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Utility maximization in an insider influenced market

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HAL Id: inria-00070624

https://hal.inria.fr/inria-00070624

Submitted on 19 May 2006

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Utility maximization in an insider influenced market

Arturo Kohatsu-Higa, Agnès Sulem

To cite this version:

Arturo Kohatsu-Higa, Agnès Sulem. Utility maximization in an insider influenced market. [Research Report] RR-5379, INRIA. 2004, pp.31. �inria-00070624�

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ISRN INRIA/RR--5379--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

Utility maximization in an insider influenced market

Arturo Kohatsu-Higa and Agnès Sulem

N° 5379

October, 8 2004

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Unité de recherche INRIA Rocquencourt

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(9)

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{W(p)(T)>0} ›®MI‹p’§tMXƒI‹p|X

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(10)

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q1gt`—XZªEgt™cašgtW£ HJI‹p|X¬acW…3›®MkWZp’VžXƒI4p|XNp¡q1gr`YXZªEgr™šašg

p ašW¤XwprLNMacª

W(p)(t) > 0 ªEgt`·pr™c™

t[0, T]£ – a°XƒI·XƒI4acWf`ZMWYXƒ`YašvXƒašgr[¬›®M!v’pr[N[4g6›.q4pr`ƒprLNMXZ`Zac«MJgt]4`fqd`Zgt„4™cML ]4WZac[4Œ‚XƒI4M ªE`ƒprvXƒašgr[grª4›®M’p|™cXƒIaš[~§tMWYXƒMu¬aš[¤XZI4M`YašWZ¦~V¤prWYWZMX

π(t) =π(t, ω) =p1(t)S(t)/W(p)(t)

ªEgt`Qp|™š™

t [0, T]£ – M¤u4M¶‹[dM·pr[ prudL/acWZWZac„4™šMqsgt`YXYªEgt™šacg/p|Wpr[

Ht

µºpru‹prqGXƒMu€q4`ZgAvMWZW

π Œta°§Aaš[4Œ‚XƒIdM # ") g|ª­XƒI4M!Xƒg|Xwpr™4›®M’pr™°XƒI

W(t)grªp|[prŒtM[~X#ac[^§rMW—XƒMu¯aš[¬XƒI4M\`ZacWZ¦~V prWZWYMXJpiX#XƒacL/M

t…Gpr[4uWZp|XƒacWYª¸VAaš[dŒWZgrL/M pru4u4a°Xƒacgt[‹pr™‹vgt[4u4a°Xƒašgr[4WfXƒI‹p|X®›!aš™c™‹„1M q4`ZMvašWYMu

™´p|XZM`?2;WYMM¤XZI4Mu4M¶‹[dacXƒacgt[kgrª

AH

:³£HJI4MvprWZM›!I4M[

Ht ⊆ Ft ⊆ Gt ⊆ F acW\W—Xƒ]4udašMu aš[

5

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HJIdM¤udVA[‹p|L/acvWJgrª¹XƒI4Mu4acWZvgt]4[~XƒMuk›®M’pr™°XƒIkq4`ZgAvMWZW

X(t) = Xπ(t) = exp(−

Z t

0

ρ(s)ds)W(π)(t)

vgt`Y`ZMWYq1gr[4u4aš[dŒ¬XZg·XƒI4Mqsgt`YXYªEgt™šacg

π ašWXƒIdM[

dX(t) =X(t)h

(µ(t)ρ(t))π(t)dt+π(t)σ(t)dB(t)i

X(0) =x >0. 2Ý{/:

HJI4ašWMb^]‹p|Xƒacgt[ašWhY]4WYXZac¶‹Mu„^V¯]4WZac[4Œ·prWJ„sMªEgr`ZMQXZI4Mu4M¶‹[4acXZašgt[¯grªªEgt`Y›#pr`Zuac[~XƒMŒr`ƒpr™cW

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5

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›#Mgt„dXƒpraš[XZI4MWZgt™c]dXƒacgt[grª 2Ý{/:

X(π)(T) =xexpn

T

Z

0

((µ(t)ρ(t))π(t)

12π2(t)σ2(t))dt+

T

Z

0

π(t)σ(t)dB(t)o .

