Proof of The Control and Filter Riccati Families of the Ising Equilibrium in Closed Form

Throughout write

and $N=\mathsf{n}\mathsf{n}^{\top}$. Since $\chi>0$ and $\psi>0$ we have $1+\chi\psi>1$, so $\Delta>1$, $y_{+}>0>y_{-}$ and $\rho=(\Delta-1)/(\Delta+1)\in(0,1)$; since $\tilde{d}>0$ we have $\Gamma>2$, $p_{+}>0>p_{-}$ and $\rho_{\Pi}\in(0,1)$. Note $y_{+}=-\rho\,y_{-}$ and $p_{+}=-\rho_{\Pi}\,p_{-}$, and, from $\Gamma^{2}-4=2\tilde{d}$, that $p_{+}=(\Gamma-2)/\bigl((\Gamma-2)(\Gamma+2)\bigr)=1/(\Gamma+2)$. Two factorisations will be used: by the quadratic formula the roots of $8\chi y^{2}+4y-\tfrac12\psi$ are $(-1\pm\Delta)/(4\chi)$, that is $y_{\pm}$, so

and likewise the roots of $2\tilde{d}\,p^{2}+4p-1$ are $(-2\pm\Gamma)/(2\tilde{d})$, that is $p_{\pm}$, so

For a continuous integrand on a compact interval the Riemann and Lebesgue integrals agree, by Restricted Lebesgue Measure and Integral Toolkit on a Compact Interval, so the integral equations below may be read in either sense.

In the two instantiations below, the symbols $l$ and $k$ are the dimension parameters of the cited Riccati results and are set to $1$; they are unrelated to the number $l=2$ of states of the present model; that model parameter is left as $l$ rather than renamed. Claim 1. Existence and uniqueness. Apply Global Existence for the Backward Riccati Equation under Convexity Conditions with $l=k=1$, horizon $T$, and the constant data $A(t)=-2$, $B(t)=\sqrt{2}$, $Q(t)=\tfrac12\psi$, $V(t)=0$, $R(t)=(4\chi)^{-1}$ and $F=0$. All entries are continuous; $Q(t)=\tfrac12\psi\ge0$ is symmetric, $R(t)>0$ is symmetric positive definite, $F=0$ is symmetric positive semidefinite, and $Q(t)-V(t)R(t)^{-1}V(t)^{\top}=\tfrac12\psi\ge0$. The corollary's equation reads, in this one-dimensional case,

which is the displayed equation. The corollary provides exactly one continuous solution on $[0,T]$; call it $z$.

The closed form. Put $g_{t}=\exp\bigl(-4\Delta(T-t)\bigr)=\exp(-4\Delta T)\exp(4\Delta t)$ and $\hat{z}_{t}=y_{+}(1-g_{t})/(1+\rho g_{t})$. The exponential is positive, so $g_{t}>0$; and $g_{t}\le1$ with equality exactly at $t=T$, because $-4\Delta(T-t)\le0$ there and the exponential is increasing. In particular $1+\rho g_{t}\ge1$, so $\hat{z}$ is well defined and continuous. By Derivative and Continuity of the Scaled Exponential Function, $g$ is differentiable with $g'_{t}=4\Delta\,g_{t}$. By the reciprocal rule Reciprocal Rule for One-Dimensional Derivatives and the sum and product rules of Sum, Constant Multiple, and Product Rules for One-Dimensional Derivatives,

the middle step because $(1+\rho g)+\rho(1-g)=1+\rho$. On the other hand, using $y_{-}=-y_{+}/\rho$,

so that

Since $8\chi y_{+}=2(\Delta-1)$ and $(1+\rho)/\rho=2\Delta/(\Delta-1)$, we have $8\chi y_{+}(1+\rho)/\rho=4\Delta$, and the last display equals $4\Delta(1+\rho)y_{+}g_{t}/(1+\rho g_{t})^{2}=-\hat{z}'(t)$. By the factorisation recorded above this says

The right-hand side is continuous in $t$, so by Fundamental Theorem of Calculus, Part II, on a Closed Real Interval applied on $[t,T]$, $\hat{z}_{T}-\hat{z}_{t}=\int_{[t,T]}\hat{z}'(r)\,dr$; and $\hat{z}_{T}=0$ because $g_{T}=1$. Rearranging gives exactly the integral equation of the claim, so $\hat{z}=z$ by the uniqueness above.

