Rational Expectations Models with Multiplicative Noise

Abstract

This paper is concerned with the multiplicative-noise rational expectations (MRE) problem. By resorting to a linear quadratic optimal control (LQOC) problem, the approximate solution to the MRE problem can be obtained. It is worth highlighting that this approximate solution can be highly close to the exact solution by adjusting weighted matrices in the cost function of the LQOC problem. Since the conditional expectation of the state is involved in the cost function, the LQOC problem is an optimization control with an additional measurability restriction, which is very involved. The orthogonal decomposition technique is used to deal with this measurability restriction. The solvability condition and the optimal decision are given by developing generalized Riccati-type equations. Numerical experiments are provided to illustrate the effectiveness of the achieved results.

Appendices

Proof of Lemma 2.1

It follows from the definition in Notations that we have

$$\begin{aligned} {\hat{u}}_{k|k-2}&={\mathbb {E}}\{u_k|{\mathcal {F}}_{k-2}\}, \end{aligned}$$

(74)

$$\begin{aligned} {\hat{u}}_{k}|^{k-1}_{k-2}&=u_k-{\mathbb {E}}\{u_k|{\mathcal {F}}_{k-2}\}, \end{aligned}$$

(75)

where ${\mathcal {F}}_{k-2}=\sigma \{\omega _0,\omega _1,\ldots ,\omega _{k-2},\nu _0,\nu _1,\ldots ,\nu _{k-2}\}$. Theorem 4.1.14 in [18] implies that for every ${\mathcal {F}}_{k-2}$-measurable W, there holds

$$\begin{aligned} {\mathbb {E}}\{W'u_k|{\mathcal {F}}_{k-2}\}=W'{\mathbb {E}}\{u_k|{\mathcal {F}}_{k-2}\}. \end{aligned}$$

(76)

Taking expectation gives

$$\begin{aligned} {\mathbb {E}}\{W'(u_k-{\mathbb {E}}\{u_k|{\mathcal {F}}_{k-2}\})\}=0. \end{aligned}$$

(77)

Associating (75) with (77), we know that ${\hat{u}}_{k}|^{k-1}_{k-2}$ is orthogonal to any ${\mathcal {F}}_{k-2}$-measurable vector. It is evident ${\hat{u}}_{k|k-2}$ is ${\mathcal {F}}_{k-2}$-measurable, so ${\hat{u}}_{k|k-2}$ and ${\hat{u}}_{k}|^{k-1}_{k-2}$ are orthogonal.

Proof of Theorem 2.1

Considering the truncated cost function (13) for any $k,0\le k\le K$, we have

$$\begin{aligned} J_k&=\sum ^K_{j=k}{\mathbb {E}}\{(u_j-{\hat{x}}_{j+2|j-1})'Q(u_j-{\hat{x}}_{j+2|j-1})\nonumber \\&\quad +({\hat{x}}_{j+2|j-1}-x_{j+2})'R({\hat{x}}_{j+2|j-1}-x_{j+2})\}. \end{aligned}$$

(78)

Consistent with (78), we assume that the prediction process by the economic agents ends at $k=K$. Therefore, we set $u_k=0$ for $k>K$. From the system (11), (78) can be written as

$$\begin{aligned} J_k&={\mathbb {E}}\Big \{\sum ^K_{j=k}u_j'Qu_j+\sum ^{K-1}_{j=k}[-2u_j'Q{\hat{x}}_{j+2|j-1}+x_{j+2}'Rx_{j+2}\nonumber \\&\quad +{{\hat{x}}_{j+2|j-1}}'(Q-R){\hat{x}}_{j+2|j-1}]\nonumber \\&\quad -2u_K'QA{\hat{x}}_{K+1|K-1}+{{\hat{x}}_{K+1|K-1}}'A'(Q-R)A{\hat{x}}_{K+1|K-1}\nonumber \\&\quad +x_{K+1}'(A'RA+\tau A_0'RA_0)x_{K+1}+\eta tr(R)\Big \}. \end{aligned}$$

(79)

