥η e^{−T K}.

!"#$ ! #$ !

_!xA^_]Aj¨x¶j¨[u_]qlxAu¢£_&^6r£mil_!!jlr£xAm

@

¸×±

_®(qlr©j¨_

E

φ⁰(ST)∂ST

∂β 1JT=n

=E

E

φ⁰(ST)∂ST

1JT=n

=E(hn(T1, . . . , Tn)1J_T=n) ,

®(rtj¨[

hn(t1, . . . , tn)

=E

φ⁰(sT(t1, . . . , tn,∆1, . . . ,∆n))∂sT

∂β (t1, . . . , tn,∆1, . . . ,∆n)

.

_uº

n≥1 ^mu t1, . . . , tn

F =sT(t1, . . . , tn,∆1, . . . ,∆n) ^m] G=∂βsT(t1, . . . , tn,∆1, . . . ,∆n)^±

E»h _!^^u±tA<®å_ AJmuxX® jl[Yj

σF

r£irtm¥A_&qµj¨rt]¢£_m]jl[Yj·?¸5[]xA¢£ui5j¨qlou_m] j¨[]_!m®»_!xA^`]ouj¨_

Hn(F, G) = G γFLF −γFhDF, DGi −GhDF, DγFi. ^·µ ^I ^¸

(18)

_%[Y¥A_

σF =

n

X

i=1

|∂aisT(t₁, . . . , tn,∆₁, . . . ,∆n)|²

≥ |∂ansT(t1, . . . , tn,∆1, . . . ,∆n)|² ≥η²e^{−2T K},

@

¸×±

}2xqåj¨[]_%!xA^uoujjlr£xAm x¦§jl[]_j¨_&qk^i(rtmP¥xA¢t¥_&r£m j¨[u_qkr£A[j[]m]ilr£u_x¦·,

I

¸×

®å_%o]ik_·µW¸»¦DxAq

DF ^mu γF

·µ ¸»¦DxAq

G·µW¸5m]´·µ¸»¦DxAq

LF·µWA¸×·µ&P¸

m]´·µWA¸»¦DxAq

DγF

aj

r£m´·µ ¸»j¨x¡xAuj¨r£m

DG^±

# 5§ §

-

(67 $5

-

(

St =β+

Jt

X

i=1

c(Ti,∆i, S_T⁻

i ) + Z t

0

b(u, Su)du+ Z t

0

σ(u, Su)dWu,

®([]_&qk_

®å_ikilo]^`_

¿wÂ ¾ Ò

63,- )

x→σ(u, x) ⁰ -7-6 !661 '(5; 6 )<'0

; 6 0 ! - 6

C >0 ^6-

i)|σ(u, x)| ≤C(1 +|x|), ii)|∂xσ(u, x)|+

∂_x²σ(u, x) ≤C , iii)|σ(u, x)| ≥η .

^xjlr£xAmm]j¨[]_¹ykou^)^u¢£rtjlo]]_!i&±¬gEou]Lxil_cjl[Yjåjl[]_¬ykou^)jlr£^`_&i

T1 < . . . < Tn

ql_nArt¥_&m·Dr£m`j¨[]_Cd²xAmj¨_×Äµ¼5qk¢£xH¢£AxAqkrtjl[]^jl[]r£i»^_&m]i¬j¨[Yj»®å_ª[Y¥A_¢£ql_&uh`ilrt^¡Ä

o]¢¤jl_&

T₁, . . . , Tn

¸×± _rtm]&¢to]]_Cj¨[u_&^.rtmj¨[]_]rtil!ql_!jlr0L&j¨rtxAm Aqlrt B ikx¡®»_&xAm]ikr£]_!q