!
Processus(aléatoires,(calcul(stochastique(:(exercices(((((((((((((((((((((((((((((((((((((((((((((((ENSM7SE(2009(
!
"#$%&!'&(!)*+&(!'&!,-*+./".*&++&!
!
!
!
!
!
!
!
!
!
PROCESSUS(ALEATOIRES(:(
MARTINGALES,(MOUVEMENT(BROWNIEN,(CALCUL(STOCHASTIQUE(
!
"0&1#*#&(!
!
!
!
!
!
!
!
!
!
!
!
!
2-+3*&1!4556!7!8%*3*&1!9$:(.-+.!
!
Processus(aléatoires,(calcul(stochastique(:(exercices(((((((((((((((((((((((((((((((((((((((((((((((ENSM7SE(2009(
!
MOUVEMENT!BROWNIEN!
!
;< )$+.1&1!%=>?:*3-%&+#&!&+.1&!%&(!4!@1$@1*>.>(!(:*3-+.&(!@$:1!:+!@1$#&((:(!A!B!
!
-< A! &(.! :+! @1$#&((:(! C! -##1$*((&D&+.(! *+'>@&+'-+.(! &.! (.-.*$++-*1&(! EFGH,<I! &.!
@$:1!.$:.!.I!A.!&(.!'&!%$*!+$1D-%&!JE5I.<!
K< A!&(.!:+!@1$#&((:(!L-:((*&+!#&+.1>I!-3&#!#$3EA(I!A.<!M!D*+E(I.<!
!
N:&!O-:./*%!1-P$:.&1!@$:1!?:&!A!($*.!:+!D$:3&D&+.!K1$Q+*&+!R!
!
!
4< ,$*.! A! M! EA.<5S.S;! ! :+! D$:3&D&+.! K1$Q+*&+! (.-+'-1'! (:1! T5I! ;U! E$+! (=-11V.&! C! ;<W!
)$+.1&1! ?:&! %&! @1$#&((:(! X! M! EA;/.! 7!A
;<5S.S;! &(.! -:((*! :+! D$:3&D&+.! K1$Q+*&+!
(.-+'-1'!(:1!T5I!;UW!Y-*1&!:+!'&((*+W!
!
!
Z< 8+! -@@&%%&! pont( brownien! %&! @1$#&((:(! EF.<5S.S;! $K.&+:! C! @-1.*1! ':! D$:3&D&+.!
K1$Q+*&+!(.-+'-1'!&+!*D@$(-+.!%&(!#$+.1-*+.&(!'=*+.&1@$%-.*$+!F5!M!F;!M!5W!
Y$1D&%%&D&+.I!(*!EA.<.[5!&(.!:+!D$:3&D&+.!K1$Q+*&+!(.-+'-1'I!$+!-!B!
!
F.!M!A.!7!.A;I!!!!5S.S;!
!
)$+.1&1! ?:&! EF.<! &(.! :+! @1$#&((:(! L-:((*&+! #&+.1>! &.! #-%#:%&1! (-! (.1:#.:1&! '&!
#$3-1*-+#&W!"+!'>':*1&!'&(!K-+'&(!'&!#$+O*-+#&!C!6\]W!H+.&1@1>.&1W!
!
!
Figure(1.(Quelques(trajectoires(du(pont(brownien(et(les(bandes(de(confiance(à(95%.(
!
!
^< ,$*&+.!A!M!EA;I!A4<!:+!D$:3&D&+.!K1$Q+*&+!K*/'*D&+(*$++&%! (.-+'-1'I! #=&(./C/'*1&!
.&%!?:&!A;!&.!A4!($+.!'&:0!D$:3&D&+.(!K1$Q+*&+(!(.-+'-1'!*+'>@&+'-+.(W!!
)$+.1&1!?:&!A!&(.!*+3-1*-+.!@-1!1$.-.*$+!&+!%$*I!#=&(./C/'*1&!?:&!@$:1!.$:.!θ∈T5I4πUI!%&!