Now, we handle the optimization problem by dynamic programming procedure. In the case of $k=K-1$, we get

$$\begin{aligned} \min _{u_K}J_{K-1}&=\min _{u_K}{\mathbb {E}}\{u_{K-1}'Qu_{K-1}+u_K'Qu_K-2u_{K-1}'Q{\hat{x}}_{K+1|K-2}+{{\hat{x}}_{K+1|K-2}}'(Q\nonumber \\&\quad -R){\hat{x}}_{K+1|K-2}-2u_K'QA{\hat{x}}_{K+1|K-1}+{{\hat{x}}_{K+1|K-1}}'A'(Q-R)A{\hat{x}}_{K+1|K-1}\nonumber \\&\quad +x_{K+1}'(R+A'RA+\tau A_0'RA_0)x_{K+1}+\eta tr(R)\}, \end{aligned}$$

(80)

in virtue of (11) and (80), one has

$$\begin{aligned} \min _{u_K}J_{K-1}&=\min _{u_K}{\mathbb {E}}\{{{\hat{u}}_{K|K-2}}'B'(Q-R)B{\hat{u}}_{K|K-2}-2{{\hat{u}}_{K|K-2}}'B'[(Q-R)A{\hat{x}}_{K|K-2}\nonumber \\&\quad +Qu_{K-1}]+{{\hat{x}}_{K|K-2}}'A'(Q-R)A{\hat{x}}_{K|K-2}+u_K'Xu_K+2u_K'Yx_K\nonumber \\&\quad +x_K'Zx_K+u_{K-1}'Qu_{K-1}-2u_{K-1}'QA{\hat{x}}_{K|K-2}\nonumber \\&\quad +\eta tr(R+A'RA+\tau A_0'RA_0)+\eta tr(R)\}, \end{aligned}$$

(81)

where

$$\begin{aligned} X&=Q+B'(R+A'QA+\tau A_0'RA_0)B\\&\quad +\tau B_0'(R+A'RA+\tau A_0'RA_0)B_0-QAB-B'A'Q,\\ Y&=B'(R+A'QA+\tau A_0'RA_0)A+\tau B_0'(R+A'RA+\tau A_0'RA_0)A_0-QAA,\\ Z&=A'(R+A'QA+\tau A_0'RA_0)A+\tau A_0'(R+A'RA+\tau A_0'RA_0)A_0. \end{aligned}$$

It can be noticed that (81) contains the ${\mathcal {F}}_{K-1}$-measurable $u_{K}$ and the ${\mathcal {F}}_{K-2}$-measurable $u_{K|K-2}$ simultaneously and they are correlated, which explicitly shows that the converted optimization problem $\min J_{K-1}$ is an optimization problem with extra measurability constraint. To eliminate the constraint, we decompose the control input orthogonally. The new inputs are no longer correlated, which guarantees dynamic programming procedures can proceed smoothly.

According to Lemma 2.1, $u_K$ and $x_K$ in (81) can be orthogonally decomposed as

$$\begin{aligned} u_K&={\hat{u}}_{K|K-2}+{\hat{u}}_{K}|^{K-1}_{K-2}, \end{aligned}$$

(82)

$$\begin{aligned} x_K&={\hat{x}}_{K|K-2}+{\hat{x}}_{K}|^{K-1}_{K-2}. \end{aligned}$$

(83)

Inserting (82)-(83) into (81) yields

$$\begin{aligned} J_{K-1}&={\mathbb {E}}\{{{\hat{u}}_{K|K-2}}'\varUpsilon ^{2}_K{\hat{u}}_{K|K-2}+2{{\hat{u}}_{K|K-2}}'(\varPsi ^{2}_K{\hat{x}}_{K|K-2}-B'Qu_{K-1})\nonumber \\&\quad +{{\hat{x}}_{K|K-2}}'[A'(Q+M^2_{K+1})A+\tau A_0'(R+M^1_{K+1})A_0]{\hat{x}}_{K|K-2}\nonumber \\&\quad +{\hat{u}}'_{K}|^{K-1}_{K-2}\varUpsilon ^1_K{\hat{u}}_{K}|^{K-1}_{K-2}+2{\hat{u}}'_{K}|^{K-1}_{K-2}\varPsi ^1_K{\hat{x}}_{K}|^{K-1}_{K-2}\nonumber \\&\quad +{\hat{x}}'_{K}|^{K-1}_{K-2}[A'(R+M^{2}_{K+1})A+\tau A_0'(R+M^1_{K+1})A_0]{\hat{x}}_{K}|^{K-1}_{K-2}\nonumber \\&\quad -2u_{K-1}'S_{K+1}{\hat{x}}_{K|K-2}+u_{K-1}'T_{K+1}u_{K-1}\nonumber \\&\quad +\eta tr(R+M^1_{K+1})+\eta tr(R)\}, \end{aligned}$$

(84)

where we have used the equalities (17)-(18) with $k=K$ and the initial value $M^1_{K+1}=A'RA+\tau A_0'RA_0$, $M^2_{K+1}=A'QA+\tau A_0'RA_0$, $S_{K+1}=QA$, and $T_{K+1}=Q$.

