0.1 Introduction 0.2 Modèle de Black

θ σ

dXt=α(β−Xt)dt +σ√XtdWt.
dXt=α(β−Xt)dt +σXδ
tdWt, δ ∈]1
2,1].
(Xt, t ∈
[0, T ])
dXt=b(t, Xt, θ)dt +σ(t, Xt, θ)dWt;t∈[0, T ], X0=η,
(Ω,A,(Ft),P)WF
RmηF0
νθ.
dXt=Xtbdt+, XtσdWt, X0=x,
θ(b, σ)
b σ
b:R+×Rd×Θ→Rd,
σ:R+×Rd×Θ→Rd⊗Rm
θ
θ
FT(ξt, , t ∈[0, T ])
ˆ
θ θ0

Gn
Gn=∂θlog L,
Gn(θ, xi) = 0
θ
ξ t1,··· , tN
0≤t1≤,··· ,≤tN≤T.
N−(ξt1,··· , ξtN).
ti=i∆, T =N∆,∆
∆N
ξ
[0, T ].
ξ: [0, T ]→E, θ
T
N∆

N∆
0, T
∆N=T
N, N → ∞,∆N→0.
N T ∆N
0
∆N=a
√N, TN=a√N.
b
σ.
ε > 0, εσ(t, x)
σ
dx(t) = b(t, x(t), θ)dt,
dxε
t=b(t, xε t, θ)dt +εσ(t, xε t)dWt,
ε0.
T
∆ 0.
−
−
n(Yi)
dY i
t= [a(t, θ)Yi t+b(t, θ)]dt +dW i t;t∈[0, T ], Y i 0=y∈R,
Win
b(t, x, θ) = a(t, θ)x+b(t, θ),1.
¯
Yt= (Y1
t+··· +Yn t)/n,
d¯ Yt= [a(t, θ)¯ Yt+b(t, θ)]dt +1
√ndBn
t;t∈[0, T ],¯
Y0=y∈R,
Bn=W1+···+Wn
√n¯
Y
1
√nn

d
(Ω,A,(Ft),P)
n−W.
S1,··· , Sd
dSi
t=Si t(bi(t, St)dt +σi j(t, St)dW j
t), i = 1, ..., d,
b σ
h > 0, Rh
tt h, (1 + Rh t=Pt+h
Pt)
Rh
th0, rt,
drt=α(ν−rt)dt +ρdWt,
drt=κ(θ−rt)dt +σ√rtdWt.
σ
St=bt +σWt, t ∈[0, T ], W .