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– M¤`ZMW—Xƒ`YašvX\Xƒg/™šgtŒtpr`Za°XƒI4LNašv]GXƒaš™cacXºV/ªE]4[4vXZašgt[dWpr[4u vgt[dWZašudM` XZI4M¤ªEgr™š™šg6›!ac[4Œ¬q1M`YªEgt`èµ

L¯pr[dvM‚v`YacXZM`Zacgt[

J(π)E[ lnX(π)(T)]lnx=

=Eh

T

Z

0

((µ(t)ρ(t))π(t) 12π2(t)σ2(t))dt+

T

Z

0

π(t)σ(t)dB(t)i .

2E‰ :

¼  ! ")+!

AH

)+? #.)+!++!+

π(t) !

)+

π(t) ), / Ht

/ / !,

π(t) !ªEgr`Y›#pr`Zukaš[~XƒMŒt`ƒp|„4™šM #

Y(t) := Rt

0 σ(s)dB(s) 2Ýz:

E[

T

R

0

π(t)dY(t) ] +E[

T

R

0

(|µ(t)ρ(t)|.|π(t)|+σ2(t)π2(t))dt]<∞.

HJI4M¬Œtg~pr™3acW‚Xƒg¶‹[4u XƒI4M¬gtqGXƒašL/pr™%q1gt`—XZªEgt™cašg

π ∈ AH

ªEgt`XƒI4M¬™cgtŒ~pr`YacXƒIdL/acv¤]dXZaš™cacXºV

q1gt`—XZªEgt™cašg¤q4`Zgt„d™šML

sup

π∈AH

Ex[ln(X(π)(T))] =Ex[ln(X)(T))]. 2 /:

+¤"©+  ¹ |

nº[ XZI4ašWQWZMvXƒacgt[ ›#M¬Œta°§tM¬pXƒI4Mgt`ZMLTXZI‹p|XvwI‹pr`ZprvXZM`Yaš«MWgtqdXƒacL¯p|™%qsgt`YXYªEgt™šacgtW£HJI4acW

XƒI4Mgt`ZML ›!aš™c™1„sM‚]4WYMuaš[pr™c™1M³²dprL/qd™šMWXZg·u4MXZM`YL/ac[4MQXƒI4p|XXƒIdM‚q4`ZgtqsgtWYMuq1gr`YXZªEgr™šašg

ašWJgtqdXZašL/pr™Â£

¼AÀ ‘ ¼ ! +!# ?#.!! &"!

2;a: !#+! ! #

π ∈ AH

#"!'#+$# ! 2 /:

2;aša:% !#+! !

π ∈ AH

&() @' ! #.)+!++

Mπ(t) := Eh

t

Z

0

(µ(s)ρ(s)σ2(s)π(s))ds+

t

Z

0

σ(s)dB(s)|Ht

i 2Ý}:

(12)

?

Ht

# /! #

P

2;ašaca: !#.!=!

π ∈ AH

& ) @ #

t, ω3 ! &"*)

s7→E[ Z s

0

σ(u)dB(u)|Ht] ;s > t

?#& ! )+ & &

d dsE[

Z s

0

σ(u)dB(u)|Ht] =Eh

µ(s)ρ(s)σ2(s)π(s)|Ht] ; s > t 2;†/:

2;a:

2Eaša: fUG]dq4q1grWZM 2;a:©I4gt™šudW£©UGac[4vM

π ∈ AH

ašWJgtqGXƒašL/pr™Ý…A›®M¤I4p6§rM

J)J(π+rβ)

ªEgt`!pr™š™

β ∈ AH

pr[4u

rR£fHJI4M`ZMªEgt`ZM

d

drJ(π+rβ)

r=0 = 0.

HJI4ašWJŒta°§tMW

Eh

T

Z

0

{µ(t)ρ(t)σ2(t)π(t)}β(t)dt+

T

Z

0

β(t)(σ(t)dB(t))i

= 0 2;/:

ªEgt`!pr™š™

β ∈ AH

£©nº[kq‹pr`—Xƒacv]4™špr`…dprq4q4™°VGac[4Œ˜XƒIdašWXƒg

β(u) =β0(t)1[t,s](u)

ªEgt`

0 t < s T… u [t, s]… ›!I4M`ZM β0(t) ašW Ht

µºLNM’prWY]4`ƒpr„d™šMr… „sgt]4[4udMu­…1WZLNgGgrXZI

pr[4uWƒp|XZašW—¶‹MW

Ds+β(t) =Dsβ(t) pG£ëpd£ s [0, T]…G›#Mgt„dXƒpraš[

Eh (

s

Z

t

{µ(u)ρ(u)σ2(u)π(u)}du+ Z s

t

σ(u)dB(u))β0(t)i

= 0. 2èŠ6ˆ/:

