Proof of Regularity and Derivative Bounds of the Extended Aggregate State Drift

Throughout fix $\gamma\in\{1,\dots,l\}$, write $x=(\Sigma,\alpha)$ for points of $U\times\mathbb{R}^m$, and recall from the extension definition its clauses 1-4. On $\Delta^l\times\mathbb{R}^m$ we use, from the probability simplex: $\Sigma^\sigma\ge0$ for all $\sigma$ and $\sum_{\sigma=1}^{l}\Sigma^\sigma=1$; and from clause 1 together with the transition-rate family bounds: $0\le\bar{\beta}(\sigma,\gamma',x)\le B$ for $x\in\Delta^l\times\mathbb{R}^m$ and all admissible index pairs.

(i). For $\sigma\in\{1,\dots,l\}$ let $\pi_\sigma:U\times\mathbb{R}^m\to\mathbb{R}$ be the coordinate function $\pi_\sigma(x)=\Sigma^\sigma$. Directly from the definition of the partial derivative (the difference quotients being constant), $\partial_i\pi_\sigma$ exists and equals the constant $\delta_{i\sigma}$, so $\pi_\sigma$ is a $C^1$ map, and each constant function is again $C^1$ with vanishing partial derivatives. By the definition of the extended aggregate state drift,

Fix $i$ and a point of $U\times\mathbb{R}^m$. The set of slice parameters $s$ for which the point shifted by $s$ along the $i$-th coordinate remains in the open set $U\times\mathbb{R}^m$ is an open subset of $\mathbb{R}$ containing the given parameter value; choose an open interval $I_0$ around that value contained in it. On $I_0$, every factor above restricts to a one-variable function whose derivative exists and is given by the corresponding partial derivative (this is the definition of the partial derivative), so the one-dimensional sum and product rules, stated for functions on an interval, give that $\partial_i\bar{b}^\gamma$ exists at the given point and equals the first displayed formula of the statement. Each term of that formula is continuous: $\bar{\beta}(\sigma,\gamma,\cdot,\cdot)$ is continuous because it is differentiable at every point (clause 2 with $C^1$ implies differentiable, and the inequality in the definition of differentiability at a point forces $f(a+h)\to f(a)$ as $h\to0$); its partial derivatives are continuous by clause 2 and the definition of a $C^1$ map; the coordinate functions and constants are continuous; and finite sums and products of continuous real functions are continuous (immediate from the sequential formulation of continuity at a point). Hence $\partial_i\bar{b}^\gamma$ is continuous for every $i$, so $\bar{b}^\gamma$ is a $C^1$ map. Applying the same slice-interval argument to the first displayed formula - a finite sum of products of coordinate functions, constants, the functions $\bar{\beta}(\cdot)$, and their first partials, all of which are $C^1$ by clause 2 - gives that $\partial_j\partial_i\bar{b}^\gamma$ exists and equals the second displayed formula (using $\partial_j\pi_\sigma=\delta_{j\sigma}$ and $\partial_j\delta_{i\sigma}=0$), and this expression is continuous by the same reasoning; hence each $\partial_i\bar{b}^\gamma$ is a $C^1$ map. Finally, for $x\in\Delta^l\times\mathbb{R}^m$, clause 1 allows replacing every $\bar{\beta}$ by $\beta$ in the defining formula of $\bar{b}^\gamma(x)$, which then coincides with the defining formula of the aggregate state drift of $\beta$ at $x$; hence $\bar{b}$ agrees with it on $\Delta^l\times\mathbb{R}^m$.

(ii). Let $x\in\Delta^l\times\mathbb{R}^m$ and $i\in\{1,\dots,l+m\}$. Estimating the four groups of terms of the first displayed formula separately: $\sum_{\sigma\neq\gamma}\delta_{i\sigma}|\bar{\beta}(\sigma,\gamma,x)|\le B$ (at most one $\sigma$ equals $i$); $\sum_{\sigma\neq\gamma}\Sigma^\sigma|\partial_i\bar{\beta}(\sigma,\gamma,x)|\le K\sum_{\sigma\neq\gamma}\Sigma^\sigma\le K$ by clause 3; $\delta_{i\gamma}\sum_{\sigma\neq\gamma}|\bar{\beta}(\gamma,\sigma,x)|\le(l-1)B$; and $\Sigma^\gamma\sum_{\sigma\neq\gamma}|\partial_i\bar{\beta}(\gamma,\sigma,x)|\le(l-1)K$. Altogether $|\partial_i\bar{b}^\gamma(x)|\le lB+lK=l(B+K)$.

