principles of financial economics leroy and werne phần 1 pps

... { (1, 1) } and x = (1, 2). The orthogonal projection xZisxZ= (1, 2) · (1, 1) (1, 1) · (1, 1) (1, 1) =74 (1, 1) = (7/4, 7/4). (17 .19 )✷ 17 .6 Diagrammatic Methods in Hilbert SpacesOne of ... (h 1 , h2). Substituting 17 .39 in 17 .37 and 17 .38 we obtain23= 1 3h 1 + 1 3(h 1 + h2), (17 .40) and 23= 1 3(h 1 + h2) + 1 3h2. (17 . 41) The solution is h 1 = h2= 2/3 which giveske=23,43,23. ... KERNELS and  y 1 2=  xZ 1 − xZ22+  y22. (17 .15 )Eqs. 17 .14 and 17 .15 imply that xZ 1 − xZ22= 0 (17 .16 )so, by the strict positivity of inner products, xZ 1 = xZ2.✷If...

... con-straints 21. 9 and 21. 10 are written as equalities since utility functions are assumed to be increasing.Budget constraints 21. 9 and 21. 10 can be written asc0= w0− p0h0( 21. 11) and ct= ... x(ξt +1 ))∂ξt +1 u∂ξtu( 21. 15)with typical elementpj(ξt) =ξt +1 ⊂ ξt(pj(ξt +1 ) + xj(ξt +1 ))∂ξt +1 u∂ξtu. ( 21. 16)Eq. 21. 16 says that the price of security j in event ξtequals ... consumption-portfolio choice problem 21. 8 are∂ξtu −λ(ξt) = 0 , ∀ξtt = 0, . . . , T, ( 21. 13)λ(ξt)p(ξt) =ξt +1 ⊂ξt(p(ξt +1 ) + x(ξt +1 ))λ(ξt +1 ), ∀ξtt = 0, . . . , T − 1, ( 21. 14)where λ(ξt)...