UGaš[4vMXƒI4acW!I4gt™šudWªEgt`!pr™š™1WZ]4vwI

β0(t) ›®Mvgt[4v™š]4udMXZI‹p|X

Eh (

s

Z

t

{µ(u)ρ(u)σ2(u)π(u)}du+ Z s

t

σ(u)dB(u))|Ht

i

= 0. 2èŠtŠ :

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Kπ(t) :=

t

Z

0

{µ(u)ρ(u)σ2(u)π(u)}du+ Z t

0

σ(u)dB(u)

Wƒp|XZašWY¶4MW

E[Kπ(s)|Ht] =E[Kπ(t)|Ht] ªEgr`!pr™š™

st. 2èŠ*‡:

m `YgtL XZI4ašW›®MŒtMX…dªEgt`

st

E[Mπ(s)|Ht] =E[E[Kπ(s)|Hs]|Ht] =E[Kπ(s)|Ht] =E[Kπ(t)|Ht] =Mπ(t),

›!I4ašvwIkašW 2Eaša:³£

2Eaša:

2Eašaca: +UG]4qdq1gtWYM 2Eaša:©I4gt™cu4W£çHJI4M[­…4ªEgt`

st…

E[Kπ(s)|Ht] =E[E[Kπ(s)|Hs]|Ht] =E[Mπ(s)|Ht] =Mπ(t) =E[Kπ(t)|Ht].

l M[dvM 2—Š*‡:©µ®p|[4uXƒI4M[€pr™cWZg2—ŠrŠ :©µ©I4gt™cu4W£fj [4u 2èŠtŠ :#v™šM’p|`Z™cV/ašLNq4™cašMW 2;acaša:³£

2Eašaca:

2;a: çUA]4q4qsgtWZM 2;acaša:fI4gt™cu4W£

HJIdM[.ac[^XZMŒt`Zp|Xƒac[4Œ 2݆/:³…®›®M€ŒtMX2èŠtŠ :w…#›!I4acvwI.prŒ~p|aš[ ašLNq4™šacMW 2èŠ6ˆ/:w£ >®V Xwpr¦^ac[4Œ

™šaš[dM’pr`vgrL˜„4aš[4p|Xƒacgt[žgrª 2èŠ6ˆ/:…%›#Mgr„dXwprac[žXƒI4p|X 2;/:Idgt™šu4WªEgt`˜p|™š™

β ∈ AH

grªXZI4M

ªEgt`ZL

β(u) = ΣNi=1βi(ti)1(ti,ti+1](u)

›!I4M`YM

0 = t0 < t1 < . . . < tN+1 = T… i = ti+1 ti

… p|[4u

βi(ti) ašW Hti

µ

L/MprWZ]4`Zpr„4™cMr…4„sgt]4[4udMu­… WZLNgAgrXƒIkpr[4ukWƒpiXƒašW—¶‹MW

Ds+βi(t) =Dsβi(t) pG£ëpd£ s[0, T]£

>®V 2Âz:³…d›®M¤I‹p’§rMªEgt`!p|™š™

β ∈ AH T

Z

0

β(t)(σ(t)dB(t)) =

T

Z

0

β(t)dY(t) = lim

i→0ΣNi=1β(ti)

ti+1

Z

ti

σ(s)dB(s),

pr[4uI4M[dvM 2Ý/:#I4gr™šu4WJªEgt`!pr™c™

β ∈ AH

£

HJIdašWL/Mpr[4WXZI‹p|X‚XƒIdMNudaš`ZMvXZašgt[‹p|™+u4M`Zac§|p|XZac§rM·grª

J p|X π ›!a°XƒI `ZMWZqsMvXXZgXZI4M u4aš`YMvXZašgt[

β…4udM[4grXZMu „^V

DβJ) ašW 0…‹a£Mr£

DβJ(π) := lim

r→0

J(π+)J)

r = 0 ;β ∈ AH. 2—Š’{/:

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O grXZMXƒI‹p|X

J :AHR ašWJvgr[4v’p’§tM|… aš[XƒI4MWYM[4WYMXZI‹p|X

J(λα+ (1λ)β)λJ(α) + (1λ)J(β); λ[0,1], α, β ∈ AH.

HJI4M`YMªEgt`YMr…4ªEgt`!pr™c™

α, β ∈ AH

pr[4u

ε(0,1)…G›#MI‹p’§tM

J(α+εβ)J(α) =J((1ε) α

1ε +εβ)J(α)

(1ε)J( α

1ε) +εJ(β)J(α)

=J( α

1ε)J(α) +ε(J(β)J( α 1ε)).

