let Prob be Probability of Special_SigmaField2 ; for r being Real st r > 0 holds
for jpi being pricefunction
for G being sequence of (set_of_random_variables_on (Special_SigmaField2,Borel_Sets)) st ( for d being Nat holds
( G . d = {1,2,3,4} --> ((jpi . d) * (1 + r)) & Change_Element_to_Func (G,d) is_integrable_on P2M Prob & Change_Element_to_Func (G,d) is_simple_func_in Special_SigmaField2 ) ) holds
( Risk_neutral_measure_exists_wrt G,jpi,r & ( for s being Nat holds jpi . s = expect ((Real_RV (r,(Change_Element_to_Func (G,s)))),Prob) ) )
let r be Real; ( r > 0 implies for jpi being pricefunction
for G being sequence of (set_of_random_variables_on (Special_SigmaField2,Borel_Sets)) st ( for d being Nat holds
( G . d = {1,2,3,4} --> ((jpi . d) * (1 + r)) & Change_Element_to_Func (G,d) is_integrable_on P2M Prob & Change_Element_to_Func (G,d) is_simple_func_in Special_SigmaField2 ) ) holds
( Risk_neutral_measure_exists_wrt G,jpi,r & ( for s being Nat holds jpi . s = expect ((Real_RV (r,(Change_Element_to_Func (G,s)))),Prob) ) ) )
assume A0:
r > 0
; for jpi being pricefunction
for G being sequence of (set_of_random_variables_on (Special_SigmaField2,Borel_Sets)) st ( for d being Nat holds
( G . d = {1,2,3,4} --> ((jpi . d) * (1 + r)) & Change_Element_to_Func (G,d) is_integrable_on P2M Prob & Change_Element_to_Func (G,d) is_simple_func_in Special_SigmaField2 ) ) holds
( Risk_neutral_measure_exists_wrt G,jpi,r & ( for s being Nat holds jpi . s = expect ((Real_RV (r,(Change_Element_to_Func (G,s)))),Prob) ) )
let jpi be pricefunction ; for G being sequence of (set_of_random_variables_on (Special_SigmaField2,Borel_Sets)) st ( for d being Nat holds
( G . d = {1,2,3,4} --> ((jpi . d) * (1 + r)) & Change_Element_to_Func (G,d) is_integrable_on P2M Prob & Change_Element_to_Func (G,d) is_simple_func_in Special_SigmaField2 ) ) holds
( Risk_neutral_measure_exists_wrt G,jpi,r & ( for s being Nat holds jpi . s = expect ((Real_RV (r,(Change_Element_to_Func (G,s)))),Prob) ) )
let G be sequence of (set_of_random_variables_on (Special_SigmaField2,Borel_Sets)); ( ( for d being Nat holds
( G . d = {1,2,3,4} --> ((jpi . d) * (1 + r)) & Change_Element_to_Func (G,d) is_integrable_on P2M Prob & Change_Element_to_Func (G,d) is_simple_func_in Special_SigmaField2 ) ) implies ( Risk_neutral_measure_exists_wrt G,jpi,r & ( for s being Nat holds jpi . s = expect ((Real_RV (r,(Change_Element_to_Func (G,s)))),Prob) ) ) )
assume A01:
for d being Nat holds
( G . d = {1,2,3,4} --> ((jpi . d) * (1 + r)) & Change_Element_to_Func (G,d) is_integrable_on P2M Prob & Change_Element_to_Func (G,d) is_simple_func_in Special_SigmaField2 )
; ( Risk_neutral_measure_exists_wrt G,jpi,r & ( for s being Nat holds jpi . s = expect ((Real_RV (r,(Change_Element_to_Func (G,s)))),Prob) ) )
for s being Nat holds jpi . s = expect ((Real_RV (r,(Change_Element_to_Func (G,s)))),Prob)
proof
let s be
Nat;
jpi . s = expect ((Real_RV (r,(Change_Element_to_Func (G,s)))),Prob)
set RV =
Change_Element_to_Func (
G,
s);
(
Change_Element_to_Func (
G,
s)
= {1,2,3,4} --> (In (((jpi . s) * (1 + r)),REAL)) &
Change_Element_to_Func (
G,
s)
is_integrable_on P2M Prob &
Change_Element_to_Func (
G,
s)
is_simple_func_in Special_SigmaField2 )
by A01;
hence
jpi . s = expect (
(Real_RV (r,(Change_Element_to_Func (G,s)))),
Prob)
by A0, ThArb;
verum
end;
hence
( Risk_neutral_measure_exists_wrt G,jpi,r & ( for s being Nat holds jpi . s = expect ((Real_RV (r,(Change_Element_to_Func (G,s)))),Prob) ) )
; verum