Bounds and monotonicity. From $0<g_{t}\le1$ we get $0\le1-g_{t}<1\le1+\rho g_{t}$, hence $0\le z_{t}<y_{+}$, with $z_{T}=0$. The displayed formula for $\hat{z}'$ is negative at every $t$, so $z$ is nonincreasing. Finally, from the first displayed factorisation of $\hat{z}_{t}-y_{+}$, $\rho<1$ and $1+\rho g_{t}\ge1$,

Claim 2. Apply Global Existence and Uniqueness for the Kalman Covariance Riccati Equation on $[0,T]$ with $k=1$ and the constant data $A(t)=-2$, $C(t)=1$, $D(t)=2\tilde{d}$ and $P_{0}=0$: the entries are continuous, and $C(t)$, $D(t)$, $P_{0}$ are positive semidefinite because $1>0$, $2\tilde{d}>0$ and $0\ge0$. Its equation reads

which is the displayed equation, and the theorem provides exactly one continuous solution; call it $p$.

Put $h_{t}=\exp(-2\Gamma t)$ and $\hat{p}_{t}=p_{+}(1-h_{t})/(1+\rho_{\Pi}h_{t})$, so that $0<h_{t}\le1$ with equality exactly at $t=0$, and $h'_{t}=-2\Gamma h_{t}$ by Derivative and Continuity of the Scaled Exponential Function. Exactly as in claim 1,

using $p_{-}=-p_{+}/\rho_{\Pi}$. Since $2\tilde{d}\,p_{+}=\Gamma-2$ and $(1+\rho_{\Pi})/\rho_{\Pi}=2\Gamma/(\Gamma-2)$, we get $2\tilde{d}p_{+}(1+\rho_{\Pi})/\rho_{\Pi}=2\Gamma$, so the two displayed quantities are equal; by the second factorisation, $\hat{p}'(t)=-4\hat{p}_{t}-2\tilde{d}\hat{p}_{t}^{2}+1$, a continuous function of $t$. By Fundamental Theorem of Calculus, Part II, on a Closed Real Interval on $[0,t]$ and $\hat{p}_{0}=0$ we obtain the integral equation, so $\hat{p}=p$. The bounds $0\le p_{t}<p_{+}=(\Gamma+2)^{-1}$, the monotonicity ($\hat{p}'>0$) and the estimate $0\le p_{+}-p_{t}=p_{+}(1+\rho_{\Pi})h_{t}/(1+\rho_{\Pi}h_{t})\le2p_{+}h_{t}$ follow as in claim 1.

Claim 3. We use the rank-one algebra $v^{\top}v=2$, $\mathsf{n}^{\top}v=0$, hence

Each $Z_{t}$ is symmetric because $vv^{\top}$ and $N$ are, and its entries are continuous in $t$ because $z$ is continuous and $t\mapsto\mu(T-t)$ is. At $t=T$ both terms vanish, so $Z_{T}=0=\hat{F}$ by claim The Fluctuation LQG Data of the Ising Equilibrium and Its Joint Coercivity §coefficients.