Making complete square over ${\hat{u}}_{K|K-2}$ and ${\hat{u}}_{K}|^{K-1}_{K-2}$ in (84) results in

$$\begin{aligned} J_{K-1}&={\mathbb {E}}\{({\hat{u}}_{K|K-2}-\bar{{\hat{u}}}_{K|K-2})'\varUpsilon ^2_{K}({\hat{u}}_{K|K-2}-\bar{{\hat{u}}}_{K|K-2})\nonumber \\&\quad +({\hat{u}}_{K}|^{K-1}_{K-2}-\bar{{\hat{u}}}_{K}|^{K-1}_{K-2})'\varUpsilon ^1_{K}({\hat{u}}_{K}|^{K-1}_{K-2}-\bar{{\hat{u}}}_{K}|^{K-1}_{K-2})\nonumber \\&\quad +{{\hat{x}}_{K|K-2}}'M^2_{K}{\hat{x}}_{K|K-2}+{\hat{x}}'_{K}|^{K-1}_{K-2}M^1_{K}{\hat{x}}_{K}|^{K-1}_{K-2}-2u_{K-1}'S_K{\hat{x}}_{K|K-2}\nonumber \\&\quad +u_{K-1}'T_Ku_{K-1}+\eta tr(R+M^1_{K+1})+\eta tr(R)\}, \end{aligned}$$

(85)

where

$$\begin{aligned} \bar{{\hat{u}}}_{K|K-2}&=-L^2_{K}{\hat{x}}_{K|K-2}-K_{K}u_{K-1}, \end{aligned}$$

(86)

$$\begin{aligned} \bar{{\hat{u}}}_{K}|^{K-1}_{K-2}&=-L^1_{K}{\hat{x}}_{K}|^{K-1}_{K-2}. \end{aligned}$$

(87)

In the above, (15)-(20), and (23)-(24) are utilized with $k=K$.

From (86)–(87) and the system (11), the optimal $u_K$ admits (46) with $k=K$. The optimal cost of (85) is given as

$$\begin{aligned} \min _{u_K}J_{K-1}&={\mathbb {E}}\{u_{K-1}'T_Ku_{K-1}-2u_{K-1}'S_K{\hat{x}}_{K|K-2}+{{\hat{x}}_{K|K-2}}'M^2_{K}{\hat{x}}_{K|K-2}\nonumber \\&\quad +{\hat{x}}'_{K}|^{K-1}_{K-2}M^1_{K}{\hat{x}}_{K}|^{K-1}_{K-2}+\eta tr(R+M^1_{K+1})+\eta tr(R)\}. \end{aligned}$$

(88)

In the case of $k=K-2$, we have

$$\begin{aligned} \min _{u_{K-1},u_K}J_{K-2}&=\min _{u_{K-1}}{\mathbb {E}}\Big \{u_{K-2}'Qu_{K-2}-2u'_{K-2}Q{\hat{x}}_{K|K-3}+{{\hat{x}}_{K|K-3}}'Q{\hat{x}}_{K|K-3}\nonumber \\&\quad +\sum ^2_{n=1}{\hat{x}}'_K|^{K-n}_{K-n-1}R{\hat{x}}_K|^{K-n}_{K-n-1}+\min _{u_{K}}J_{K-1}\Big \}. \end{aligned}$$

(89)

It together with (88) means that one only needs to resolve

$$\begin{aligned}&\min _{u_{K-1}}{\mathbb {E}}\Big \{u_{K-2}'Qu_{K-2}+u_{K-1}'T_Ku_{K-1}-2u'_{K-2}Q{\hat{x}}_{K|K-3}-2u_{K-1}'S_K{\hat{x}}_{K|K-2}\nonumber \\&\quad +{\hat{x}}_{K|K-3}'(Q+M^2_{K}){\hat{x}}_{K|K-3}+\sum ^{2}_{n=1}{\hat{x}}'_{K}|^{K-n}_{K-n-1}(R+M^n_{K}){\hat{x}}_{K}|^{K-n}_{K-n-1}\nonumber \\&\quad +\eta tr(R+M^1_{K+1})+\eta tr(R)\Big \}. \end{aligned}$$