For the Lipschitz estimate let $x,y\in\Delta^l\times\mathbb{R}^m$. The segment from $x$ to $y$ stays in $\Delta^l\times\mathbb{R}^m$: a convex combination of two points of the simplex has nonnegative entries summing to $1$, hence lies in the simplex. The segment therefore lies in the open set $U\times\mathbb{R}^m$, and the Taylor expansion lemma, part (i), applied to the $C^1$ map $\bar{b}^\gamma$ with $n=l+m$ and $M_1=l(B+K)$ gives $|\bar{b}^\gamma(x)-\bar{b}^\gamma(y)|\le\sqrt{l+m}\,l(B+K)\,d(x,y)$.

(iii). For $x\in\Delta^l\times\mathbb{R}^m$, estimating the six groups of the second displayed formula with clause 3 and $\sum_{\sigma\neq\gamma}\Sigma^\sigma\le1$, $\Sigma^\gamma\le1$: the terms with $\delta_{i\sigma}$ and $\delta_{j\sigma}$ contribute at most $K$ each; the terms $\Sigma^\sigma\partial_j\partial_i\bar{\beta}(\sigma,\gamma,x)$ contribute at most $K$ in total; the terms with $\delta_{i\gamma}$ and $\delta_{j\gamma}$ contribute at most $(l-1)K$ each; and the terms $\Sigma^\gamma\partial_j\partial_i\bar{\beta}(\gamma,\sigma,x)$ contribute at most $(l-1)K$ in total. Hence $|\partial_j\partial_i\bar{b}^\gamma(x)|\le3K+3(l-1)K=3lK$.

For the uniform continuity claim, let $\varepsilon>0$ and set $C^*=2\,l\,\sqrt{l+m}\,K+2\,l\,K$ (a convenient over-bound: the exact coefficient collected below is $2l\sqrt{l+m}\,K+2(l-1)K$, and we over-estimate $l-1$ by $l$). By clause 4 there is $\delta_1>0$ such that $|\partial_j\partial_i\bar{\beta}(\sigma,\gamma',x)-\partial_j\partial_i\bar{\beta}(\sigma,\gamma',y)|\le\varepsilon/(2l)$ for all admissible indices whenever $d(x,y)\le\delta_1$. Set $\delta=\delta_1$ if $C^*=0$ and $\delta=\min(\delta_1,\varepsilon/(2C^*))$ otherwise. Let $x,y\in\Delta^l\times\mathbb{R}^m$ with $d(x,y)\le\delta$, and take the difference of the second displayed formula at $x$ and at $y$ term by term. For the first-derivative factors: each $\partial_j\bar{\beta}(\sigma,\gamma',\cdot,\cdot)$ is a $C^1$ map (clause 2) whose partial derivatives are bounded by $K$ on $U\times\mathbb{R}^m$ (clause 3), so part (i) of the Taylor expansion lemma along the segment from $x$ to $y$ (which lies in the simplex product, as above) gives $|\partial_j\bar{\beta}(\sigma,\gamma',x)-\partial_j\bar{\beta}(\sigma,\gamma',y)|\le\sqrt{l+m}\,K\,d(x,y)$. For the product terms, writing $y=(\Sigma',\alpha')$:

and similarly with $\gamma$ in place of $\sigma$. Summing all contributions: the $\delta_{i\sigma}$ and $\delta_{j\sigma}$ groups give at most $2\sqrt{l+m}\,K\,d(x,y)$; the $\delta_{i\gamma}$ and $\delta_{j\gamma}$ groups give at most $2(l-1)\sqrt{l+m}\,K\,d(x,y)$; the two product groups give at most $2(l-1)K\,d(x,y)$ from the first summands (using $|\Sigma^\sigma-\Sigma'^\sigma|\le d(x,y)$ for every $\sigma$) plus $\tfrac{\varepsilon}{2l}\big(\sum_{\sigma\neq\gamma}\Sigma'^\sigma+(l-1)\Sigma'^\gamma\big)\le\tfrac{\varepsilon}{2l}\cdot l=\tfrac{\varepsilon}{2}$ from the second summands. Altogether the difference is at most $C^*d(x,y)+\varepsilon/2\le\varepsilon/2+\varepsilon/2=\varepsilon$, uniformly over $i,j,\gamma$, as claimed.