@1$#&((:(! _! '>O*+*! @-1! _.! M! 9θA.! ! &(.! :+! D$:3&D&+.! K1$Q+*&+! K*/'*D&+(*$++&%!
(.-+'-1'I!-3&#!9θ!%-!D-.1*#&!E#$(Eθ<!(*+Eθ<!`!/(*+Eθ<!#$(Eθ<<W!
!
0.0 0.2 0.4 0.6 0.8 1.0
-1.0 -0.5 0.0 0.5 1.0
t
Pt
!
Processus(aléatoires,(calcul(stochastique(:(exercices(((((((((((((((((((((((((((((((((((((((((((((((ENSM7SE(2009(
!
VARIATION!QUADRATIQUE!‐!COVARIATION!
!
;< ,$*&+.!X!&.!a!'&:0!@1$#&((:(!C!.1-P&#.$*1&(!'&!#%-((&(!b;W!!
-W )$+.1&1!?:&!!c!X!d.!M!c!a!d.!M!5I!@:*(!?:&!c!X!I!a!d.!M5!!E.<W!!
KW )$+.1&1!?:&!!c!X!I!e!d.!M5I!!@$:1!.$:.!@1$#&((:(!e!.&%!?:&!!c!e!d.!c!f!!E.<W!!
!
4< ,$*.!A!:+!D$:3&D&+.!K1$Q+*&+!(.-+'-1'W!8+!1-@@&%%&!?:&!c!A!d.!M!.I!.W!
-W ,$*.!X!:+!@1$#&((:(!C!.1-P&#.$*1&(!'&!#%-((&(!b;W!b-%#:%&1!c!X!I!Ad.!!
KW ,$*.!_!:+!D$:3&D&+.!K1$Q+*&+!-3&#!'>1*3&!_.!M!µ.!f!A.W!b-%#:%&1!c!_!d.!
!
Z< 8+!1&@1&+'!%&(!+$.-.*$+(!'&!%=&0&1#*#&!Z<!'&!%-!(&#.*$+!g!)$:3&D&+.!K1$Q+*&+!hW!
-W )$+.1&1!?:&!!c!A;I!A4!d.!M!5W!
KW "0@1*D&1!!c!A;I!_;!d.!!&+!O$+#.*$+!'&!#$1EA;.I!_;.<W!
!
MARTINGALES!
(
;< ,$*.!E_.<.[5!:+!D$:3&D&+.!K1$Q+*&+!(.-+'-1'W!)$+.1&1!?:&!B!
-W E_.4<.[5!&(.!:+&!($:(/D-1.*+L-%&I!&.!E_.4!/!.<.[5!&(.!:+&!D-1.*+L-%&W!
KW E&0@E_.<<.[5! &(.! :+&! ($:(/D-1.*+L-%&I! &.! E&0@Eα_.! 7!iα4×.<<.[5! &(.! :+&!
D-1.*+L-%&!@$:1!.$:.!1>&%!αW!!
!
!
4< TGestion(de(portefeuille(et(martingaleU!
8+!#$+(*'j1&!k!-#.*O(!O*+-+#*&1(I!'$+.!%&(!@1*0!C!%-!'-.&!t!($+.!+$.>(!EX;.! I! lI! Xk.<W! 8+!
(:@@$(&!?:&!#m-?:&!@1$#&((:(!EXn<!&(.!:+&!martingale!E;SnSk<W!!
J$.$+(!θ.!M!Eθ;.I!lI!θk.<!%&!+$DK1&!'&!@-1.(!'>.&+:!'-+(!#m-?:&!-#.*O!&+.1&!.!&.!./;W!o-!
3-%&:1!':!@$1.&O&:*%%&!&(.!-%$1(! p.!M!θ;.!X;.!f! l!f!θk.!Xk.W!8+! (:@@$(&!+-.:1&%%&D&+.!