(90)

Because the right hand of (81) can be written as

$$\begin{aligned}&\min _{u_K}{\mathbb {E}}\Big \{u_{K-1}'Qu_{K-1}+u_K'T_{K+1}u_K-2u'_{K-1}Q{\hat{x}}_{K+1|K-2}\nonumber \\&\quad -2u_K'S_{K+1}A{\hat{x}}_{K+1|K-1}+{\hat{x}}_{K+1|K-2}'(Q+M^2_{K+1}){\hat{x}}_{K+1|K-2}\nonumber \\&\quad +\sum ^{2}_{n=1}{\hat{x}}'_{K+1}|^{K+1-n}_{K-n}(R+M^n_{K+1}){\hat{x}}_{K}|^{K+1-n}_{K-n}+\eta tr(R)\Big \}, \end{aligned}$$

(91)

which has the same form of (90), thus we complete the induction argument. The optimal cost of (13) is as in (25).

Proof of Lemma 3.1

It follows from the definition in Notations that for $i=1,\ldots ,d$ and $j=1,\ldots ,d-1$, we have

$$\begin{aligned} {\hat{u}}_{k|k-i}&={\mathbb {E}}\{u_k|{\mathcal {F}}_{k-i}\}, \end{aligned}$$

(92)

$$\begin{aligned} {\hat{u}}_{k}|^{k-j}_{k-1-j}&={\mathbb {E}}\{u_k|{\mathcal {F}}_{k-j}\}-{\mathbb {E}}\{u_k|{\mathcal {F}}_{k-j-1}\}. \end{aligned}$$

(93)

From Theorem 4.1.13 in [18], (93) can be rewritten as

$$\begin{aligned} {\hat{u}}_{k}|^{k-j}_{k-1-j}={\mathbb {E}}\{u_k|{\mathcal {F}}_{k-j}\}-{\mathbb {E}}\{{\mathbb {E}}\{u_k|{\mathcal {F}}_{k-j}\}|{\mathcal {F}}_{k-j-1}\}. \end{aligned}$$

(94)

Theorem 4.1.14 in [18] implies that for every ${\mathcal {F}}_{k-1-j}$-measurable W, $j=1,2,\ldots ,d-1$, there holds

$$\begin{aligned} {\mathbb {E}}\{W'{\mathbb {E}}\{u_k|{\mathcal {F}}_{k-j}\}|{\mathcal {F}}_{k-j-1}\}=W'{\mathbb {E}}\{{\mathbb {E}}\{u_k|{\mathcal {F}}_{k-j}\}|{\mathcal {F}}_{k-j-1}\}. \end{aligned}$$

(95)

Taking expectation yields

$$\begin{aligned} {\mathbb {E}}\{W'({\mathbb {E}}\{u_k|{\mathcal {F}}_{k-j}\}-{\mathbb {E}}\{{\mathbb {E}}\{u_k|{\mathcal {F}}_{k-j}\}|{\mathcal {F}}_{k-j-1}\})\}=0. \end{aligned}$$

(96)

Associating (94) with (96), we know that ${\hat{u}}_{k}|^{k-j}_{k-1-j}$ is orthogonal to any ${\mathcal {F}}_{k-1-j}$-measurable vector. It is evident ${\hat{u}}_{k|k-i}$ are ${\mathcal {F}}_{k-1-j}$-measurable as $j\le i$, so ${\hat{u}}_{k|k-i}$ and ${\hat{u}}_{k}|^{k-j}_{k-1-j}$ are orthogonal for $1\le j<i$, i.e.,

$$\begin{aligned} {\mathbb {E}}\big \{{\hat{u}}_{k|k-i}'{\hat{u}}_{k}|^{k-j}_{k-1-j}\big \}=0, 1\le j<i. \end{aligned}$$

(97)

