Probabilistic modeling in finance and biology - Limit theorems and applications

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(1)Probabilistic modeling in finance and biology - Limit theorems and applications Julien Guyon. To cite this version: Julien Guyon. Probabilistic modeling in finance and biology - Limit theorems and applications. Mathematics [math]. Ecole des Ponts ParisTech, 2006. English. �pastel-00001995�. HAL Id: pastel-00001995 https://pastel.archives-ouvertes.fr/pastel-00001995 Submitted on 7 Nov 2006. HAL is a multi-disciplinary open access archive for the deposit and dissemination of scientific research documents, whether they are published or not. The documents may come from teaching and research institutions in France or abroad, or from public or private research centers.. L’archive ouverte pluridisciplinaire HAL, est destinée au dépôt et à la diffusion de documents scientifiques de niveau recherche, publiés ou non, émanant des établissements d’enseignement et de recherche français ou étrangers, des laboratoires publics ou privés..

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(72) i0 • N"./H(jW./7<79b* X\./)"&:A!@ +X ,J.lBc!)"X\89*j)!@ ./7<89)7u,-8<X$B'/)":.;&! s#N)Kc[V   aB).;89s#+7u,-!"8<X\.;89'(l*\))"!#)"\) *!$X?@ %'^*6#%X?@ )8<s#\[$7<+X\'* 7< U!@ '/)8<s#wBc'(#)“*2B)"'^&!"#—./7[!@ .;'(89)!0 • ,-!5y7<.Ž)./89'( B'/#)Y7™./s#7<79U" X$g#%8<“!^n "#89y8< ^!@ )Z@ .Ž 7u,z./C./7Lh^"—*—7u,-!))"!#)A*y*89&!)@ 8<. 8<'/ZX\Bc'()7<792Bc'(#)U7<!\!@ s#C.;8<'( *%894!@ ) 8<7<79!q5'&:C./58<s#!0feF)&:!)"&:!1#)q*!FB)"'(B)"8[@ ^!@ œC%!F*#"&:A!@ X\.g*,JŠF#^n 79!)DIy7<!#%)g./BB798<&!.;89'(?. 7i,z"8<X\.;89'(e*%!g*79:.;ywH .;X\X\./*,z'/B%89'(y#)'(BU!@ !%!D '/SB)89A7™. ]{'()X$—*,-#+./)589&!7<  

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(79) .;8<);D;,J.;8R8< )"'^*#%89 79w&!'(&!B%—* $"&:C.|Vm9!*fe./) ;'|K P8L]{#)&!./ - &DB)"'(#%KMp@ s#p&![ ]

(80) ./X$8<7<79—* B)"'^&!5n "#A5'&:C./58<s#S. 8<5]

(81) ./89M*%!UU!@ '/)!! X$!U7<89X\8L!F'/)8<H/8<C./#‡s#M,z./8c./BB%7<8<s#U@ U#%89 .k  #›X\'* 7<G.;#%'()"!H()"!"89] .EK/!&jP8L]{#)&!.;89'(DM%œCO&!'/%]{)'( ^j@ ./#^‡a*'(U!@ !0Y‹2'/ "!#79!X$! X\“)!@ "#79.;“&'(%œC)"X\27u,-8< #898<'(\27<!Y&!./7<&#7<YB%)!@ 798<X$8<C.;8<)A*2P%8<'(79';n H/8<"!DCXZ./89Y8<79S&!'(%"8L# y./#%8v#%p"'(7<89*gPC./" ).;89'(7<79yB'/#)“X\!)“*wX\./8u ) 5h"^@ XZ. 8<s#679!b@ #* *%$*^h^./X\89s#&!7<7<#%7™./89)/D*6B79#g!NB79#g'(XrP)#!0‰2 [@ #*%7<8<X$8<./8<)"/DX$!U@ &!'(797<&8LK(X\Ž./# • Š1fbT“= • Œ   D.W[@ ^@ B%#P7<8[!@    *./! " #$&% ' 

