where r=T-t,A(T)=4()=-4(T-,and B(T)is similarly defined. Or Duffie and Kan [1996 show that,in order for (15)to hold for any Yi,it is sufficient that6 1.u(Yi,t)be affine in Yi:u(Yi,t)=a+by:,where a be a N x 1 vector and b be a N x N matrix. 2.a(Y,t)a(Y,t)be affine in :(t)a(,t)=ho+,where ho and h,j=1,2,...,N,are N x N matrices. 3.A(T)and B(T)satisfy the following ODEs: Aa+d'B(T)-jB(T)'hoB(T), (16) B=B+6B(r)-2, (17) where vi(T)=B(T)'hB(T). For suitable choices of (a,B;a,b;ho,h:1 <j<N),the Ricatti equations (16)-(17)admit a unique solution (A(T),B(T))under the initial conditions A(0)=0 and B(0)=ONx1.It is easy to verify that the solution has the property A'(0)=a and B'(0)=B. Dai and Singleton 2000 examine multi-factor affine models with the following structure: Tt 80+6Yi, dYi K (0-Y:)dt+vS:dw, where St is a diagonal matrix with [Stli=+Bi.Letting B be the Nx N matrix with ith column given by B,they construct admissible affine models- models that give unique,well-defined solutions for D(t,T)-by restricting the parameter vector(o,d,K,Θ，∑，a,B).Specifically,for given m=rank(B), they introduce the canonical model with the structure BB mxm 0mx(N-m） DB (N-m)×m K@-m)x(N-m)] 6A more general mathematical characterization of affine models is presented in Duffie, Filipovic,and Schachermayer [2001].See Gourieroux,Monfort,and Polimenis [2002]for a formal development of multi-factor affine models in discrete time. 14where τ ≡ T − t, A˙(τ ) = ∂A(τ) ∂τ = −∂A(T −t) ∂t , and B˙(τ ) is similarly defined. Duffie and Kan [1996] show that, in order for (15) to hold for any Yt, it is sufficient that6 1. µ(Yt, t) be affine in Yt: µ(Yt, t) = a + bYt, where a be a N × 1 vector and b be a N × N matrix. 2. σ(Yt, t)σ(Yt, t)0 be affine in Yt: σ(Yt, t)σ(Yt, t)0 = h0+PN j=1 hj 1Y j t , where h0 and hj 1, j = 1, 2,... ,N, are N × N matrices. 3. A(τ ) and B(τ ) satisfy the following ODEs: A˙ = α + a0 B(τ ) − 1 2 B(τ ) 0 h0B(τ ), (16) B˙ = β + b0 B(τ ) − 1 2 v(τ ), (17) where vj (τ ) ≡ B(τ )0 hj 1B(τ ). For suitable choices of (α, β; a, b; h0, hj 1 : 1 ≤ j ≤ N), the Ricatti equations (16)–(17) admit a unique solution (A(τ ), B(τ )) under the initial conditions A(0) = 0 and B(0) = 0N×1. It is easy to verify that the solution has the property A0 (0) = α and B0 (0) = β. Dai and Singleton [2000] examine multi-factor affine models with the following structure: rt = δ0 + δ0 Yt, dYt = K (Θ − Yt) dt + ΣpSt dWQ t , where St is a diagonal matrix with [St]ii = αi+Y 0 t βi. Letting B be the N ×N matrix with i th column given by βi, they construct admissible affine models– models that give unique, well-defined solutions for D(t, T)– by restricting the parameter vector (δ0, δ, K, Θ, Σ, α, B). Specifically, for given m = rank(B), they introduce the canonical model with the structure K =  KBB m×m 0m×(N−m) KDB (N−m)×m KDD (N−m)×(N−m)  , 6A more general mathematical characterization of affine models is presented in Duffie, Filipovic, and Schachermayer [2001]. See Gourieroux, Monfort, and Polimenis [2002] for a formal development of multi-factor affine models in discrete time. 14