Similarly, ${\hat{u}}_{k}|^{k-j}_{k-1-j}$ is orthogonal to any ${\mathcal {F}}_{k-1-j}$-measurable vector. From ${\mathcal {F}}_{k-i}\subseteq {\mathcal {F}}_{k-j-1}\subset {\mathcal {F}}_{k-j}$, ${\hat{u}}_{k}|^{k-i}_{k-1-i}$ with $j<i\le d$ is ${\mathcal {F}}_{k-j-1}$-measurable. Accordingly, there holds

$$\begin{aligned} {\mathbb {E}}\big \{{\hat{u}}'_{k}|^{k-j}_{k-1-j}{\hat{u}}_{k}|^{k-i}_{k-1-i}\big \}=0, j<i\le d. \end{aligned}$$

(98)

Now (97) together with (98) shows that ${\hat{u}}_{k|k-1-d}$, ${\hat{u}}_{k}|^{k-1}_{k-2},\ldots , {\hat{u}}_{k}|^{k-d}_{k-d-1}$ are orthogonal to each other.

Proof of Theorem 3.1

Considering the truncated cost function (59) for any $k,1\le k\le K$, we have

$$\begin{aligned} {\bar{J}}_k&=\sum ^K_{j=k}{\mathbb {E}}\{(u_j-{\hat{x}}_{j+d|j-1})'Q(u_j-{\hat{x}}_{j+d|j-1})\nonumber \\&\quad +({\hat{x}}_{j+d|j-1}-x_{j+d})'R({\hat{x}}_{j+d|j-1}-x_{j+d})\}. \end{aligned}$$

(99)

Consistent with (99), we assume that the prediction process by the economic agents ends at $k=K$. Therefore, we set $u_k=0$ for $k>K$. From the system (11), (99) can be written as

$$\begin{aligned} {\bar{J}}_k&={\mathbb {E}}\Big \{\sum ^K_{j=k}u_j'Qu_j+\sum ^{K-d+1}_{j=k}[-2u_j'Q{\hat{x}}_{j+d|j-1}\nonumber \\&\quad +x_{j+d}'Rx_{j+d} +{{\hat{x}}_{j+d|j-1}}'(Q-R){\hat{x}}_{j+d|j-1}]\nonumber \\&\quad +\sum ^K_{j=K-d+2}[-2u_j'X_j{\hat{x}}_{K+1|j-1}+x_{K+1}'Y_jx_{K+1}\nonumber \\&\quad +{{\hat{x}}_{K+1|j-1}}'Z_j{\hat{x}}_{K+1|j-1}+\eta tr(H_j)]\Big \}, \end{aligned}$$

(100)

where $X_j=QA^{d-K-1+j}$, $Y_j=A'Y_{j-1}A+\tau A_0'Y_{j-1}A_0$ with the initial value $Y_{K-d+1}=R$, $Z_j=(A^{d-K-1+j})'(Q-R)A^{d-K-1-j}$, and $H_j=\sum ^{j+d-1}_{k=K+1}\eta (A^{d-k-1+j})'RA^{d-k-1+j}$. Now, we handle the optimization problem by dynamic programming procedure.

In the case of $k=K-d+1$, we get

$$\begin{aligned} \min _{u_K}{\bar{J}}_{K-d+1}&=\min _{u_K}{\mathbb {E}}\Big \{\sum ^K_{j=K-d+1}(u_j'Qu_j-2u_j'X_j{\hat{x}}_{K+1|j-1}+x_{K+1}'Y_jx_{K+1}\nonumber \\&\quad +{{\hat{x}}_{K+1|j-1}}'Z_j{\hat{x}}_{K+1|j-1})+\sum ^K_{j=K-d+2}\eta tr(H_j)\Big \}, \end{aligned}$$

(101)

by virtue of (11) and (101), one has

$$\begin{aligned} \min _{u_K}{\bar{J}}_{K-d+1}&=\min _{u_K}{\mathbb {E}}\Big \{\sum ^{K}_{j=K-d+1}[{u_j}'Qu_j-2{u_j}'X_jA{\hat{x}}_{K|j-1}-2{u_j}'X_jB{\hat{u}}_{K|j-1}\nonumber \\&\quad +x_K'(A'Y_jA+\tau A_0Y_jA_0)x_K+u_K'(B'Y_jB+\tau B_0Y_jB_0)u_K\nonumber \\&\quad +2u_K'(B'Y_jA+\tau B_0Y_jA_0)x_K+{{\hat{x}}_{K|j-1}}'A'Z_jA{\hat{x}}_{K|j-1}\nonumber \\&\quad +{{\hat{u}}_{K|j-1}}'B'Z_jB{\hat{u}}_{K|j-1}+2{{\hat{u}}_{K|j-1}}'B'Z_jA{\hat{x}}_{K|j-1}]+\sum ^{K}_{j=K-d+2}\eta tr(H_j)\Big \}. \end{aligned}$$

