Mar 11, 2026
x = (1, v, s), X(\omega) = (R, (\omega - K)^+, \omega), \omega\in\Omega.
\min_{p,\lambda} \frac{1}{2}(1 - \sum_i p_i^2) - \lambda\cdot(x - \sum_i X(\omega_i) p_i
d/dp_j, 0 = -p_j + \lambda\cdot X(\omega_j).
p_j = \lambda\cdot X(\omega_j)
x = \sum_i X(\omega_i))\lambda\cdot X(\omega_j) = (\sum_i X(\omega_i)X(\omega_i)^T)\lambda
\lambda = (\sum_i X(\omega_i)X(\omega_i)^T)^{-1}x.
x = (1, v, s), X(\omega) = (R, (\omega - K)^+, \omega), \omega\in\{L,K,H\}.
p_j = (\sum_i X(\omega_i)X(\omega_i)^T)^{-1}x \cdot X(\omega_j) = (\sum_i X(\omega_i)X(\omega_i)^T)^{-1}X(\omega_j)^T x.
X(L) = (R, 0, L), X(K) = (R, 0, K), X(H) = (R, H-K, H)
\begin{bmatrix} R^2 & 0 & RL \\ 0 & 0 & 0 \\ LR & 0 & L^2 \\ \end{bmatrix} + \begin{bmatrix} R^2 & 0 & RK \\ 0 & 0 & 0 \\ KR & 0 & K^2 \\ \end{bmatrix} + \begin{bmatrix} R^2 & R(H-K) & RH \\ R(H-K) & (H-K)^2 & (H-K)H \\ HR & (H-K)H & H^2 \\ \end{bmatrix}
\begin{bmatrix} 3R^2 & R(H-K) & R(L + K + H) \\ R(H-K) & (H-K)^2 & (H-K)H \\ R(L+K+H) & (H-K)H & L^2 + K^2 + H^2 \\ \end{bmatrix}
x = (1, v_1,\ldots,v_n, s), X(\omega) = (R, (\omega - K_1)^+,\ldots,(\omega - K_n)^+, \omega), \omega\in\Omega.
X(L) = (R, 0,\ldots, 0, L)
X(K_1) = (R, 0,\ldots, 0, K_1)
X(K_2) = (R, K_2 - K_1,\ldots, 0, K_2)
\dots
X(H) = (R, H - K_1,\ldots, 0, H)
x = X(L) D_L + X(K_1) D_{K1} \cdots X(H) D_H
x = X(L) D_L + X_1 D_1 \cdots X(H) D_H
\begin{bmatrix} 1 \\ v_1 \\ v_2 \\ v_3 \\ \vdots \\ v_n \\ s \\ \end{bmatrix} = \begin{bmatrix} R & R & R & R &\cdots & R & R \\ 0 & 0 & K_2 - K_1 & K_2 - K_1 &\cdots & K_n - K_1 & H - K_1 \\ 0 & 0 & 0 & K_3 - K_2 &\cdots & K_n - K_2 & H - K_2 \\ 0 & 0 & 0 & 0 &\cdots & K_n - K_3 & H - K_3 \\ \vdots & \vdots &\vdots &\vdots &\ddots &\vdots &\vdots \\ 0 & 0 & 0 & 0 &\cdots & K_n - K_{n-1} & H - K_{n-1} \\ 0 & 0 & 0 & 0 &\cdots & 0 & H - K_n \\ L & K_1 & K_2 & K_3 &\cdots & K_n & H \\ \end{bmatrix} \begin{bmatrix} D_L \\ D_1 \\ D_2 \\ D_3 \\ \vdots \\ D_n \\ D_H \\ \end{bmatrix}
\begin{bmatrix} 1 \\ s \\ v_1 \\ v_2 \\ v_3 \\ \vdots \\ v_n \\ \end{bmatrix} = \begin{bmatrix} R & R & R & R &\cdots & R & R \\ L & K_1 & K_2 & K_3 &\cdots & K_n & H \\ 0 & 0 & K_2 - K_1 & K_2 - K_1 &\cdots & K_n - K_1 & H - K_1 \\ 0 & 0 & 0 & K_3 - K_2 &\cdots & K_n - K_2 & H - K_2 \\ 0 & 0 & 0 & 0 &\cdots & K_n - K_3 & H - K_3 \\ \vdots & \vdots &\vdots &\vdots &\ddots &\vdots &\vdots \\ 0 & 0 & 0 & 0 &\cdots & K_n - K_{n-1} & H - K_{n-1} \\ 0 & 0 & 0 & 0 &\cdots & 0 & H - K_n \\ \end{bmatrix} \begin{bmatrix} D_L \\ D_H \\ D_1 \\ D_2 \\ D_3 \\ \vdots \\ D_n \\ \end{bmatrix}
Let K_0 = 0 and K_{n+1} = H.
