The Regularised Entropic Rate Cost

Let $\underline{a}$ and $\bar{a}$ be real numbers with $0<\underline{a}<1<\bar{a}$. Define the profile $\varpi:\mathbb{R}\to\mathbb{R}$ by

$$\varpi(u)=\frac{\vartheta(u)}{\max\{u,\underline{a}\}},\qquad \vartheta(u)=\min\Bigl\{\,1,\ \frac{\max\{u,0\}}{\underline{a}},\ \max\Bigl\{0,\ 2-\frac{u}{\bar{a}}\Bigr\}\Bigr\}\qquad(u\in\mathbb{R}),$$

which is well defined because $\max\{u,\underline{a}\}\ge\underline{a}>0$ for every $u\in\mathbb{R}$.

Throughout, the derivative of a real-valued function on $\mathbb{R}$ is the derivative at an interior point of $\mathbb{R}$, every point of $\mathbb{R}$ being interior to $\mathbb{R}$; $\varpi'$, $\phi'$ and $\phi''$ denote such derivatives, and being of class $C^{2}$ on $\mathbb{R}$ means being of class $C^{2}$ on the open subset $\mathbb{R}$ of $\mathbb{R}^{1}$. A map $f:\mathbb{R}\to\mathbb{R}$ is called Lipschitz with constant $C$ for the metric of the real line on domain and codomain. Write $\int_{[c,d]}f(r)\,dr$ for the Lebesgue integral over the compact interval $[c,d]$, defined whenever $c\le d$ and $f$ is continuous on $[c,d]$.

Then the following hold.

1. (The profile.)(i) $\varpi$ is Lipschitz with constant $2\,\underline{a}^{-2}$, and hence continuous on $\mathbb{R}$. Moreover

$$0\le\varpi(u)\le\underline{a}^{-1}\ \ (u\in\mathbb{R}),\qquad \varpi(u)=0\ \ (u\le0\ \text{or}\ u\ge2\bar{a}),\qquad \varpi(u)=u^{-1}\ \ (\underline{a}\le u\le\bar{a}),$$

and $\varpi(u)>0$ for every $u$ with $0<u<2\bar{a}$.

2. (The first antiderivative.)(ii) Define $\Psi:\mathbb{R}\to\mathbb{R}$ by

$$\Psi(u)=\int_{[1,u]}\varpi(r)\,dr\ \ (u\ge1),\qquad \Psi(u)=-\int_{[u,1]}\varpi(r)\,dr\ \ (u<1).$$

Then $\Psi$ is Lipschitz with constant $\underline{a}^{-1}$, $\Psi(1)=0$, $\Psi(u)\ge0$ for $u\ge1$, $\Psi(u)\le0$ for $u\le1$, $|\Psi(u)|\le2\,\bar{a}\,\underline{a}^{-1}$ for every $u\in\mathbb{R}$, and $\Psi$ is differentiable at every $u\in\mathbb{R}$ with $\Psi'(u)=\varpi(u)$.

3. (The regularised entropic rate cost.)(iii) Define $\phi:\mathbb{R}\to\mathbb{R}$ by

$$\phi(u)=\int_{[1,u]}\Psi(r)\,dr\ \ (u\ge1),\qquad \phi(u)=-\int_{[u,1]}\Psi(r)\,dr\ \ (u<1),$$

called the regularised entropic rate cost with cut-offs $\underline{a}$ and $\bar{a}$. Then $\phi$ is of class $C^{2}$ on $\mathbb{R}$, with $\phi'=\Psi$ and $\phi''=\varpi$ at every point of $\mathbb{R}$, and $\phi(1)=0$, $\phi'(1)=0$. It is the only function of class $C^{2}$ on $\mathbb{R}$ whose second derivative is $\varpi$ and which satisfies $\phi(1)=0$ and $\phi'(1)=0$.

4. (Sign, convexity and bounds.)(iv) $\phi(u)\ge0$ for every $u\in\mathbb{R}$, and $\phi(u)>0$ for every $u\neq1$; consequently $1$ is the unique minimiser of $\phi$ on $\mathbb{R}$. The function $\phi$ is convex on $\mathbb{R}$. Its derivatives satisfy $|\phi'(u)|\le2\,\bar{a}\,\underline{a}^{-1}$ and $0\le\phi''(u)\le\underline{a}^{-1}$ for every $u\in\mathbb{R}$, $\phi'$ is Lipschitz with constant $\underline{a}^{-1}$, and $\phi''$ is Lipschitz with constant $2\,\underline{a}^{-2}$.