(102)

According to (61), ${\hat{u}}_{K|j-1}$, ${\hat{x}}_{K|j-1}$ and $u_j,j=K-d+1,\ldots ,K$ in (102) can be orthogonally decomposed as

$$\begin{aligned} {\hat{u}}_{K|j-1}&={\hat{u}}_{K|K-d}+\sum ^{d-1}_{n=K-j+1}{\hat{u}}_K|^{K-n}_{K-n-1}, \end{aligned}$$

(103)

$$\begin{aligned} {\hat{x}}_{K|j-1}&={\hat{x}}_{K|K-d}+\sum ^{d-1}_{n=K-j+1}{\hat{x}}_K|^{K-n}_{K-n-1}, \end{aligned}$$

(104)

$$\begin{aligned} u_j&={\hat{u}}_{j|K-d}+\sum ^{d-1}_{n=K-j+1}{\hat{u}}_j|^{K-n}_{K-n-1}. \end{aligned}$$

(105)

Inserting (103)-(105) into (102) yields

$$\begin{aligned} {\bar{J}}_{K-d+1}&={\mathbb {E}}\Big \{{{\hat{u}}_{K|K-d}}'\varUpsilon ^{d}_K{\hat{u}}_{K|K-d}+2{{\hat{u}}_{K|K-d}}'\Big (\varPsi ^{d}_K{\hat{x}}_{K|K-d}-\sum ^{K-1}_{j=K-d+1}B'(A^{d-K-1+j})'\nonumber \\&\quad \times Q{\hat{u}}_{j|K-d}\Big )+{{\hat{x}}_{K|K-d}}'[A'(Q+M^d_{K+1})A+\tau A_0'(R+M^1_{K+1})A_0]{\hat{x}}_{K|K-d}\nonumber \\&\quad +\sum ^{d-1}_{n=1}{\hat{u}}'_{K}|^{K-n}_{K-n-1}\varUpsilon ^n_K{\hat{u}}_{K}|^{K-n}_{K-n-1}+2\sum ^{d-1}_{n=1}{\hat{u}}'_{K}|^{K-n}_{K-n-1}\Big (\varPsi ^n_K{\hat{x}}_{K}|^{K-n}_{K-n-1}\nonumber \\&\quad -\sum ^{K-1}_{j=K-n+1}B'(A^{d-K-1+j})'Q{\hat{u}}_j|^{K-n}_{K-n-1}\Big )+\sum ^{d-1}_{n=1}{\hat{x}}'_{K}|^{K-n}_{K-n-1}[A'(R+M^{n+1}_{K+1})A\nonumber \\&\quad +\tau A_0'(R+M^1_{K+1})A_0]{\hat{x}}_{K}|^{K-n}_{K-n-1}\nonumber \\&\quad -2\sum ^{K-1}_{j=K-d+1}\Big ({{\hat{u}}_{j|K-d}}'S^{d,j}_{K+1}A{\hat{x}}_{K|K-d}+\sum ^{d-1}_{n=K-j+1}{\hat{u}}'_j|^{K-n}_{K-n-1}S^{n+1,j}_{K+1}A{\hat{x}}_{K}|^{K-n}_{K-n-1}\Big )\nonumber \\&\quad +\sum ^{K-1}_{j=K-d+1}\Big ({{\hat{u}}_{j|K-d}}'T^{d,j}_{K+1}{\hat{u}}_{j|K-d}+\sum ^{d-1}_{n=K-j+1}{\hat{u}}'_j|^{K-n}_{K-n-1}T^{n+1,j}_{K+1}{\hat{u}}_j|^{K-n}_{K-n-1}\Big )\nonumber \\&\quad +\eta tr(R+M^1_{K+1})+\sum ^{K}_{j=K-d+2}\eta tr(H_j)\Big \}, \end{aligned}$$

(106)

where we have used the equalities (64)-(65) with $k=K$ and the initial value $M^{n}_{K+1}=\sum ^K_{j=K-d+2}Y_j+\sum ^K_{j=K-n+2}Z_j$, $T^{n,j}_{K+1}=Q$ and $S^{n,j}_{K+1}=X_j$.

