The Fluctuation Linear-Quadratic Cost Functional

Let $l$, $m$, $\mathcal{A}$, $\beta$ with rate bound $B$, $(U,V,\bar{\beta})$ with derivative bound $K$, $(L,G)$, $(W,\bar{L},\bar{G})$ with second-derivative bound $K_c$, $T>0$, and $(S,A)$ be as in the definition of a stationary mean-field triple, let $\bar{b}$ be the extended aggregate state drift of $(U,V,\bar{\beta})$, let $P$ be a stationary co-state for these data, and adopt the partial-derivative notation $\partial_j\partial_i$ of the extension definitions. Throughout, a real-valued function on a subinterval $I$ of the real numbers $\mathbb{R}$ is called continuous on $I$ when it is continuous relative to $I$, both $I$ and the codomain $\mathbb{R}$ carrying the metric of the real line. Define, for $t\in[0,T]$ and $i,j\in\{1,\dots,l+m\}$, the fluctuation Hessian coefficients

$$H_{ij}(t)=\partial_j\partial_i\bar{L}(S_t,A_t)-\sum_{\delta=1}^{l}P^\delta_t\,\partial_j\partial_i\bar{b}^\delta(S_t,A_t),\qquad\qquad F_{\gamma\delta}=\partial_\delta\partial_\gamma\bar{G}(S_T)\quad(\gamma,\delta\in\{1,\dots,l\}),$$

where the second-order partial derivatives of $\bar{b}$ exist and are continuous by part (i) of the regularity of the extended aggregate state drift, and those of $\bar{L}$ and $\bar{G}$ exist and are continuous by clause 2 of the cost extension definition together with clauses 1 and 2 of the $C^k$ definition. Each $H_{ij}$ is continuous on $[0,T]$, by continuity of compositions and continuity of sums and products (both applied pointwise on $[0,T]$, the Euclidean and metric notions of continuity for real-valued maps agreeing by claim 1 of the continuity agreement lemma, and the vector map $t\mapsto(S_t,A_t)$ being continuous into $\mathbb{R}^{l+m}$ in the Euclidean sense because $d((S_s,A_s),(S_t,A_t))\le\sum_{\gamma=1}^{l}|S^\gamma_s-S^\gamma_t|+\sum_{j=1}^{m}|A^j_s-A^j_t|$ by claim 1 of the componentwise estimates, each summand being controlled by the continuity of the components in clause 1 of the trajectory-pair definition) along the continuous trajectory pair and the continuous co-state; by the extreme value theorem it attains a maximum and a minimum on $[0,T]$, and is therefore bounded.

Let $(x_t)_{t\in[0,T]}$ and $(a_t)_{t\in[0,T]}$ be families of $\mathbb{R}^l$-valued and $\mathbb{R}^m$-valued random vectors on a common probability space $(\Omega,\mathcal{F},\mathbb{P})$ (each component a random variable), and write $z_t=(x_t,a_t)$, identified with an $\mathbb{R}^{l+m}$-valued map with components $z^1_t,\dots,z^{l+m}_t$, so that $x_t$ has components $x^1_t,\dots,x^l_t$ and $a_t$ has components $a^1_t,\dots,a^m_t$ (the symbol $x_t$ here is unrelated to the generic point $x=(\Sigma,\alpha)$ of the extension definitions), such that:

(i) there is an event $\Omega_1$ of probability $1$ such that, for each $i\in\{1,\dots,l+m\}$, the map $(t,\omega)\mapsto\mathbf{1}_{\Omega_1}(\omega)\,z^i_t(\omega)$ on $[0,T]\times\Omega$ is measurable with respect to the product $\sigma$-algebra of the trace Borel $\sigma$-algebra on $[0,T]$ and $\mathcal{F}$, where $\mathbf{1}_{\Omega_1}$ is the function equal to $1$ on $\Omega_1$ and $0$ off $\Omega_1$;

