1. Processus de Poisson 2. Ecriture des fonctions valeur

Surfhvvxv gh Srlvvrq hw ìfulwxuh ghv irqfwlrqv ydohxu
Slhuuh Fdkxf hw Dqguì ]|oehuehuj/ 53351
41 Surfhvvxv gh Srlvvrq
Rq vxssrvh vrxyhqw gdqv ohv prgëohv hq whpsv frqwlqx txh ghv ìyìqhphqwv doìdwrluhv vxlyhqw xq
surfhvvxv gh Srlvvrq1 Rq glw tx*xq ìyìqhphqw [ +sdu h{hpsoh o*dssdulwlrq g*xq fkrf gh surgxfwly0
lwì, vxlw xq surfhvvxv gh Srlvvrq dyhf xq wdx{ g*duulyìh vl od gxuìh W tx*lo hvw qìfhvvdluh g*dwwhqguh
srxu txh [ vxuylhqqh hvw xqh yduldeoh doìdwrluh txl d xqh irqfwlrq gh glvwulexwlrq h{srqhqwlhooh gh
sdudpëwuh > vrlw
I +W , Sure i[ vxuylhqqh dydqw W j @ 4 h3bA =
Fhwwh ìtxdwlrq lpsoltxh txh od ghqvlwì gh suredelolwì gh W hvw
i +W , @ I +W , @ h3bA =
Fhfl vljqlh txh od suredelolwì txh o*ìyìqhphqw [ vxuylhqqh vxu xq shwlw lqwhuydooh gh whpsv
vlwxì hqwuh W hw W . gw hvw dssur{lpdwlyhphqw ìjdoh ã h3bA gw= Sdu frqvìtxhqw/ od suredelolwì txh
o*ìyìqhphqw vxuylhqqh vxu xq shwlw lqwhuydooh gh whpsv gw ã sduwlu gh o*lqvwdqw suìvhqw +W @ 3, ydxw
dssur{lpdwlyhphqw gw= Oh wdx{ g*duulyìh shxw grqf v*lqwhusuìwhu frpph od suredelolwì g*dssdulwlrq
gh o*ìyìqhphqw [ vxu xq shwlw lqwhuydooh gh whpsv gw txh o*rq dsshooh hqfruh oh x{ gh suredelolwì gh
o*ìyìqhphqw1
51 Hfulwxuh ghv irqfwlrqv ydohxu vrxv iruph lqwìjudoh
Fkhufkrqv ã gìwhuplqhu oh sul{ g*xq dfwli ã wrxwh gdwh w> txh qrxv qrwhurqv +w,= Fhw dfwli udssruwh
xq uhyhqx lqvwdqwdqì $+w, hw ã xq wdx{ gh ghvwuxfwlrq txl vxlw xq surfhvvxv gh Srlvvrq gh sdudpëwuh
w,= Rq vxssrvh dxvvl txh ohv uhyhqxv vrqw
A 3= O*dfwli gìwuxlw ã od gdwh w d xqh ydohxu qrwìh +
dfwxdolvìv dx wdx{ u A 3=
Sdu gìqlwlrq/ od ydohxu gh o*dfwli v*ìfulw=
w
+ , @
HA
] |nA
|
w . W,
$+{,h3oE%3|g{ . h3oA +
+4,
rü HA gìvljqh o*hvsìudqfh gh W= Frpph od irqfwlrq gh ghqvlwì gh W d srxu h{suhvvlrq h3bA > rq
shxw ìfuluh +w, gh od pdqlëuh vxlydqwh=
w
+ , @
] " ] |nA
f
|
$+{,h3oE%3|g{
h3bA gW .
] "
f
w . W , h3bA gW
h3oA +
+5,
Od suhplëuh
lqwìjudoh shxw v*lqwìjuhu sdu sduwlhv/ hq srvdqw=
x+W , @ U||nA $+{,h3oE%3|g{> fh txl lpsoltxh gx+W , @ $+W . w,h3oA gW>
hw gy +W , @ h3bA gW> txl hqwudñqh/ y +W , @ h3bA =
U
U
Rq rewlhqw/ hq dssoltxdqw od irupxoh g*lqwìjudwlrq sdu sduwlhv/ xgy @ xy ygx =
] " ] |nA
f
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$+{,h3oE%3|g{ h3bA gW @ h3bA
] |nA
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$+{,h3oE%3|g{
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f
$+w . W ,h3EonbA gW
Od suhplëuh lqwìjudoh gx phpeuh gh gurlwh d xqh ydohxu qxooh vl od frqglwlrq Olp
$+w,h3o| @ 3> fh
|<"
txh qrxv dgphwwurqv1 Hq idlvdqw oh fkdqjhphqw gh yduldeoh W @ w . W> rq shxw ìfuluh o*ìtxdwlrq +5,
gh od pdqlëuh vxlydqwh=
w
+ , @
]"
|
$+W ,h3EonbEA 3|gW .
]"
|
W ,gW h3EonbEA 3|+
+6,
Hq gìulydqw fhwwh ghuqlëuh h{suhvvlrq/ rq rewlhqw=
+w, @ $+w, . +u . ,
] "
|
$+W ,h3EonbEA 3|gW .
]"
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W ,gW h3EonbEA 3|+
w,
+7,
+
O*h{suhvvlrq hqwuh ohv furfkhwv gx phpeuh gh gurlwh fruuhvsrqg h{dfwhphqw ã fhooh gh
ìtxdwlrq +6,1 Rq rewlhqw grqf qdohphqw hq vxevwlwxdqw=
w, +w, . +w,>
u+w, @ $+w, . +
w grqqìh
+ ,
+8,
fh txl fruuhvsrqg ã od iruph g*ìtxdwlrq ghv irqfwlrqv ydohxuv xwlolvìhv gdqv o*hqvhpeoh gh o*rxyudjh1
61 Hfulwxuh ghv irqfwlrqv ydohxu sdu dssur{lpdwlrq
Rq shxw uhwurxyhu od irupxoh ghv irqfwlrqv ydohxu hq whpsv frqwlqx gh pdqlëuh soxv lqwxlwlyh/ hq
surfìgdqw sdu dssur{lpdwlrq1 Hq hhw/ hq dgphwwdqw txh o*dfwli udssruwh xq x{ gh uhyhqx z+w,
vxu xq shwlw lqwhuydooh gh whpsv gw> hw txh oh x{ gh suredelololwì txh o*dfwli vrlw gìwuxlw vxu fh shwlw
lqwhuydooh gh whpsv gw vrlw gw> od ydohxu gh o*dfwli shxw v*ìfuluh=
w . gw, . +4 gw,+w . gw,
h3o_| $+w,gw . gw+
hq dgphwwdqw o*dssur{lpdwlrq h3o_| @ no_| > srxu gw $ 3> rq rewlhqw=
w
+ , @
+w . gw, +w,
w . gw, +w . gw, .
u+w, @ $+w, . +
gw
Fhwwh ghuqlëuh fruuhvsrqg ã +8, oruvtxh gw $ 3=
5