By completion of squares over ${\hat{u}}_{K|K-d}$ and ${\hat{u}}_{K}|^{K-n}_{K-n-1},i=1,2,\ldots ,d-1$ in (106) results in

$$\begin{aligned} {\bar{J}}_{K-d+1}&={\mathbb {E}}\Big \{({\hat{u}}_{K|K-d}-\bar{{\hat{u}}}_{K|K-d})'\varUpsilon ^{d}_{K}({\hat{u}}_{K|K-d}-\bar{{\hat{u}}}_{K|K-d})\nonumber \\&\quad +\sum ^{d-1}_{n=1}({\hat{u}}_{K}|^{K-n}_{K-n-1}-\bar{{\hat{u}}}_{K}|^{K-n}_{K-n-1})'\varUpsilon ^n_{K}({\hat{u}}_{K}|^{K-n}_{K-n-1}-\bar{{\hat{u}}}_{K}|^{K-n}_{K-n-1})\nonumber \\&\quad +{{\hat{x}}_{K|K-d}}'M^d_{K}{\hat{x}}_{K|K-d}+\sum ^{d-1}_{n=1}{\hat{x}}'_{K}|^{K-n}_{K-n-1}M^n_{K}{\hat{x}}_{K}|^{K-n}_{K-n-1}\nonumber \\&\quad -2\sum ^{K-1}_{j=K-d+1}\Big ({{\hat{u}}_{j|K-d}}'S^{d,j}_K{\hat{x}}_{K|K-d}+\sum ^{d-1}_{n=K-j+1}{\hat{u}}'_j|^{K-n}_{K-n-1}S^{n,j}_K{\hat{x}}_{K}|^{K-n}_{K-n-1}\Big )\nonumber \\&\quad +\sum ^{K-1}_{j=K-d+1}\Big ({{\hat{u}}_{j|K-d}}'T^{d,j}_K{\hat{u}}_{j|K-d}+\sum ^{d-1}_{n=K-j+1}{\hat{u}}'_j|^{K-n}_{K-n-1}T^{n,j}_K{\hat{u}}_j|^{K-n}_{K-n-1}\Big )\nonumber \\&\quad +\eta tr(R+M^1_{K+1})+\sum ^{K}_{j=K-d+2}\eta tr(H_j)\Big \}, \end{aligned}$$

(107)

where

$$\begin{aligned} \bar{{\hat{u}}}_{K|K-d}&=-L^d_{K}{\hat{x}}_{K|K-d}-\sum ^{K-1}_{j=K-d+1}K^{d,j}_{K}{\hat{u}}_{j|K-d}, \end{aligned}$$

(108)

$$\begin{aligned} \bar{{\hat{u}}}_{K}|^{K-n}_{K-n-1}&=-L^n_{K}{\hat{x}}_{K}|^{K-n}_{K-n-1}-\sum ^{K-1}_{j=K-n+1}K^{n,j}_{K}{\hat{u}}_{j}|^{K-n}_{K-n-1}. \end{aligned}$$

(109)

In the above, (62)-(67), and (70)-(71) are utilized with $k=K$. From (108)-(109), the optimal $u_K$ admits (46) with $k=K$. The optimal cost of (107) is given as

$$\begin{aligned} \min _{u_K}{\bar{J}}_{K-d+1}&={\mathbb {E}}\Big \{\sum ^{K-1}_{j=K-d+1}\Big ({{\hat{u}}_{j|K-d}}'T^{d,j}_K{\hat{u}}_{j|K-d}+\sum ^{d-1}_{n=K-j+1}{\hat{u}}'_j|^{K-n}_{K-n-1}T^{n,j}_K{\hat{u}}_j|^{K-n}_{K-n-1}\Big )\nonumber \\&\quad -2\sum ^{K-1}_{j=K-d+1}\Big ({{\hat{u}}_{j|K-d}}'S^{d,j}_K{\hat{x}}_{K|K-d}+\sum ^{d-1}_{n=K-j+1}{\hat{u}}'_j|^{K-n}_{K-n-1}S^{n,j}_K{\hat{x}}_{K}|^{K-n}_{K-n-1}\Big )\nonumber \\&\quad +{{\hat{x}}_{K|K-d}}'M^d_{K}{\hat{x}}_{K|K-d}+\sum ^{d-1}_{n=1}{\hat{x}}'_{K}|^{K-n}_{K-n-1}M^n_{K}{\hat{x}}_{K}|^{K-n}_{K-n-1}\nonumber \\&\quad +\eta tr(R+M^1_{K+1})+\sum ^{K}_{j=K-d+2}\eta tr(H_j)\Big \}. \end{aligned}$$

