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Proposition 16.28.2022685BEFF668

Proposition 16.28.20.22685BEFF668 For every $a\in \mathbb C^ n$ and $\tau >0$, the right derivative

\[ D_\tau (a)=\lim _{h\downarrow 0}\frac{F_{\tau +h}(a)-F_\tau (a)}{h} \]

exists and is finite. The difference quotients decrease as $h>0$ increases, and

89B57A4461F9 16.28.21
\begin{equation} V_ K^*(a)\leq \frac{F_{\tau +h}(a)-F_\tau (a)}h \leq D_\tau (a)\leq \frac{F_\tau (a)-m}{\tau }. \end{equation}

As $h\to \infty $, they converge to $V_ K^*(a)$, with the error bound

A8634DE3239C 16.28.22
\begin{equation} 0\leq \frac{F_{\tau +h}(a)-F_\tau (a)}h-V_ K^*(a) \leq \frac{M-m}h. \end{equation}

Moreover, $D_\tau (a)=V_ K^*(a)$ if and only if

29BB5B35B07B 16.28.23
\begin{equation} F_\delta (a)=F_\tau (a)+(\delta -\tau )V_ K^*(a) \qquad (\delta \geq \tau ). \end{equation}

Proof. For $0<h_1<h_2$, concavity of $F_\delta (a)$ gives

\[ F_{\tau +h_1}(a)-F_\tau (a) \geq \frac{h_1}{h_2}\bigl (F_{\tau +h_2}(a)-F_\tau (a)\bigr ). \]

Thus the difference quotients decrease with $h$. Lemma 16.28.18 bounds them between $V_ K^*(a)$ and $(F_\tau (a)-m)/\tau $, which proves the existence and finiteness of $D_\tau (a)$ and (16.28.21). Combining the lower bound for the increments with (16.28.17) gives (16.28.22), with a bound independent of $a$. If $D_\tau (a)=V_ K^*(a)$, then by (16.28.21) every difference quotient equals $V_ K^*(a)$, which gives (16.28.23). Conversely, that formula gives $D_\tau (a)=V_ K^*(a)$. $\square$

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