E@$:1?:$*!R<!?:&!θ!&(.!:+!@1$#&((:(!prévisible!K$1+>W!
!
8+!'*.!?:&!%&!@$1.&O&:*%%&!&(.!autofinancé!(*!B!
!
θ;.!X;.!f!l!f!θk.!Xk.!M!θ;.f;!X;.!f!l!f!θk.f;!Xk.!
!
-W H+.&1@1>.&1!#&..&!>L-%*.>!E@1&+'1&!:+!&0&D@%&!(*D@%&<W!
KW )$+.1&1! ?:&! (*! %&! @$1.&O&:*%%&! &(.! -:.$O*+-+#>I! -%$1(! p! &(.! :+&! D-1.*+L-%&W!
H+.&1@1>.&1!#&!1>(:%.-.!&+!.&1D&!'&!L&(.*$+!'&!@$1.&O&:*%%&W!
!
!
Z< [Sur(la(ruine(du(joueur]W!!
8+! #$+(*'j1&! %&! P&:! '&! @*%&! $:! O-#&! >?:*%*K1>! E@M;q4<I! &.! $+! (&! @$(&! %-! ?:&(.*$+!
(:*3-+.&!B!?:&%%&!&(.!%-!@1$K-K*%*.>!@$:1!?:&!%&!P$:&:1!1>-%*(&!:+&!@&1.&!(:@>1*&:1&!C!
:+&!?:-+.*.>!!b!d5!E1:*+&<!-3-+.!'=-3$*1!1>-%*(>!:+!K>+>O*#&!(:@>1*&:1!$:!>L-%!C!!ad5!R!
!
8+! +$.&! X;I! lI! X+! %&! K>+>O*#&! 1>-%*(>! C! #m-?:&! @-1.*&I! >3&+.:&%%&D&+.! +>L-.*OI! &.!
,+MX;flfX+W! ,$*.! r! M! *+Os+JI! ,+! [! -! $:! ,+! S! /KtI! &.! ! @1:*+&! ! %-! @1$K-K*%*.>! #m&1#m>&W!
u-+(!%-!(:*.&I!$+!@1&+'!-!&.!K!&+.*&1(W!
!
!
Processus(aléatoires,(calcul(stochastique(:(exercices(((((((((((((((((((((((((((((((((((((((((((((((ENSM7SE(2009(
!
-W )$+.1&1!?:&!r!&(.!:+!.&D@(!'=-11V.I!&.!&0@1*D&1!!@1:*+&!!&+!O$+#.*$+!'&!rW!
KW 8+! -'D&.! ?:&! r! &(.! O*+*I! *W&W! FErcf<M;! E3$*1! @-1! &0W! T_*%%*-D(U! @$:1! :+&!
'>D$+(.1-.*$+! '&! #&! 1>(:%.-.I! :.*%*(-+.! -:((*! %&! .mW! '=-11V.<W! "+! '>':*1&! ?:&!
@1:*+&! M! -qE-fK<I! @-1! -@@%*#-.*$+! ':! .m>$1jD&! '=-11V.! '&! u$$K! EIndication(:(
utiliser(le(théorème(3,(puis(le(théorème(de(convergence(dominée(de(Lebesgue<W!
H+.&1@1>.&1W!b-%#:%&1!@1:*+&!(*!-!&.!K!+&!($+.!@-(!&+.*&1(W!
#W N:&! O-:./*%! D$'*O*&1! @$:1! #-%#:%&1! %-! @1$K-K*%*.>! '&! 1:*+&! &+! .&D@(! #$+.*+:I!
'$+#!(*!E,.<!&(.!%&!D$:3&D&+.!K1$Q+*&+!(.-+'-1'!R!!
!
!
^< [Sur(la(décomposition(de(Doob]W!
-W u>D$+.1&1! %&! .m>$1jD&! '&! '>#$D@$(*.*$+! '&! u$$KW! 2:(.*O*&1! %=*+.&1@1>.-.*$+!