(110)

In the case of $k=K-d$, we have

$$\begin{aligned} \min _{u_{K-1},u_K}{\bar{J}}_{K-d}&=\min _{u_{K-1}}{\mathbb {E}}\Big \{{u_{K-d}}'Qu_{K-d}-2u_{K-d}Q{\hat{x}}_{K|K-d-1}+{{\hat{x}}_{K|K-d-1}}'Q{\hat{x}}_{K|K-d-1}\nonumber \\&\quad +\sum ^d_{n=1}{\hat{x}}'_K|^{K-n}_{K-n-1}R{\hat{x}}_K|^{K-n}_{K-n-1}+\min _{u_{K}}J_{K-d+1}\Big \}. \end{aligned}$$

(111)

This, together with (110), means that one only needs to resolve

$$\begin{aligned}&\min _{u_{K-1}}{\mathbb {E}}\Big \{\sum ^{K-1}_{j=K-d}\Big ({{\hat{u}}_{j|K-d-1}}'T^{d,j}_K{\hat{u}}_{j|K-d-1}+\sum ^{d}_{n=K-j+1}{\hat{u}}'_j|^{K-n}_{K-n-1}T^{n,j}_K{\hat{u}}_j|^{K-n}_{K-n-1}\Big )\nonumber \\&\quad -2\sum ^{K-1}_{j=K-d}\Big ({{\hat{u}}_{j|K-d-1}}'S^{d,j}_K{\hat{x}}_{K|K-d-1}+\sum ^{d}_{n=K-j+1}{\hat{u}}'_j|^{K-n}_{K-n-1}S^{n,j}_K{\hat{x}}_{K}|^{K-n}_{K-n-1}\Big )\nonumber \\&\quad +{{\hat{x}}_{K|K-d-1}}'(Q+M^d_{K}){\hat{x}}_{K|K-d-1}+\sum ^{d}_{n=1}{\hat{x}}'_{K}|^{K-n}_{K-n-1}(R+M^n_{K}){\hat{x}}_{K}|^{K-n}_{K-n-1}\nonumber \\&\quad +\eta tr(R+M^1_{K+1})+\sum ^{K}_{j=K-d+2}\eta tr(H_j)\Big \}, \end{aligned}$$

(112)

where $S^{d,K-d}_K=T^{d,K-d}_K=Q$. Because the right hand of (101) can be written as

$$\begin{aligned}&\min _{u_{K-1}}{\mathbb {E}}\Big \{\sum ^{K}_{j=K-d+1}\Big ({{\hat{u}}_{j|K-d}}'T^{d,j}_{K+1}{\hat{u}}_{j|K-d}+\sum ^{d}_{n=K-j+2}{\hat{u}}'_j|^{K+1-n}_{K-n}T^{n,j}_{K+1}{\hat{u}}_j|^{K+1-n}_{K-n}\Big )\nonumber \\&\quad -2\sum ^{K}_{j=K-d+1}\Big ({{\hat{u}}_{j|K-d}}'S^{d,j}_{K+1}{\hat{x}}_{K|K-d}+\sum ^{d}_{n=K-j+2}{\hat{u}}'_j|^{K+1-n}_{K-n}S^{n,j}_{K+1}{\hat{x}}_{K}|^{K+1-n}_{K-n}\Big )\nonumber \\&\quad +{{\hat{x}}_{K+1|K-d}}'(Q+M^d_{K+1}){\hat{x}}_{K+1|K-d}+\sum ^{d}_{n=1}{\hat{x}}'_{K+1}|^{K+1-n}_{K-n}(R+M^n_{K+1}){\hat{x}}_{K+1}|^{K+1-n}_{K-n}\nonumber \\&\quad +\sum ^{K}_{j=K-d+2}\eta tr(H_j)\Big \}, \end{aligned}$$

(113)

which has the same form of (112), thus we complete the induction argument.