Integration by parts formula for locally smooth laws and applications to sensitivity computations

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Submitted on 19 May 2006

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Integration by parts formula for locally smooth laws and applications to sensitivity computations

Vlad Bally, Marie-Pierre Bavouzet, Marouen Messaoud

To cite this version:

Vlad Bally, Marie-Pierre Bavouzet, Marouen Messaoud. Integration by parts formula for locally smooth laws and applications to sensitivity computations. [Research Report] RR-5567, INRIA. 2005, pp.54. �inria-00070439�

ISRN INRIA/RR--5567--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

Integration by parts formula for locally smooth laws and applications to sensitivity computations

Vlad Bally , Marie-Pierre Bavouzet and Marouen Messaoud

N° 5567

May, 12 2005

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ª

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e')1·! #%e!#% !*~ »! .!–‡..#D.'

! #"%$'&)(+*,"

¡ c²Š'|Kqamov±‰ac!u6„«†aup|K¦a„‡¦av±§xvxn¶l mo†]„‡Š!c

(Ω,F, P)^{’%„ mosa¦} σ−„‡§±˜‡c'¦aup„

G ⊆ F ^„‡qa‰º„

mpc'wVsac!qaŠ'c,|‡¨up„‡qa‰a|‡b£„‡upv±„‡¦a§xc'm

V_i, i∈N

¿ ¡ c‰ac!qa|‡n©c

Gi =G ∨σ(V_j, j 6=i)^{¿ sau}

„‡v±b vxmenp|‘moc!npnp§±c3„‡q vxqXn©c'˜‡u©„npv±|KqO¦Xl«†]„‡uonpm,¨I|Kuob>sa§±„ ¨I|‡ue¨IsaqyŠ!n©vx|Kq]„‡§xm|‡¨

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σ^´ „§±˜Kc!¦au©„

G

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¡ c ª v±§x§

ª

|Ku>…|Kqmp|Kb$c{mpcn

A∈ G^{ª _av±Š‹_ ª v±§x§‡¦c} µc'‰n©_yup|Ksa˜‡_np_av±mRmpc'Šn©vx|Kq ¿ ¡ c

‰ac'qa|n©c

L (A) np_ac,mo†]„‡Š!c,|‡¨n©_acup„‡qa‰a|Kb*£„‡uov¥„‡¦y§±c'mmosaŠ‹_²np_]„n

E (|F|^p 1 )<∞

¨I|Ku„§±§

p ∈ N^’„‡qa‰ L(p+)(A) ^{ª v±§x§} ¦c3np_ac·mo†]„‡Š!c·|‡¨np_ac·up„‡qa‰a|Kb £„‡uov¥„‡¦a§xc'm

F ^¨I|Ku

ª

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δ >0 mpsaŠ‹_ n©_a„n

E

|F|^p+δ 1A

<∞^{¿ ¡} c„‡mompsab$cgnp_]„n

Â{Å œ Á ›:›

Vi ∈L(∞)(A) i∈N

4|‡u$c'„‡Š‹_

i ∈ N ª

c²Š'|Kqamov±‰ac!u·mo|Kb3c

k_i ∈ N „‡qa‰°mp|Kb$c

Gi

´¶b$cG„mpsaup„‡¦a§±c.up„‡qa‰a|Kb

£‡„upv¥„¦a§±c!m

a_i(ω) =t⁰_i(ω)< t¹_i(ω)< . . . < t^k_iⁱ(ω)< t^k_iⁱ⁺¹(ω) =b_i(ω).

¡ c‰ac!qa|‡n©c

Di(ω) =

ki

[

j=0

t^j_i(ω), t^j+1_i (ω)¿½dr|‡n©vxŠ'cMnp_]„n

ª

cb3„Gl·n‹„ >‡c

ai =−∞ ^„‡qa‰ bi =∞^¿

¡ c ª

v±§x§

ª

|Ku<>

ª

vxn©_¨IsaqaŠn©vx|Kqam ‰ac ]qyc'‰ |Kq

(a_i(ω), b_i(ω))^ª _avxŠ‹_6„‡uoc…mob3|N|‡np_3c yŠ!c'†yn

¨I|Kurnp_ac†|KvxqVnpm

t^j_i, j = 1, . . . , ki

¿ ¡ c‰ac]qac

C^k(Di) np|·¦c,n©_ycmpcnM|‡¨n©_ac,b3c'„‡mpsau ´

„‡¦a§±c>¨IsyqaŠ!npv±|Kqam

f : Ω×R→R ^mpsaŠ‹_ n©_]„n'’0¨I|Kuc!£Kc!uol

ω^’ y → f(ω, y) ^v±m k ^n©vxb3c!m

‰avx³Tc'uoc'qXn©v±„‡¦a§±c,|Kq

D_i(ω) „‡qa‰É¨I|‡uMcG„‡Š‹_

j = 1, . . . , k_i’Rn©_ac¼§xc!¨An_a„‡qa‰ mpvx‰ac„‡qa‰Énp_ac upv±˜‡_VnD_]„‡qy‰ mpv±‰ycg§±vxb3v¯n©m