2—Š‰ :

O g6›…4›!acXZI 1

1ε = 1 +η ›#MI‹p’§tM

ε→0lim 1

ε(J( α

1ε)J(α)) = lim

η→0

1 +η

η (J(α+ηα)J(α)) = DαJ(α).

#gtL˜„4ac[4aš[4Œ˜XZI4ašW›!a°XƒI 2—Š‰ :®›#MŒtMX

DβJ(α) = lim

ε→0

1

ε(J+εβ)J(α))DαJ(α) +J(β)J(α).

– Mvgr[4v™c]4u4M‚XƒI‹piX

J(β)J(α)DβJ(α)DαJ(α) ;α, β∈ AH.

nº[¯q‹pr`—Xƒacv]4™špr`…^prqdq4™cVAaš[dŒXZI4ašWçXƒg

α=π p|[4u]4WYaš[4ŒXƒI‹piX

DβJ) = 0 „~V 2—Š6{:…^›#M

ŒtMX

J(β)J)0 ªEgt`!p|™š™

β∈ AH,

›!I4ašvwIkq4`Zg6§rMW!XƒI‹p|X

π acWJgtqdXƒacL¯p|™Â£

2

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L˜]4™´pªEgr`XƒI4MgtqdXZašL/pr™sWYXƒ`Zp|XƒMŒrV

π£

À ‘ À E“4‘ ½ (& ! ? #

π ∈ AH

# ! #.$# ! 2 /:

! ! '&/3

π(t)E2(t)|Ht] =E[(µ(t)ρ(t))|Ht] +a(t). 2èŠ*z:

!#+!

a(t) lim

h→0+

1 hE[

Z t+h

t

σ(s)dB(s)|Ht]. 2—Š /:

(15)

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R M`—Xƒgt[¡qd`Zgt„4™cMLk£!l M`ZM¤XZI4M`Zp|XƒM¤grªçprqdq4`ZMva´piXƒašgr[€pr[du€§tgt™šp|Xƒac™ša°XºVpr`YM¤`YMq4™šprvMu¡„~V

XƒI4Maš`·„sMWYXNMWYXƒacL¯piXƒgt`YW…fXƒI4Mvgt[4u4a°Xƒašgr[‹pr™#M³²Gq1MvXwpiXƒašgr[4W£ HJI4M`YM acW¬pr[ M²AXƒ`Zp XZM`YL

a(t) ›!IdašvwIprqdq1Mpr`ZW®u4]4MQXZg¤XƒIdM‚pr[~Xƒacvašq4p|Xƒa°§tM![‹p|XZ]4`ZM grªXƒI4M gt`ZacŒtaš[4pr™4Mb^]‹piXƒašgr[­£fjQ[

aš[~XƒM`Zq4`YMXwpiXƒašgr[grª%XZI4ašWXZM`YL ašWJŒta°§tM[ ac[€UAMvXZašgt[d£

¼ ž“4‘

Gt+δ ⊆ Ht

δ > 0= ! ),!+ a(t) "!+ * !

#!+),& ! @&() @),!

1 hE[

Z t+h

t

σ(s)dB(s)|Ht] = 1 h

Z t+h

t

σ(s)dB(s) # hδ.

(#

Ht =Ft+δ

a(t) "!+ * ! " #.! !, ! ")

&() "!#. $# 3 ! & !, !=#.-/! /

## # "! # ! "!#

³s

nº[XZI4ašWWZMvXZašgt[›#M©vgtLNq4]dXZMfXƒI4Mç§|pr™š]4MfªE]4[dvXƒacgt[‚›!I4M[¤XƒI4MçgtqdXZašL/pr™tqsgt`YXYªEgt™šacgM²GacWYXƒW£

– MW—Xwpr`—X„~Vp¤™šMLNL/p 2EWZMM2#gt`Ygt™š™špr`YV/‡G£°Š6ˆ/:çaš[

5

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©¼ ž“ (& !

f : [0, T]× R -/#. ! # # != ), /

#.!+ !# '+&" !

Dt+f(t) := lim

s→t+Dsf(t)

! #?

t[0, T]

T

Z

0

|Dt+f(t)|dt <∞.

!

Eh

T

Z

0

f(t)dB(t)i

=Eh

T

Z

0

Dt+f(t)dti ,

#+ "!, ' !! !+) !

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