(82)  " "  )(  iD#”./)589&!7<;DIB)!@ B#%P7<8[ @  ' uD.`[@ ^@ "'(#X$8<y[w#”./#%)"/D *%!"8<U@ .6 7™.Ž&'(X\Xr#C./#%^g@ *!P89'(7<'(H/8<"!D^"2! &!'(#)"“*w)!@ *./&8<'(0 ‹'(#—B%)!@ "! '/yX\./8< C./ 2B7<#!b*A@ ./8<7R79!“)'/8<BC./)"8<2* 7™.6 ;DG'(# [4I'/);&Eƒ .;Ž*7<Ž)B7™.;&!!)p*C./r7<!#%)r&!'/[‡;DF!)!@ #%XZ./ 79!p)!@ "#79.;rB)89&!89BC./#^‡D [Z}*U!@ H ./H/E./ \s#%!7<s#ZB89"!6* )!&:%!)&:%b]{#%#%)/01‹'(#6 #8LK('($BC./$7u,-'()"*) &:%)'('/7<'(H(89s#/Dg&'(X\X$!>&!ƒ '(—&!")"K#**U[@ :./897>B./)y7u,z./C.;79hŽ*%Ž7u,-!)")!#%)—]

(83) ./8<P%7< *%#e"&:A!@ X\.$*I,JŠF#79!)“B'/#)Y7<@ s#C.;89'(“*8L4@ )! 8<!797<S"'^&:C.;"89s#!0. (.

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(91)        ! –R!“X$'^*! 7<SœCC./%&!8<)Y79!YB79#Y#%897<89!@ A*C./“7<YPC./s#2£‡^B)89X\“79gB)8 ‡*I,z#% '/B%89'(&!'(X$X$7™.bK;./7<#)X$'|h/!%e89&!'(%#%      “*,-#+s#C./ 8L^  @     -rs#8 *U!@ Bc!*l*%#X\B  wp*#l././)" * 0   )B)@ 7™.K;./7<#).e 7u,-8<%":.;  w*$7u,z./&[89] "'(#‚n9./&!  I.;&89'(Dc8<*%8<&!pP'/#)"8<!)D[&;0–RŽB)"'^&!"#     "g"'(#%K/!  X$'*A!@ 798<w@ BC.;)#%r*8L4I#%8<'/DC&/,-!5"n!.| no*89) 7™.6'(79#%89'(+*,z#%k!@ s#C. 8<'/e*8L4@ )8<7<7<w5';n &:./"8<s#  

(92) ŠM~wŒ [0C†o7R,-!5g./79'()YB./2Bc'("8<P%7<w*w&E./79&!#79!)“[‡%./&[X\      ‰7i,z'( .\C.;#%)!797<X\2)&!'(#%)\.Z 7<.\"8<XŽ#7<.;89'(+#X?@ )89s#%  {K/'(8<)6.Z &!p# "[g7u,-[‡^X\B79 )./8L^ @ .;#&:C./B%89)" ( [0v_‰'(#% *,z./P'/)*#%\*%8<&)[@ 8<.;89'(jX\Bc'()7<79*\7<.*%hC./X$8<s#$*  Bc!)"X\[r*%\H>@ U!@ )"!) #$K/.;)8™.;P7<6./7[E@ .;'(8<)"   *'( r7<. 79'(8F!5rB%)'&:\*%Z&!7<79$*%   0 y.;BB7<89s#6!"#897™.+X?[@ '*$*f '( [n • ./)79' ‰7™.+X\'h/!./)8LX?@ 8<s#6*  &'(B89!8<%*A!@ Bc!*./ !6*+7<.eK;./)8<./P7<     p&!'( K/!)H/ K/!)       u0 • ]

(93) ./8<./ EDq'/ &'(X$X\[A*#^‡\!)")!#%) #—!))"!#)U*—*%8<&)[@ 8<.;89'(6!X$Bc'()7<7<;D      "!     uD *%89 "   "$#)   &% # "[D6#%+!))"!#)5:.;8<58<s#*%#l.N 7i,‚@ &:C./ 897<79'(C./H( $fe'/[n • ./)79' &0 • [ *%!)%8u! )"/DFP89!?&!'/#/DM*+7u,-'()"*)*% ('*) ”DF!"$s#C./ 89œA@  B./)7<_YU@ '()! X\ •  ).;7–R8<X$89/0cfe—).K;./#‡IDc*U[@ :./897<7[@ —.;#G&:C./B8L)" D&!'/&!)w7i,z))"!#)]