f(ω, t^j_i(ω)−), f(ω, t^j_i(ω)+) ^c yvxmon)„qa‰ „‡uoc ]qav¯n©cF?I¨I|Ku

j = 0

„‡qa‰

j =ki+ 1 ^ª c>„‡mompsab$cn©_]„n)np_ac,uov±˜K_Xnr_]„‡qy‰‘mpvx‰ac‡’4uoc'mp†c'Šn©v¯£Kc'§¯l n©_ace§±c¨Anr_]„‡qa‰

mpv±‰yc§±v±b$vxnDcµvxmon©m…„‡qa‰.vxm ]qavxnpc(A2¿ ¡ c‰ac!qa|‡n©c

Γi(f) =

ki

X

j=1

f(ω, t^j_i(ω)−)−f(ω, t^j_i(ω)+)

+f(ω, bi(ω)−)−f(ω, ai(ω)+). ^{? “ A}

4|‡u

f, g ∈ C1(D_i)’µn©_acv±qXnpc'˜Kup„n©vx|Kq ¦Xl †]„‡uˆn©m¨I|Kupbsa§¥„>˜Kv¯£Kc!m

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(ai,bi)

f g⁰(ω, y)dy= Γi(f g)− Z

(ai,bi)

f⁰g(ω, y)dy , ^{?ٔ A}

mp| Γi

upc'†yupc'moc'qXn©mrn©_acŠ!|KqXn©uov±¦asynpv±|Kq |¨np_ac¦T|‡up‰ac!u)npc'uob3m5´½|‡u'’a†asµn)vxn)|‡np_ac'u

ª

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f ^|Ku g^¿

¾0cn n, k ∈ N

¿ ¡ c·‰yc'qa|‡npc·¦Xl

Cn,k

np_ac·Š!§¥„‡mom|‡¨%np_ac

G × B(Rⁿ) b3c'„‡mpsaup„‡¦a§xc

¨IsaqaŠ!npv±|Kqym

f : Ω×Rⁿ→R mosaŠ‹_²np_]„n

Ii(f)∈ C^k(Di)^’ i= 1, . . . , n^{’ ª _ac!upc} I_i(f)(ω, y) :=f(ω, V₁, . . . , V_i−1, y, V_i+1, . . . , V_n).

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α= (α1, . . . , αk)∈ {1, . . . , n}^{k’ ª} c‰ac!qa|‡n©c

∂_α^kf = ∂^k

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f^¿

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ª

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α = (α₁, . . . , α_p)∈ {1, . . . , n}^p’

∂_α^pf(V1, . . . , Vn)∈L(∞)(A)^¿

^_ycM†T|‡v±qXn©m

t^j_i, j = 1, . . . , ki

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ª

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Â{Å œ Á ›:› F6

i∈ N ^{)- '832)73! !#/} Vi

/))- N88'

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& 5(!*:!* '8)

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,+)- )- N4 &

Gⁱ×B(R),: <5!& " '7

pi =pi(ω, x) ^{'8- )-} E(Θf(Vi)1(ai,bi)(Vi)) = E

Θ

Z

R

f(x)pi(ω, x)1(ai,bi)(x)dx

,

6 ()7)6

Gⁱ," <5!&F <32" 6 5!&

Θ ^{32 6 ())6 +,}

: <5!&" '7

f :R→R

8:,4-

pi ∈ C¹(Di) ³² ∂ylnpi(ω, y)∈L_(∞)(A)

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ª

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Vi ^ª

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|ଠpi

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ª

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ª

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ª

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pi

v±m mpb$|N|‡n©_$|Kq3np_ac

ª

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R

’ ª

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D_i =R

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]|KurcG„Š‹_

i ∈N ª

c>Š!|Kqamov±‰ac!uM„

Gⁱ× B(R)´¶b$cG„‡mosau©„‡¦y§±c,„‡qa‰É†|Kmpv¯n©v¯£Kc¨IsaqaŠ!npv±|Kq

πi : Ω×R → R₊ mpsaŠ‹_On©_a„n

πi(ω, y) = 0 ^¨I|‡u y /∈ (ai, bi) ^„‡qa‰ πi ∈ C1(Di)^{¿ ¡ c}

„‡mpmosab3c

Â{Å œ Á ›:›

πi ∈L_(∞)(A) ³² π_i⁰ ∈L₍₁₊₎(A)

