Appendix
This section contains the proofs of all the lemmas.
Proof of Lemma 3.3
Let us extend the function g to \({\mathbf {R}}^{n}\) as \(g({\mathbf {x}})=0\) for \({\mathbf {x}}\notin {\mathbf {R}}_{+}^{n} \) and the function \(\upsilon \) to \({\mathbf {R}}^{n}\times {\mathbf {R}}_{+}^{n}\) as \(\upsilon ({\mathbf {x}},\varvec{\alpha })=0\) for \(\left( {\mathbf {R}}^{n} \times {\mathbf {R}}_{+}^{n}\right) \diagup B\), where B is defined in (2.4). Using the expressions (2.1 ) and the change of variables formula for finite variation processes, and calling \({\mathbf {z}}_{s}={\mathbf {X}}_{s}^{{\mathbf {L}}}{}\) and \({\breve{\mathbf {z}} }_{s}={\check{\mathbf {X}}}_{s}^{{\mathbf {L}}}\), we can write
$$\begin{aligned} \begin{array} [c]{l} g({\mathbf {z}}_{\tau })e^{-c\tau }-g({\mathbf {x}})\\ \begin{array} [c]{ll} &{}= \int \nolimits _{0}^{\tau }\mathbf {p\cdot }\nabla g({\mathbf {z}}_{s-} )e^{-cs}ds-c\int \nolimits _{0}^{\tau }g({\mathbf {z}}_{s-})e^{-cs}ds\\ &{}\quad -\int \nolimits _{0}^{\tau }e^{-cs}\left( \nabla g({\mathbf {z}}_{s-} )\mathbf {\cdot }d{\mathbf {L}}_{s}^{c}\right) +\sum \limits _{{\mathbf {L}}_{s} \ne {\mathbf {L}}_{s-},~s\le \tau }\left( g({\mathbf {z}}_{s})-g({\breve{\mathbf {z}} }_{s})\right) e^{-cs}\\ &{} \quad + \sum \limits _{{\breve{\mathbf {z}}}_{s}\ne {\mathbf {z}}_{s-},~s \le \tau }\left( g({\breve{\mathbf {z}}}_{s})-g({\mathbf {z}}_{s-})\right) e^{-cs}. \end{array} \end{array} \end{aligned}$$
(7.1)
Note that \({\mathbf {z}}_{s}\in {\mathbf {R}}_{+}^{n}\) for \(s\le \tau \) except in the case that \(\tau =\tau ^{{\mathbf {L}}}\). Since \({\mathbf {z}}_{s}=\) \({\breve{\mathbf {z}}}_{s}-\Delta {\mathbf {L}}_{s},\)
$$\begin{aligned}&-\int \nolimits _{0}^{\tau }e^{-cs}\nabla g({\mathbf {z}}_{s-})\mathbf {\cdot }d{\mathbf {L}}_{s}^{c}+\sum \limits _{{\mathbf {L}}_{s}\ne {\mathbf {L}}_{s-},s\le \tau }\left( g({\mathbf {z}}_{s})-g({\breve{\mathbf {z}}}_{s})\right) e^{-cs}\nonumber \\&\quad = -\int \nolimits _{0}^{\tau }e^{-cs}\nabla g({\mathbf {z}}_{s-})\mathbf {\cdot }d{\mathbf {L}}_{s}^{c} -\sum \limits _{{\mathbf {L}}_{s}\ne {\mathbf {L}}_{s-},s\le \tau }e^{-cs}\left( \int \nolimits _{0}^{1}\left( \nabla g\left( {\breve{\mathbf {z}}}_{s}-\gamma \Delta {\mathbf {L}}_{s}\right) \mathbf {\cdot }\Delta {\mathbf {L}} _{s}\right) d\gamma \right) \nonumber \\&\quad = -\int _{0-}^{\tau }e^{-cs}{\mathbf {a}}\cdot d{\mathbf {L}}_{s}+\int \nolimits _{0}^{\tau }e^{-cs}\left( {\mathbf {a}}-\nabla g({\mathbf {z}} _{s-})\right) \mathbf {\cdot }d{\mathbf {L}}_{s}^{c}\nonumber \\&\qquad + \sum \limits _{{\mathbf {L}}_{s}\ne {\mathbf {L}}_{s-},s\le \tau }e^{-cs} \int \nolimits _{0}^{1}\left( {\mathbf {a}}-\nabla g\left( {\breve{\mathbf {z}}} _{s}-\gamma \Delta {\mathbf {L}}_{s}\right) \right) \mathbf {\cdot }\Delta {\mathbf {L}}_{s}d\gamma . \end{aligned}$$
