let Omega be non empty set ; for F being SigmaField of Omega
for G being sequence of (set_of_random_variables_on (F,Borel_Sets))
for phi being Real_Sequence
for d being Nat holds RVPortfolioValueFutExt (phi,F,G,d) is random_variable of F, Borel_Sets
let F be SigmaField of Omega; for G being sequence of (set_of_random_variables_on (F,Borel_Sets))
for phi being Real_Sequence
for d being Nat holds RVPortfolioValueFutExt (phi,F,G,d) is random_variable of F, Borel_Sets
let G be sequence of (set_of_random_variables_on (F,Borel_Sets)); for phi being Real_Sequence
for d being Nat holds RVPortfolioValueFutExt (phi,F,G,d) is random_variable of F, Borel_Sets
let phi be Real_Sequence; for d being Nat holds RVPortfolioValueFutExt (phi,F,G,d) is random_variable of F, Borel_Sets
let d be Nat; RVPortfolioValueFutExt (phi,F,G,d) is random_variable of F, Borel_Sets
defpred S1[ Nat] means RVPortfolioValueFutExt (phi,F,G,$1) is random_variable of F, Borel_Sets ;
ElementsOfPortfolioValueProb_fut (F,(G . 0)) is random_variable of F, Borel_Sets
by FINANCE2:28;
then A1:
(phi . 0) (#) (ElementsOfPortfolioValueProb_fut (F,(G . 0))) is random_variable of F, Borel_Sets
by FINANCE2:26;
RVPortfolioValueFutExt (phi,F,G,0) is random_variable of F, Borel_Sets
proof
for
w being
Element of
Omega holds
(RVPortfolioValueFutExt (phi,F,G,0)) . w = ((phi . 0) (#) (ElementsOfPortfolioValueProb_fut (F,(G . 0)))) . w
proof
let w be
Element of
Omega;
(RVPortfolioValueFutExt (phi,F,G,0)) . w = ((phi . 0) (#) (ElementsOfPortfolioValueProb_fut (F,(G . 0)))) . w
(RVPortfolioValueFutExt (phi,F,G,0)) . w = PortfolioValueFutExt (
0,
phi,
F,
G,
w)
by Def2;
then (RVPortfolioValueFutExt (phi,F,G,0)) . w =
(RVPortfolioValueFutExt_El (phi,F,G,w)) . 0
by SERIES_1:def 1
.=
(RVElementsOfPortfolioValue_fut (phi,F,G,0)) . w
by FINANCE2:def 6
.=
(phi . 0) * ((ElementsOfPortfolioValueProb_fut (F,(G . 0))) . w)
by FINANCE2:def 5
;
hence
(RVPortfolioValueFutExt (phi,F,G,0)) . w = ((phi . 0) (#) (ElementsOfPortfolioValueProb_fut (F,(G . 0)))) . w
by VALUED_1:6;
verum
end;
hence
RVPortfolioValueFutExt (
phi,
F,
G,
0) is
random_variable of
F,
Borel_Sets
by A1, FUNCT_2:63;
verum
end;
then J0:
S1[ 0 ]
;
J1:
for n being Nat st S1[n] holds
S1[n + 1]
proof
let n be
Nat;
( S1[n] implies S1[n + 1] )
assume B1:
S1[
n]
;
S1[n + 1]
C0:
for
w being
Element of
Omega holds
(RVPortfolioValueFutExt (phi,F,G,(n + 1))) . w = ((RVPortfolioValueFutExt (phi,F,G,n)) . w) + ((RVElementsOfPortfolioValue_fut (phi,F,G,(n + 1))) . w)
proof
let w be
Element of
Omega;
(RVPortfolioValueFutExt (phi,F,G,(n + 1))) . w = ((RVPortfolioValueFutExt (phi,F,G,n)) . w) + ((RVElementsOfPortfolioValue_fut (phi,F,G,(n + 1))) . w)
(RVPortfolioValueFutExt (phi,F,G,(n + 1))) . w = PortfolioValueFutExt (
(n + 1),
phi,
F,
G,
w)
by Def2;
then D1:
(RVPortfolioValueFutExt (phi,F,G,(n + 1))) . w = ((Partial_Sums (RVPortfolioValueFutExt_El (phi,F,G,w))) . n) + ((RVPortfolioValueFutExt_El (phi,F,G,w)) . (n + 1))
by SERIES_1:def 1;
(RVPortfolioValueFutExt (phi,F,G,n)) . w = PortfolioValueFutExt (
n,
phi,
F,
G,
w)
by Def2;
hence
(RVPortfolioValueFutExt (phi,F,G,(n + 1))) . w = ((RVPortfolioValueFutExt (phi,F,G,n)) . w) + ((RVElementsOfPortfolioValue_fut (phi,F,G,(n + 1))) . w)
by FINANCE2:def 6, D1;
verum
end;
K2:
RVElementsOfPortfolioValue_fut (
phi,
F,
G,
(n + 1)) is
random_variable of
F,
Borel_Sets
by FINANCE2:30;
C2:
dom (RVPortfolioValueFutExt (phi,F,G,n)) = Omega
by FUNCT_2:def 1;
dom (RVElementsOfPortfolioValue_fut (phi,F,G,(n + 1))) = Omega
by FUNCT_2:def 1;
then
(
dom (RVPortfolioValueFutExt (phi,F,G,(n + 1))) = (dom (RVPortfolioValueFutExt (phi,F,G,n))) /\ (dom (RVElementsOfPortfolioValue_fut (phi,F,G,(n + 1)))) & ( for
c being
object st
c in dom (RVPortfolioValueFutExt (phi,F,G,(n + 1))) holds
(RVPortfolioValueFutExt (phi,F,G,(n + 1))) . c = ((RVPortfolioValueFutExt (phi,F,G,n)) . c) + ((RVElementsOfPortfolioValue_fut (phi,F,G,(n + 1))) . c) ) )
by C0, FUNCT_2:def 1, C2;
then
RVPortfolioValueFutExt (
phi,
F,
G,
(n + 1))
= (RVPortfolioValueFutExt (phi,F,G,n)) + (RVElementsOfPortfolioValue_fut (phi,F,G,(n + 1)))
by VALUED_1:def 1;
hence
S1[
n + 1]
by B1, K2, FINANCE2:23;
verum
end;
for n being Nat holds S1[n]
from NAT_1:sch 2(J0, J1);
hence
RVPortfolioValueFutExt (phi,F,G,d) is random_variable of F, Borel_Sets
; verum