^_ac'moc

ª

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ª

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pi

b·„Gl3_]„G£Kc‰avxmpŠ'|‡qVnpv±qVsav¯n©v±c!m5vxq

t^j_i, j = 1, . . . , ki

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ª

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πi

v±q3|Kup‰ac!u

n©|ŠG„‡qyŠ'c'§0np_ac'moce¦|Kuo‰ac'un©c!upb$mH?:„‡m c'§x§„‡m%n©_ac¦|Kup‰yc'un©c!upb$mv±q

a ^„qa‰ b ^A¿

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ª

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Á

Ÿ‡œ

Å

ž ?›

¿Èup„‡qa‰a|Kb £„‡uov¥„‡¦a§xc

F v±mDŠG„‡§x§±c!‰6„mov±b$†a§±c)¨IsaqaŠn©v±|‡q]„‡§vx¨

n©_ac!upcMc yvxmonpm%mp|Kb$c

n „‡qa‰6mo|Kb3c

G × B(Rⁿ)^´ b3c'„‡mpsyu©„‡¦a§xcr¨IsyqaŠ!npv±|Kq

f : Ω×Rⁿ→R

mpsaŠ‹_n©_]„n

F =f(ω, V1, ..., Vn).

¡ c…‰ac'qa|n©cM¦Xl

S(n,k)

np_acMmo†]„‡Š'cg|‡¨0n©_ac…mpvxb3†a§xcr¨IsyqaŠ!npv±|Kq]„§±m mpsaŠ‹_ n©_a„n

f ∈ Cn,k

„‡qa‰

S(n,k)(A) ^ª vx§±§T¦Tcn©_ycemo†]„‡Š!ce|¨np_acmpv±b$†a§xcg¨IsaqaŠn©v±|‡q]„‡§±m5mpsaŠ‹_n©_]„n

f ∈ Cn,k(A)^¿

¡ c ª

v±§x§Rsamocn©_acqa|‡n©„n©vx|Kq

∂ViF := ∂f

∂xi

(ω, V1, . . . , Vn)^’ i= 1, . . . , n^¿

Á  Å Ÿ Á ›G› Á › ¿½Çmpvxb3†y§±cM†auo|µŠ!c'mom…|‡¨§±c!qa˜‡np_

n vxm)„moc'wVsac!qaŠ'c>|¨up„‡qa‰a|Kb

£‡„upv¥„¦a§±c!m

U = (Ui)i≤n

mosaŠ‹_²np_]„n

Ui(ω) =ui(ω, V1(ω), . . . , Vn(ω)),

ª

_ac'uoc

ui : Ω×Rⁿ→R, i∈N ^„‡uoc G × B(Rⁿ)´¶b$cG„‡mosau©„¦a§±c¨IsaqaŠn©vx|Kqam'¿ ¡ cr‰ac'qa|n©c

¦Vl P(n,k)

?Iupc'mo†Tc!Š!npvx£Kc!§xl

P(n,k)(A)) n©_ac3mp†]„Š'c6|‡¨np_ac·mov±b$†a§±c$†auo|µŠ!c'mpmoc'm¼|‡¨%§±c'qy˜‡n©_

n mosaŠ‹_¸n©_]„~n

u_i ∈ Cn,k

’ i = 1, . . . , n ??uoc'mo†Tc!Š!n©v¯£Kc!§xl

u_i ∈ Cn,k(A)^’ i = 1, . . . , n)^¿

d…|‡npv±Š'c np_]„n3vx¨

U ∈ P^{(n,k) np_ac'q} Ui ∈ S^{(n,k) „‡qa‰°vx¨} U ∈ P^(n,k)(A) ^n©_ac!q Ui ∈ S(n,k)(A)^¿

q.n©_acmo†]„‡Š'c|‡¨n©_ycemov±b$†a§±cg†aup|NŠ'c!mpmpc!m

ª

cŠ'|Kqamov±‰ac!un©_acmpŠ'„‡§¥„u†aup|N‰asaŠn

hU, Vi_π :=

Xn i=1

πi(ω, Vi)Ui(ω)Vi(ω).

