Modélisation stochastique 1ère année Isup

Y
X
X Y
•Xj
X= (X1, ..., Xn−1)Y=Xn,
n
Y
•Xj
X= (X1, ..., Xn−1)Y=Xn
L(Y|X)Y X.

B.BY.
YB=σ(X1, ..., Xn−1),
σ((Uk)k∈K)Uk.
L2,
L1
L2,
L1−L2.
(Ω,F,P),
ΩFΩ,P
(Ω,F).
L2(F)ZF−
(R,BR)BRR
E[Z2]<∞.
< X, Y > =ZX(ω)Y(ω)dP(ω) = E[XY ].
kXk2=E[X2]1/2.

H H =L2(F)F
H. pF
HF Z ∈ H, pF(Z)
kpF(Z)−Zk2= min
W∈FkW−Zk2.
Z F
F Z k·k2,
kpF(Z)−Zk2Z F.
Z
F.
BΩB ⊂ F
ZB− B∈ BR,
{Z∈ BR} ∈ B.
ZB B
Z L2(B)⊂L2(F)
L2(B)
L2(F).
L2(B), pL2(B)
L2(B).
X∈L2(F). X
B,
E[X|B] = pL2(B)(X).
B0B0={Ω,∅}.
L2(B0) =
E[X|B0] = E[X]. E[X]
L2X.
E[X]L2(B0) =
E[X|B]
E[X|B]B− X
F− E[X|B]
XB ⊂ F
X.