The two-color Riccati identities - The KLS theorem and its methods

The evolution of $r_t=s_t\abs{\delta_t}^2$ is governed by an exact Riccati identity. The matrix version is the most informative, so it is stated first.

Define the within-class covariance

R_t=A_t-B_t=p_t\Sigma_t^E+q_t\Sigma_t^F\succeq0.

(24.1)

Also put

K_t=G_t+(q_t-p_t)\delta_t\delta_t^T.

(24.2)

Proof 24.1

Let $v=v_t$ and $s=s_t$ . From (23.5), (23.6), and Ito’s product rule,

\dd v_t=(C_t^E-p_tA_t)\dd W_t-A_tv_t\dd t,

(24.5)

where

C_t^E=\int_E(x-a_t)(x-a_t)^T\dd\mu_t(x).

(24.6)

A direct covariance decomposition gives

C_t^E-p_tA_t=s_tK_t.

(24.7)

Hence $\dd v=sK\dd W-Av\dd t$ . Also

\dd s=(q-p)v\cdot\dd W-\abs v^2\dd t.

(24.8)

Since $B=vv^T/s$ , Ito’s rule gives a drift

sK^2-(AB+BA)+rB+(q-p)^2\frac rsB-(q-p)(KB+BK).

(24.9)

Substituting $K=G+(q-p)B/s$ and using $B^2=rB$ , the linear terms in $q-p$ cancel and the quadratic terms in $q-p$ sum to zero, leaving (24.3).

Subtract (24.3) from the covariance SDE (23.7). Since $B^2=rB$ ,

-A^2+AB+BA-rB=-(A-B)^2=-R^2.

(24.10)

This yields (24.4).

The damping is coercive. From $B_t\preceq A_t$ ,

\delta_t^TA_t\delta_t\ge s_t\abs{\delta_t}^4,

(24.15)

D_t=2s_t\delta_t^TA_t\delta_t-r_t^2\ge r_t^2 .

(24.16)

Thus the only positive term in the scalar Riccati identity is the source

S_t=s_t\norm{\Sigma_t^E-\Sigma_t^F}_{\HS}^2.

(24.17)