(ii) $\displaystyle\int_{[0,T]}\mathbb{E}\big[|x_t|^2+|a_t|^2\big]\,dt<\infty$, where $|\cdot|$ is the Euclidean norm (Euclidean distance to the origin) and the integral is the Lebesgue integral with value in $[0,\infty]$. Here the map $(t,\omega)\mapsto\mathbf{1}_{\Omega_1}(\omega)\big(|x_t(\omega)|^2+|a_t(\omega)|^2\big)$ is product-measurable, being a jointly continuous, hence sequentially continuous, function of the maps of (i) by measurability of sequentially continuous functions of measurable Euclidean maps (using $\mathbf{1}_{\Omega_1}^2=\mathbf{1}_{\Omega_1}$), so the integrand $t\mapsto\mathbb{E}\big[|x_t|^2+|a_t|^2\big]$ is a measurable $[0,\infty]$-valued function of $t$ by the Tonelli theorem, applicable here and in every use below because the trace Lebesgue measure on $[0,T]$ (a finite measure) and $\mathbb{P}$ are finite, hence $\sigma$-finite, measures. The integrand is unchanged when $z_t$ is replaced by $\mathbf{1}_{\Omega_1}z_t$: writing $N$ for the complement of $\Omega_1$, an event of probability zero, the functions $|x_t|^2+|a_t|^2$, $\mathbf{1}_{\Omega_1}\big(|x_t|^2+|a_t|^2\big)$, and $\mathbf{1}_{N}\big(|x_t|^2+|a_t|^2\big)$ are nonnegative random variables (sequentially continuous functions of random variables, again by the same lemma); the third is dominated pointwise by the $[0,\infty]$-valued function equal to $\infty$ on $N$ and $0$ elsewhere, which is the increasing pointwise limit of the simple functions $n\,\mathbf{1}_{N}$ and so has expectation $0$ by monotone convergence and the integral of a simple function; hence $\mathbf{1}_{N}\big(|x_t|^2+|a_t|^2\big)$ has expectation $0$ by monotonicity, and additivity gives $\mathbb{E}\big[|x_t|^2+|a_t|^2\big]=\mathbb{E}\big[\mathbf{1}_{\Omega_1}\big(|x_t|^2+|a_t|^2\big)\big]$;

(iii) $\mathbb{E}\big[|x_T|^2\big]<\infty$, with the expectation.

Write $\psi_t(\omega)=\tfrac{1}{2}\sum_{i=1}^{l+m}\sum_{j=1}^{l+m}H_{ij}(t)\,z^i_t(\omega)\,z^j_t(\omega)$ for $(t,\omega)\in[0,T]\times\Omega$. Since $\mathbf{1}_{\Omega_1}^2=\mathbf{1}_{\Omega_1}$, the map $(t,\omega)\mapsto\mathbf{1}_{\Omega_1}(\omega)\,\psi_t(\omega)$ is a jointly continuous, hence sequentially continuous, function of the measurable maps $(t,\omega)\mapsto t$ (the preimage of a Borel set $B$ being the rectangle $B\times\Omega$) and $(t,\omega)\mapsto\mathbf{1}_{\Omega_1}(\omega)\,z^i_t(\omega)$ of (i), and is therefore product-measurable by measurability of sequentially continuous functions of measurable Euclidean maps; its positive and negative parts $(\mathbf{1}_{\Omega_1}\psi)^{+}$ and $(\mathbf{1}_{\Omega_1}\psi)^{-}$ are product-measurable in turn, being its compositions with the continuous functions $u\mapsto\max(u,0)$ and $u\mapsto\max(-u,0)$, again by the same lemma. Fix a real number $C$ with $\tfrac{1}{2}\sum_{i=1}^{l+m}\sum_{j=1}^{l+m}|H_{ij}(t)|\le C$ for all $t\in[0,T]$, which exists by the boundedness of each $H_{ij}$ recorded above; then $|\mathbf{1}_{\Omega_1}\psi_t|\le C\,\mathbf{1}_{\Omega_1}\big(|x_t|^2+|a_t|^2\big)$ on $[0,T]\times\Omega$, since $|z^i_t\,z^j_t|\le|x_t|^2+|a_t|^2$ for all $i,j$.

