Stationary monitored-chain transport variance coefficient

The path in [F1] has locally piecewise-constant sample paths with finitely many jumps on bounded intervals. For the supplied measurable jump times $S_n$ and labels $A_n$, the event $a(s)=a$ is the countable union of $\{(s,\omega ):S_n(\omega )\le s<S_{n+1}(\omega ),A_n(\omega )=a\}$. Thus $(s,\omega )\mapsto v_{a(s,\omega )}$ is product-measurable and bounded by $M:={\max }_a\mathrm{\mid }{v}_a\mathrm{\mid }$. On the probability space $(\mathrm{\Omega },\mathcal{F},{\mathbb{P}}_{\pi })$ times $[0,T]$ with Lebesgue measure, its absolute integral is at most $TM<\mathrm{\infty }$; both measures are finite, so [F4] makes $Y_T$ a measurable bounded random variable and allows its expectation to pass through the time integral. It is stationary when initialized with $\pi$, since [F2] gives $\pi W=0$. For $s,t\ge 0$, stationarity and the transition semigroup give

For $t\ge s$, this is the Markov property; for $s>t$, the same scalar follows by self-adjointness of $e^{(s-t)W}$. Since $v$ is centered, ${\mathbb{E}}_{\pi }[Y_T]=0$.

On $\mathrm{\Omega }\times [0,T{]}^2$, the product-measurable integrand $v_{a(s,\omega )}{v}_{a(t,\omega )}$ has absolute value at most $M^2$, so its absolute product integral is at most $T^2{M}^2$. The probability measure and the Lebesgue measures of $[0,T]$ and $[0,T{]}^2$ are finite and hence sigma-finite. Applying [F4] to exchange expectation with the square integral, and to the two time coordinates, therefore gives

using the covariance identity from step 1.1 and splitting the square into its two triangles.

Expand $v$ in an orthonormal eigenbasis of the self-adjoint $W{\mathrm{\mid }}_{{\pi }^{\perp }}$. Its eigenvalues are at most $-{\gamma }_{\mathrm{c}\mathrm{l}}<0$, so the expansion gives $\mathrm{\mid }{C}_v(s)\mathrm{\mid }\le e^{-{\gamma }_{\mathrm{c}\mathrm{l}}s}\mathrm{\parallel }v{\mathrm{\parallel }}_{\pi }^2$. The finite matrix exponential makes $C_v$ continuous. On $[0,\mathrm{\infty })$, set $f_T(s)={\mathbf{1}}_{[0,T]}(s)(1-s\mathrm{/}T)C_v(s)$. For every sequence $T_n\to \mathrm{\infty }$, $f_{T_n}(s)\to C_v(s)$ pointwise and $\mathrm{\mid }{f}_{T_n}(s)\mathrm{\mid }\le \mathrm{\parallel }v{\mathrm{\parallel }}_{\pi }^2{e}^{-{\gamma }_{\mathrm{c}\mathrm{l}}s}$, an integrable majorant.

Applying [F3] to step 2.2 gives ${\int }_0^{T_n}(1-s\mathrm{/}{T}_n)C_v(s)ds\to {\int }_0^{\mathrm{\infty }}{C}_v(s)ds$ for every $T_n\to \mathrm{\infty }$, hence the real-$T$ limit exists and equals that integral. The same finite spectral expansion gives ${\int }_0^{\mathrm{\infty }}{e}^{sW}vds=(-W{\mathrm{\mid }}_{{\pi }^{\perp }}{)}^{-1}v$, so the integral is $⟨(-W{\mathrm{\mid }}_{{\pi }^{\perp }}{)}^{-1}v,v{⟩}_{\pi }$. Each eigencomponent contributes its squared coefficient divided by a positive gap eigenvalue, proving nonnegativity.