'&!)! #$DD&!%=&11&:1!'&! @1>3*(*$+!#:D:%>&I!&.! G!#$DD&!%=&(@>1-+#&! '&!L-*+!
#:D:%>&W!
KW u$++&1!%-!'>#$D@$(*.*$+!'&!u$$K!@$:1!%&!P&:!'&!@*%&!$:!O-#&W!
#W u$++&1!%-!'>#$D@$(*.*$+!'&!u$$K!'=:+!@1$#&((:(!G9E;<!B!X.!M!φ!X./;!f!ε.!I!-3&#!
Eε.<!:+!K1:*.!K%-+#!#&+.1>W!
!
!
\< [Loi(du(premier(temps(d’atteinte(d’une(barrière(par(un(mouvement(brownien]W!!
u-+(!#&.!&0&1#*#&I!$+!3-!#-%#:%&1!'&!4!O-v$+(!%-!%$*!':!@1&D*&1!.&D@(!'=-..&*+.&!'=:+&!
K-11*j1&!'=:+!D$:3&D&+.!K1$Q+*&+!(.-+'-1'!A!B!!
!
r-!M!*+Os.d5I!A.!M!-tI!!!-d5!
!!
F$:1!#&%-I! $+!3-!#-%#:%&1! %-!.1-+(O$1D>&!'&! o-@%-#&!'&!r-!B!"T&0@E/λr-<UI!?:*! @$:11-!
&+(:*.&!V.1&!*+3&1(>&W!
!
-W 2:(.*O*&1!?:&!r-!M!*+Os.d5I!A.![!-t!&.!?:&!r-!&(.!:+!.&D@(!'=-11V.W!
KW 8+! 1-@@&%%&! ?:&! %&(! .1-P&#.$*1&(! ':! A1$Q+*&+! ($+.! @1&(?:&! (w1&D&+.! +$+!
K$1+>&(W!N:=&+!'>':*1&!@$:1!r-!R!
#W ,$*.! ([5W! F$($+(! ).(! M! &0@E(A.! 7!(
4.q4<W! )$+.1&1! ?:&! "T)r-.UM;! E*+'*#-.*$+!B!
:.*%*(&1! %=&0&1#*#&! ;<W! "+! '>':*1&! ?:&! "T)r-UM;I! @:*(! "T&0@E/λr-<U! M! &0@E/
-E4λ<;q4<W!
!
F-1!*+3&1(*$+! '&!%-!.1-+(O$1D>&! '&!o-@%-#&I!$+! D$+.1&!?:&!r-! -'D&.!%-!'&+(*.>! '&!
@1$K-K*%*.>!B!
O-E.<M-E4π.Z</;q4!&0@E/-4q4.<!
!
8+!@&:.!1&.1$:3&1!#&!1>(:%.-.!L1x#&!-:!O-D&:0!g!@1*+#*@&!'&!1>O%&0*$+!hW!
!
'W 2:(.*O*&1!L1x#&!C!:+!-1L:D&+.!'&!(yD>.1*&!?:&!FEr-!S!.<!M!4FEA.![!-<I!.I!-d5W!!
&W 9&.1$:3&1!-%$1(!%-!'&+(*.>!'&!r-W!
!
!
Processus(aléatoires,(calcul(stochastique(:(exercices(((((((((((((((((((((((((((((((((((((((((((((((ENSM7SE(2009(
!
CALCUL!STOCHASTIQUE!
!
;< TFormule(d’intégration(par(partiesU!
u-+(!#&.!&0&1#*#&I!$+!(=*+.>1&((&!C!%-!O$1D:%&!'=*+.>L1-.*$+!@-1!@-1.*&(!@$:1!'&(!(&D*/
D-1.*+L-%&(/#$+.*+:&(!B!
!
!
!
-W u>D$+(.1-.*$+!>%>D&+.-*1&W!u>D$+.1&1!%-! O$1D:%&!'-+(! %&!#-(!@-1.*#:%*&1!$z!