(94) ./8<P79ws#% 7i,z'/j&!'(X$X\[y79'()"s#%7u,-'(j./BB%)'&:     2B./)y'/”"&:U!@ X\.*,zŠM#%7<!)0 • &:U!@ X\. "Z.\ 7™.\]{'(89279rB79#y"8<X$B7<.Z 8<X$B7[!@ X$! )2[y7<pB7<#%—&!'/#)./X$X$! y#%8<798<;@ 0†o7q&!'/89" .  $"H(!79!) &p79!A&'^ ,$&!8< S*27u,‚!@ s#C.;8<'(\*894“!@ )"! 89!797<“"'^&:C.;"89s#y! )—*%!#^‡Z*C.;!S* *%8<&)[@ 8<.;89'(0%ˆM)!@ &8<'/2&!7<.%0 .

(95)    +   , .- 0/ 1 -  3254& q8 ‡^'(*!#^‡e8<)76 98  p2#bBC./&rB)"'(PC./P%8<7<89 @ &: 9;<>= “"#)—79!s#7vK8L2#% X$'(#%K/!X$! 2P)'¡“89! 8 no*89X\89'(7@?AB?  C [0 yb'/ ;  ED &?GF  IHEJKH  F7™.yœC7L).;89'(6!H(*)!@ —B./)M&!—P%)'¡“8<D&!'(X$B7[@ ^@ “*!U!"!XŽP79!1U!@ H/7<8<H/E./P79!0 y "M*'(F#%]{'(&[8<'/  LM BLON POPPO L      Q   v#%]{'(&8<'(  D< RD0S  NT*SUT  NT  T*V    Q  W V 0ˆ>'/#)Y#XŽ#7L898<*89&!7  XZY\[  1D ]^XC]_ XNa`cb(b(bO`ZX  )!B%)!@ "! w.679'(H(#%!#) [7 dfe$!57u,-'(BU!@ ).;#)Y*8L4@ )! 8<!g7 d@h eihj'_dlk enN m b(b(bod0k e p )0 yeXr#89   *g.6'()"X\g#&!798Ln *%8<!%r  qsbBq2'(+*U!@ "8<H(% B./)  *'/Ž7<! *U@ )89KF@ !*$'(# tvuxy3w z|{    g7u,-!"!XŽP79Z*r]{'(&[89'( uxw + \Q '()*)"!S"'( p.r &)'(89./&!—B'(7Lh'(X$8™./79/D(8i0z;0!7<79!1s#yBc'(#)U'(#%} XZY\[ D^8<7[‡^89"  KY\[ 7<Ys# B'/#)S'(#^5 k<Y   D ~  6[5 ] d e  |k s]*H ~  €`qk‚q„ƒ   (0    *%'( 7<r*U@ )89KF@ !Ž*\'(# tvux w    w7u,-!"!XŽP79Z*r]{'(&[89'( u†w  + \Q '()*)"!“'/Pc'()U!@ !D8u0-/0%!7<79!Ys#x dle‡Y‡ˆ w   YBc'(#)S'(#%5 X‰Y\[  0 .

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(100) =F8<X$B798<s#“s#GLY€DG'/A7<89B&:8L¢8<!%!D&!2s#%8Bc!)"X\[S*,z./B^n B%7<8<s#)A#\U@ '()! X\2*,z£‡^8<"!&gY*I,z#%8<&!8L^@ B'/#)A79!SŠM~wŒDBC./)U[‡^!X$B79y7<“U@ '()! X\ 0 0 wBC./H( *  + ( i0  . vk. `.

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(102) ./89);D'(G./BB)"'^&:  BC.;)—'/j&:U@ XZ.Z*I,JŠF#79!)y*,-'()"*)  (D*%8<'/ 

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(117) ./&8<79y*w7™.ŽX\.|"'().;89'(#%89K;./  

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