By the Tonelli theorem applied to the product-measurable map of (ii), $\Phi(\omega)=\int_{[0,T]}\mathbf{1}_{\Omega_1}(\omega)\big(|x_t(\omega)|^2+|a_t(\omega)|^2\big)\,dt$ defines a measurable $[0,\infty]$-valued function of $\omega$; for $\omega\in\Omega_1$ the section being integrated is exactly $t\mapsto|x_t(\omega)|^2+|a_t(\omega)|^2$, since $\mathbf{1}_{\Omega_1}(\omega)=1$. Let $\Omega_2=\Omega_1\cap\{\omega\in\Omega:\Phi(\omega)<\infty\}$; by (i) alone $\Omega_2$ is an event, and by (ii) it has probability $1$: $\mathbb{E}[\Phi]$ equals the finite quantity of (ii) by the Tonelli theorem, while for every natural number $n$ we have $\Phi\ge n\,\mathbf{1}_{\{\Phi=\infty\}}$ pointwise, so $\mathbb{E}[\Phi]\ge n\,\mathbb{P}(\Phi=\infty)$ by monotonicity and the integral of a simple function, which forces $\mathbb{P}(\Phi=\infty)=0$. At each $\omega\in\Omega_2$ the section $t\mapsto\psi_t(\omega)$ is measurable, being the difference of the sections at $\omega$ of $(\mathbf{1}_{\Omega_1}\psi)^{+}$ and $(\mathbf{1}_{\Omega_1}\psi)^{-}$ (sections of product-measurable $[0,\infty]$-valued maps are measurable, as in the Tonelli theorem), and it is Lebesgue integrable over $[0,T]$, being measurable with $\int_{[0,T]}|\psi_t(\omega)|\,dt\le C\,\Phi(\omega)<\infty$ by monotonicity and the definition of integrability.

The fluctuation linear-quadratic cost of the pair $((x),(a))$ relative to the stationary mean-field triple $(S,A,P)$ and the chosen extensions is the real number

$$LQG\big[(x),(a)\big]=\mathbb{E}[I]+\mathbb{E}\Big[\tfrac{1}{2}\sum_{\gamma=1}^{l}\sum_{\delta=1}^{l}F_{\gamma\delta}\,x^\gamma_T\,x^\delta_T\Big],$$

where $I$ is the random variable that vanishes off $\Omega_2$ and at each $\omega\in\Omega_2$ equals the Lebesgue integral $\int_{[0,T]}\psi_t(\omega)\,dt$ of the preceding paragraph, and both expectations are expectations on $(\Omega,\mathcal{F},\mathbb{P})$. Both expectations are well defined and finite. Indeed, by the Tonelli theorem the integrals over $[0,T]$ of $(\mathbf{1}_{\Omega_1}\psi)^{+}$ and $(\mathbf{1}_{\Omega_1}\psi)^{-}$ define measurable $[0,\infty]$-valued functions of $\omega$, each at most $C\,\Phi$ pointwise by monotonicity, hence finite at every $\omega\in\Omega_2$; for each of the two, the function equal to it on $\Omega_2$ and to $0$ off $\Omega_2$ is a random variable, $\Omega_2$ being an event, and $I$ is the difference of these two random variables. Moreover $|I|\le C\,\Phi$ pointwise and $\mathbb{E}[\Phi]<\infty$, so $I$ is integrable and $\mathbb{E}[I]$ is defined and finite, again by monotonicity. The terminal random variable $\tfrac{1}{2}\sum_{\gamma=1}^{l}\sum_{\delta=1}^{l}F_{\gamma\delta}\,x^\gamma_T\,x^\delta_T$ is measurable by measurability of continuous functions of measurable maps, and its absolute value is at most $\tfrac{1}{2}\big(\sum_{\gamma,\delta}|F_{\gamma\delta}|\big)|x_T|^2$, which has finite expectation by (iii); so it is integrable and its expectation is defined and finite, by monotonicity once more. Finally, the value of $LQG[(x),(a)]$ does not depend on the admissible choice of $\Omega_1$: if $\Omega_1'$ is another event as in (i), with associated $\Omega_2'$ and $I'$, then $\Omega_2\cap\Omega_2'$ has probability $1$ and the two prescriptions agree there, so the nonnegative random variable $|I-I'|$ vanishes off an event of probability zero and has expectation $0$ by the domination argument of (ii), whence $|\mathbb{E}[I]-\mathbb{E}[I']|=\big|\mathbb{E}[I-I']\big|\le\mathbb{E}\big[|I-I'|\big]=0$ by the basic integral laws.