Completed notes of the course modulo 2

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u y u y y0 k k y = y0 J = E h y (t + k) y0 (t) 2i t u (t)= argmin (J) J y (t + k) y0 (t) 2 y (t + k) y (t + k)= f (u (t) ...u (t n)) + g (y (t + k 1) ...y (t n)) u (t) ,u (t 1) ... y (t + k)= y0 (t) u (t) u (t)= f (y (t + k 1) ...y (y n)) + g (u (t 1) ...u (t n)) C (z) y0 (t)= eF (z) y (t)+ B (z) E (z) u (t) k y (t + k)=ˆy (t + k)+ e (t + k)| {z } impredicibile ˆy (t + k)= y0 (t) ˆy (t + k | t)= eF (z) C (z) y (t)+ B (z) E (z) C (z) u (t) eF (z) E (z) u (t)= 1 B (z) E (z) h C (z) y0 (t) eF (z) y (t) i • B (z) • C(z) A(z) B (z) e (t) ⇠ WN (0, 1) y (t)= 1 2 y (t 1) + u (t 1) + 2u (t 2) + 2e (t) y (t)= 1 2z 1 1 1 2 z 1 z 1u (t)+ z 1 1 2 z 1 e (t) B (z) u (t) f.d.t.y0 ! u = 1 1 2 z 1 1+2 z 1 yG (t + k) yG (t + k)= P (z) y (t + k)+ u (t) Q (z) P (z) Q (z) A (z) y (t)= B (z) u (t k)+ C (z) e (t) ,e (t) ⇠ WN (0, 1) ˜y (t + k)= P (z) y (t + k) ˜y0 (t)= y0 (t) Q (z) u (t) J = E h ˜y (t + k) ey0 (t) 2i P (z)= PN (z) PD (z) Q (z)= QN (z) QD (z) ey (t)= P (z) y (t)= PN (z) PD (z) ✓ B (z) A (z) u (t k)+ C (z) A (z) e (t) ◆ ˜A (z)= PD (z) A (z) ˜B (z)= PZ (z) B (z) eC (z)= PN (z) C (z) eC (z)= eA (z) E (z)+ z kF (z) eC (z)˜y0 (t)= eF (z) ey (t)+ eB (z) E (z) u (t) PN (z) G (z) u (t)+ F (z) y (t) H (z) y0 (t)=0 G (z)= PD (z)[ B (z) QD (z) E (z)+ C (z) QN (z)] F (z)= eF (z) QD (z) H (z)= C (z) PD (z) QD (z) QN (z)=0 ,Q D (z)=1 U (t) J = E h P (z) y (t + k) y0 (t) 2i G (z)= PD (z) B (z) E (z) H (z)= C (z) PD (z) F (z)= ˜F (z) ( z)= B (z) C (z) PN (z) B (z) C (z) S (z)= z k P (z) Q (z) PN (z)= PD (z)=1 J = E h y (t + k)+ Q (z) u (t) y0 (t) 2i C (z)= A (z) E (z)+ z k ˜F (z)