XMaI! &+! :.*%*(-+.! (*D@%&D&+.! :+&! (:K'*3*(*$+! 'y-'*?:&W! "0@%*?:&1! #$DD&+.!
$+!@&:.!&+!'>':*1&!%&!1>(:%.-.!'-+(!%&!#-(!L>+>1-%W!
KW 9&.1$:3&1!%-!O$1D:%&!'=*+.>L1-.*$+!@-1!@-1.*&(!&+!:.*%*(-+.!%&!#-%#:%!'=H.{W!
!
#W ,$*.! φ! :+&! O$+#.*$+! '&! #%-((&! b;W! 2:(.*O*&1! ?:&! φ! &(.! :+&! (&D*/D-1.*+L-%&!
#$+.*+:&! E'>.&1D*+*(.&<W! "+! '>':*1&! %-! O$1D:%&! '=*+.>L1-.*$+! @-1! @-1.*&!
3-%-K%&!@$:1!:+&!(&D*/D-1.*+L-%&!&.!:+&!O$+#.*$+!'&!#%-((&!b;!B!
!
!
!
8K(&13&1!%=-+-%$L*&!-3&#!%-!O$1D:%&!'=*+.>L1-.*$+!@-1!@-1.*&(!$1'*+-*1&W!
!
!
4< TProcessus(d’Ornstein7UhlenbeckU!
o&!@1$#&((:(!'=81+(.&*+/|m%&+K&#n!&(.!%-!($%:.*$+!:+*?:&!'&!%=>?:-.*$+!B!
!
!
!
-3&#! _! :+! D$:3&D&+.! K1$Q+*&+! (.-+'-1'I! c( >( 0,(
σ
( ≥( 0( '&:0! #$+(.-+.&(! &.! 0! :+!
+$DK1&! 1>&%! ?:&%#$+?:&W! H%! (=-L*.! '$+#! '=:+&! >?:-.*$+! '*OO>1&+.*&%%&! -3&#! :+! .&1D&!
'=&11&:1!D$'>%*(>!@-1!:+!D$:3&D&+.!K1$Q+*&+W!"+!O*+-+#&I!*%!&(.!:.*%*(>!@$:1!'>#1*1&!
%-!'y+-D*?:&!'&(!.-:0!'=*+.>1V.W!
!
-W 9>($:'1&!%&!@1$K%jD&!'-+(!%&!#-(!'>.&1D*+*(.&!Eσ!M!5<W!
KW "+!:.*%*(-+.!!a.!M!X.!&#.I!!D$+.1&1!?:&!!X.!!3>1*O*&!B!
!
!
!
#W )$+.1&1! ?:&! @$:1! .$:.! .I! ! X.! ! &(.! '&! %$*! +$1D-%&! W! F%:(!
L>+>1-%&D&+.I! D$+.1&1! ?:&! ! X! ! &(.! :+! @1$#&((:(! L-:((*&+! &.! 3>1*O*&1! ?:&!
#$3EX(I!X.<!M! W!o&(!.1-P&#.$*1&(!'&!X!($+./&%%&(!#$+.*+:&(!R!
'W 2:(.*O*&1! %-! .&1D*+$%$L*&! '&! g!@1$#&((:(! '&! 1&.$:1! C! %-! D$y&++&!hI! ?:*! &(.!
:.*%*(>&!@$:1!%&!@1$#&((:(!'=81+(.&*+/|m%&+K&#nW!H+.&1@1>.&1!%&[email protected]&!#W!
!
1 / 8 100%
La catégorie de ce document est-elle correcte?
Merci pour votre participation!

Faire une suggestion

Avez-vous trouvé des erreurs dans linterface ou les textes ? Ou savez-vous comment améliorer linterface utilisateur de StudyLib ? Nhésitez pas à envoyer vos suggestions. Cest très important pour nous !