¡ c‰ac]qacqa|

ª

n©_ac‰av¯³Rc!upc'qXnpv¥„‡§|K†c'u©„~n©|Kuom

ª

_av±Š‹_„‡†a†cG„uv±qh‘„‡§x§±v±„[£NvxqÝm ŠG„§±Š'sy§±sam!¿

^{Á ž Iž°Æ Á  yžyœ Á} D:S(n,1) → P(n,0) : ^v¯¨ F =f(ω, V1, . . . , Vn)

n©_ac!q

DiF := ∂f

∂xi

(ω, V1(ω), . . . , Vn(ω))1Di(ω)(Vi), DF = (DiF)i≤n∈ P(n,0).

^{Á ž Iž­Ÿ Å yža Á Ÿ Á žyœG Evx£‡c'q} F = (F¹, . . . , F^d)^’ Fⁱ =fⁱ(ω, V1, . . . , Vn)∈ S(n,1)

np_ac,h‘„‡§±§xv¥„G£Nv±q3Š'|[£„‡upv±„‡qaŠ'cb3„n©uov·v±m

σ_F^ij =

DFⁱ, DF^j

π = Xn

p=1

πp(ω, Vp)∂pfⁱ∂pf^j(ω, V1, . . . , Vn).

^_av±mvxm…„¼mˆlµb$b$c!n©uov±ŠM†|Kmovxn©v¯£Kc‰ac]qav¯n©cb·„~n©upvT¿

Á Å  Å Å Æ Tœ Á \ž D¡ ce‰yc ]qac

δ:P(n,1) → S(n,0)

B

¨I|Ku

U = (Ui)_1≤i≤n mosaŠ‹_.n©_]„n

Ui(ω) =ui(ω, V1, . . . , Vn)^{’ ª c,‰ac]qac} δi(U) := −

∂

∂xi

(πiui) + (πiui)∂lnpi

(ω, V1, . . . , Vn), δ(U) :=

Xn i=1

δ_i(U).

^{Á š Å ~Æ Á  œ Á  Å {Á ~žµœ Å  ¿É4|‡u} F = f(ω, V1, . . . , Vn) ∈ S(n,0)

„‡qa‰

U = (u_i(ω, V₁, . . . , V_n))_i=1,...,n ∈ P(n,0) ^ª

ce‰ac]qac

[F, U]π = Xn

i=1

Γi(Ii(f ×ui)×πi×pi)

= Xn

i=1 ki

X

j=1

(f ×ui)(ω, V1, . . . , Vj−1, t^j_i−, Vj+1, . . . , Vn)(πipi)(ω, t^j_i−)

−(f×u_i)(ω, V₁, . . . , V_j−1, t^j_i+, V_j+1, . . . , V_n)(π_ip_i)(ω, t^j_i+) +

Xn i=1

(f ×ui)(ω, V1, . . . , Vj−1, bi−, Vj+1, . . . , Vn)(πipi)(ω, bi−)

− Xn

i=1

(f×ui)(ω, V1, . . . , Vj−1, ai+, Vj+1, . . . , Vn)(πipi)(ω, ai+).

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i = 1, . . . , n

E(|F δi(U)|1A) + E(πi(ω, Vi)|DiF ×Ui|1A)<∞. ^{?I— A}

- E(|[F, U]π|1A)<∞ ³²

E(hDF, Uiπ 1A) = E(F δ(U)1A) + E([F, U]π1A). ^{?:Ž A}

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E(hDF, Uiπ 1A) = E(F δ(U)1A).

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Xn i=1

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Xn i=1

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∂if ×(πiui)×pi

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j=0

Z

(t^j_i,t^j_i+1)

f×(∂i(πiui)×pi+ (πiui)×∂pi))

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ai

f ×(∂i(πiui) +πiui∂lnpi)×pi.

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R

(|ui∂if| πipi)(ω, V1, . . . , Vi−1, y, Vi+1, . . . , Vn)dy <∞, Z

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Z bi

ai

(πiui∂if)(ω, V1, . . . , Vi−1, y, Vi+1, . . . , Vn)pi(ω, y)dy

= E(F δ_i(U)| Gi) + Γ_i(I_i(f ×u_i)π_ip_i).

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<∞, i= 1, . . . , n , ^?:5A

9(, η >0 ^{-) 6} F ∈ S(n,1)(A) U ∈ P(n,1)(A)^{+ @-}

E (QhDF, Ui^π1A) = E (F δ(Q U)1A) + E ([F, Q U] 1A) . ^{?: A}