In order for (18)to hold for any Yi,it is sufficient that 1.u(Yi,t)=a+byt,where the N x 1 vector a and N x N matrix b are constants; 2.o(Y,t)=o,where o is a N x N constant matrix; 3.A(T),B(T),and C(r)satisfy the following ODEs: A=a+dB-Bad'B+Tcelo'Col, (19) B =B+B-2C'aa'B+2C'a, (20) C =+[6C+C'b]-2C'ao'C. (21)】 For suitable choices of (a,B,y;a,b;o),the Ricatti equations (19)-(21)admit a unique solution (A(T),B(T),C(T))under the initial conditions A(0)=0, B(0)=ONx1,and C(0)=ONxN.It is easy to verify that the solution has the property that A'(0)=a,B(0)=B,and C(0)=y. The canonical representation of the QG models is simpler than in the case of affine models,because shocks to Y are homoskedastic.To derive their canonical model,Ahn,Dittmar,and Gallant [2002]normalize the diagonal elements of y to unity,set B=0,have K (the mean reversion matrix for Y) being lower triangular,and have diagonal.They show that the QG models in Longstaff [1989],Constantinides [1992],and Lu [1999]are restricted special cases of their most flexible canonical model. 4 DTSMs with Jump Diffusions Suppose that rt =r(Yi,t)is a function of a jump-diffusion process Y with risk-neutral dynamics dY=μ(Y,t)dt+o(Y,t)dW+△YdZ, (22) where Zt is a Poisson counter with risk-neutral intensity At,and the jump size AYi is drawn from a risk-neutral distribution v(r)=v(x;Yi,t). No arbitrage implies that the zero-coupon bond price D(t,T)satisfies [品+A]p)-r心pen=0 (23) 16In order for (18) to hold for any Yt, it is sufficient that 1. µ(Yt, t) = a + bYt, where the N × 1 vector a and N × N matrix b are constants; 2. σ(Yt, t) = σ, where σ is a N × N constant matrix; 3. A(τ ), B(τ ), and C(τ ) satisfy the following ODEs: A˙ = α + a0 B − 1 2 B0 σσ0 B + Trace [σ0 Cσ] , (19) B˙ = β + b 0 B − 2C0 σσ0 B + 2C0 a, (20) C˙ = γ + [b 0 C + C0 b] − 2C0 σσ0 C. (21) For suitable choices of (α, β, γ; a, b; σ), the Ricatti equations (19)–(21) admit a unique solution (A(τ ), B(τ ), C(τ )) under the initial conditions A(0) = 0, B(0) = 0N×1, and C(0) = 0N×N . It is easy to verify that the solution has the property that A0 (0) = α, B0 (0) = β, and C0 (0) = γ. The canonical representation of the QG models is simpler than in the case of affine models, because shocks to Y are homoskedastic. To derive their canonical model, Ahn, Dittmar, and Gallant [2002] normalize the diagonal elements of γ to unity, set β = 0, have K (the mean reversion matrix for Y ) being lower triangular, and have Σ diagonal. They show that the QG models in Longstaff [1989], Constantinides [1992], and Lu [1999] are restricted special cases of their most flexible canonical model. 4 DTSMs with Jump Diffusions Suppose that rt = r(Yt, t) is a function of a jump-diffusion process Y with risk-neutral dynamics dYt = µ(Yt, t) dt + σ(Yt, t) dWQ t + ∆Yt dZt, (22) where Zt is a Poisson counter with risk-neutral intensity λt, and the jump size ∆Yt is drawn from a risk-neutral distribution νt(x) ≡ ν(x; Yt, t). No arbitrage implies that the zero-coupon bond price D(t, T) satisfies  ∂ ∂t + A  D(t, T) − r(Y, t) D(t, T)=0, (23) 16