By claim The Fluctuation LQG Data of the Ising Equilibrium and Its Joint Coercivity §coefficients, $E_{r}=-vv^{\top}$ is symmetric, $\mathsf{B}_{r}=-\tfrac12vv^{\top}$, $V_{r}=0$, $R_{r}^{-1}=4\chi I$ and $Q_{r}=\tfrac12\psi\,vv^{\top}+\mu N$. Hence

Therefore

Integrating entrywise over $[t,T]$ and using claim 1 for the first coefficient and $\int_{[t,T]}\mu\,dr=\mu(T-t)$ for the second gives

which, since $\hat{F}=0$, is the asserted integral equation entry by entry. The integrand is continuous in $r$, so subtracting the equations at $t$ and at $0$ gives the equivalent forward form $Z^{\gamma\delta}_{t}=Z^{\gamma\delta}_{0}+\int_{[0,t]}\dot{z}^{\gamma\delta}(r)\,dr$ with the stated continuous densities. Symmetry, continuity of the entries, the density identity and the terminal condition $Z_{T}=\hat{F}$ are precisely what hypothesis (H2) asks for.

Claim 4. The matrices $\mathcal{E}_{t}$, $\Theta^{\star}_{t}$ and $\tilde{D}_{t}$ are constant in $t$ by claims The Fluctuation LQG Data of the Ising Equilibrium and Its Joint Coercivity §lqg-data and The Fluctuation LQG Data of the Ising Equilibrium and Its Joint Coercivity §information, hence have continuous entries. For $z\in\mathbb{R}^{2}$ we have $z\cdot\Theta^{\star}_{t}z=(v\cdot z)^{2}\ge0$, so $\Theta^{\star}_{t}$ is symmetric positive semidefinite; and, writing $c=\tfrac{q}{2}+q_{0}>0$, $\tilde{D}_{t}=c^{-1}\tilde{\mathcal{E}}_{t}^{\top}\tilde{\mathcal{E}}_{t}$ is symmetric with $z\cdot\tilde{D}_{t}z=c^{-1}|\tilde{\mathcal{E}}_{t}z|^{2}\ge0$, hence positive semidefinite; the zero matrix is positive semidefinite. So Global Existence and Uniqueness for the Kalman Covariance Riccati Equation applies and furnishes a unique $\Pi$.

Put $\hat{\Pi}_{t}=p_{t}vv^{\top}$, which has continuous entries. Using $\mathcal{E}_{t}=-vv^{\top}$, the rank-one algebra of claim 3 and the identity $vv^{\top}\tilde{D}_{t}=\tilde{d}\,vv^{\top}$ of claim The Fluctuation LQG Data of the Ising Equilibrium and Its Joint Coercivity §information,

so the integrand of the Kalman equation is $\bigl(-4p_{r}-2\tilde{d}p_{r}^{2}+1\bigr)vv^{\top}$. Integrating entrywise over $[0,t]$ and using claim 2 gives $\int_{0}^{t}(\cdots)dr=p_{t}vv^{\top}=\hat{\Pi}_{t}$, and $P_{0}=0$; so $\hat{\Pi}$ solves the equation. By the uniqueness in Global Existence and Uniqueness for the Kalman Covariance Riccati Equation, $\Pi_{t}=p_{t}vv^{\top}$.

Claim 5. For real $2\times2$ matrices of the form $\lambda vv^{\top}$ and $\kappa vv^{\top}$ the entrywise pairing is

while

With $Z_{t}=z_{t}vv^{\top}+\mu(T-t)N$ and $\Theta^{\star}_{t}=vv^{\top}$ the first pairing is $4z_{t}\cdot1+\mu(T-t)\cdot0=4z_{t}$; with $\Xi_{t}=8\chi z_{t}^{2}vv^{\top}$ and $\Pi_{t}=p_{t}vv^{\top}$ the second is $4\cdot8\chi z_{t}^{2}\cdot p_{t}=32\chi z_{t}^{2}p_{t}$. Neither expression contains $\mu$. Both $z$ and $p$ are continuous by claims 1 and 2, so $t\mapsto4z_{t}+32\chi z_{t}^{2}p_{t}$ is continuous by Continuity of Sums and Products of Real-Valued Functions on a Metric Space.