\begin{bmatrix} 1 \\ s \\ v_1 \\ v_2 \\ v_3 \\ \vdots \\ v_n \\ \end{bmatrix} = \begin{bmatrix} R & R & R & R &\cdots & R & R \\ L & K_1 - K_0 & K_2 - K_0 & K_3 - K_0 &\cdots & K_n - K_0 & K_{n+1} - K_0 \\ 0 & 0 & K_2 - K_1 & K_2 - K_1 &\cdots & K_n - K_1 & K_{n+1} - K_1 \\ 0 & 0 & 0 & K_3 - K_2 &\cdots & K_n - K_2 & K_{n+1} - K_2 \\ 0 & 0 & 0 & 0 &\cdots & K_n - K_3 & K_{n+1} - K_3 \\ \vdots & \vdots &\vdots &\vdots &\ddots &\vdots &\vdots \\ 0 & 0 & 0 & 0 &\cdots & K_n - K_{n-1} & K_{n+1} - K_{n-1} \\ 0 & 0 & 0 & 0 &\cdots & 0 & K_{n+1} - K_n \\ \end{bmatrix} \begin{bmatrix} D_L \\ D_H \\ D_1 \\ D_2 \\ D_3 \\ \vdots \\ D_n \\ \end{bmatrix}
Let J = 1[i + 1 = j] so J^k = 1[i + k = j].
Let \Delta = I - J.
\Delta^{-1} = (I - J)^{-1} = \sum_{k\ge 0} J^k.
\Delta D = \operatorname{diag}(K_1, \ldots, K_n)\Delta^{-1}
u_{ij} = K_j - K_{i-1}
D = [(K_j - K_{i-1})1(i\le j)], V = D^{-1} = [v_{ij}1(i\le j)]
v_{ii} = 1/u_{ii}
v_{ij} = (-1/u_{ii}) \sum_{k = i + 1}^j u_{ik} v_{kj}
v_{i,i} = 1/(K_i - K_{i-1})
v_{i,i+1} = -1/(K_{i} - K_{i-1}) + 1/(K_{i+1} - K_{i})
v_{i,i+2} = 1/(K_{i+1} - K_i).
v_{i,j} = 0, j > 2.
A = [R], B = [R\,R\,R\cdots R\,R], C = [L\,0\,0\,0 \cdots 0\,0]^T and D = [(K_j - K_i)1(j > i)], 0 \le i < j \le n + 1.
If E = (D - CA^{-1}B)^{-1} is the Schur complement then \begin{bmatrix} A & B \\ C & D \\ \end{bmatrix}^{-1} = \begin{bmatrix} A^{-1} + A^{-1}BECA^{-1} & -A^{-1}BE \\ -ECA^{-1} & E \\ \end{bmatrix} If E = (A - BD^{-1}C)^{-1} is the Schur complement then \begin{bmatrix} A & B \\ C & D \\ \end{bmatrix}^{-1} = \begin{bmatrix} E & - EBD^{-1} \\ -D^{-1}CE & D^{-1} + D^{-1}CEBD^{-1} \\ \end{bmatrix}
\begin{bmatrix} K_1 & K_2 & \cdots & K_{n + 1} \\ 0 & K_2 & \cdots & K_{n + 1} \\ \vdots & \vdots &\ddots & \vdots \\ 0 & 0 &\cdots & K_{n+1} \\ \end{bmatrix} = \begin{bmatrix} 1 & 1 & \cdots & 1 \\ 0 & 1 & \cdots & 1 \\ \vdots & \vdots &\ddots & \vdots \\ 0 & 0 &\cdots & 1\\ \end{bmatrix} \begin{bmatrix} K_1 & 0 & \cdots & 0 \\ 0 & K_2 & \cdots & 0 \\ \vdots & \vdots &\ddots & \vdots \\ 0 & 0 &\cdots & K_{n+1} \\ \end{bmatrix} \begin{bmatrix} 1 & 1 & \cdots & 1 \\ 0 & 1 & \cdots & 1 \\ \vdots & \vdots &\ddots & \vdots \\ 0 & 0 &\cdots & 1\\ \end{bmatrix}^{-1} = \begin{bmatrix} 1 & -1 & 0 & \cdots & 0 \\ 0 & 1 & -1 & \cdots & 0 \\ \vdots & \vdots &\ddots & \vdots \\ 0 & 0 &0 &\cdots & 1\\ \end{bmatrix}^{-1} \begin{bmatrix} K_0 & K_0 & \cdots & K_0 \\ 0 & K_1 & \cdots & K_1 \\ \vdots & \vdots &\ddots & \vdots \\ 0 & 0 &\cdots & K_n \\ \end{bmatrix} = \begin{bmatrix} K_0 & 0 & \cdots & 0 \\ 0 & K_1 & \cdots & 0 \\ \vdots & \vdots &\ddots & \vdots \\ 0 & 0 &\cdots & K_n \\ \end{bmatrix} \begin{bmatrix} 1 & 1 & \cdots & 1 \\ 0 & 1 & \cdots & 1 \\ \vdots & \vdots &\ddots & \vdots \\ 0 & 0 &\cdots & 1\\ \end{bmatrix}
(A - B)^{-1} = A^{-1}\sum_{k=0}^\infty (A^{-1}B)^k
Let U = [1(i\le j)] = 1[\le] so U^{-1} = I - J, J = [1(i + 1 = j)].
A = U \operatorname{diag}(K_1,\ldots,K_{n+1}), B = \operatorname{diag}(K_0,\ldots,K_n)U
U^{-1} = I - J.
A^{-1} B = K_1^{-1}(I - J)K_0 U
(A^{-1} B)^2 = K_1^{-1}(I - J)K_0 U K_1^{-1}(I - J)K_0 U