(7.2)
Since
$$\begin{aligned} M_{1}(t)= & {} \sum \limits _{{\breve{\mathbf {z}}}\left( s-\right) \ne {\mathbf {z}} _{s-},s\le t}\left( g({\breve{\mathbf {z}}}_{s})-g({\mathbf {z}}_{s-})\right) e^{-cs}\nonumber \\&\quad -\lambda \int \limits _{0}^{t}e^{-cs}\int \limits _{{\mathbf {R}}_{+}^{n} }\left( g({\mathbf {z}}_{s-}-\varvec{\alpha })-g({\mathbf {z}}_{s-})\right) dF(\varvec{\alpha })ds \end{aligned}$$
(7.3)
and
$$\begin{aligned} M_{2}(t)=\sum \limits _{{\breve{\mathbf {z}}}\left( s-\right) \ne {\mathbf {z}} _{s-},s\le t}-\upsilon ({\breve{\mathbf {z}}}_{s-},{\mathbf {z}}(s-)-{\breve{\mathbf {z}}}_{s})e^{-cs}+\lambda \int \limits _{0}^{t}e^{-cs}\int \limits _{{\mathbf {R}} _{+}^{n}}\upsilon ({\mathbf {z}}_{s-},\varvec{\alpha })dF(\varvec{\alpha })ds\nonumber \\ \end{aligned}$$
(7.4)
are martingales with zero expectation, we have from (7.1) and (7.2)
$$\begin{aligned}{}\begin{array}[c]{l} (g({\mathbf {z}}_{\tau })I_{\{\tau <\tau ^{{\mathbf {L}}}\}}-\upsilon ({\mathbf {z}} _{\tau -},{\mathbf {z}}_{\tau -}-{\mathbf {z}}_{\tau })I_{\{\tau =\tau ^{{\mathbf {L}}} \}})e^{-c\tau }-g({\mathbf {x}})\\ \begin{array} [c]{ll} &{}= (g({\mathbf {z}}_{\tau })-\upsilon ({\mathbf {z}}_{\tau -},{\mathbf {z}}_{\tau -}-{\mathbf {z}}_{\tau }))e^{-c\tau }-g({\mathbf {x}})\\ &{}= \int \nolimits _{0}^{\tau }{\mathcal {L}}(g)({\mathbf {z}}_{s-})e^{-cs}ds-\int _{0-}^{\tau }e^{-cs}{\mathbf {a}}\cdot d{\mathbf {L}}_{s}\\ &{}\quad +\int \nolimits _{0}^{\tau }e^{-cs}\left( {\mathbf {a}}-\nabla g({\mathbf {z}} _{s-})\right) \mathbf {\cdot }d{\mathbf {L}}_{s}^{c}\\ &{}\quad + \sum \limits _{{\mathbf {L}}_{s}\ne {\mathbf {L}}_{s-},s\le \tau }e^{-cs} \int \nolimits _{0}^{1}\left( {\mathbf {a}}-\nabla g\left( {\breve{\mathbf {z}}} _{s}-\gamma \Delta {\mathbf {L}}_{s}\right) \mathbf {\cdot }\Delta {\mathbf {L}} _{s}\right) d\gamma +M(\tau ); \end{array} \end{array} \end{aligned}$$
where \(M(t)=M_{1}(t)+M_{2}(t)\). \(\square \)
In order to prove Lemma 3.6, we will use a technical lemma in which we construct a sequence of smooth functions that approximate a (possible non-smooth) viscosity supersolution. This is done in order to apply Lemma 3.3 to an approximate smooth function instead of the viscosity supersolution; we have to do that because the amount of time the controlled process spends at non-differentiable points of the viscosity supersolution could have positive Lebesgue measure. We omit the proof of this lemma because it is similar to the one-dimensional version given in Lemma 4.1 of [6]; the result is obtained by standard convolution arguments using that the function \({\mathcal {R}}\) is continuous.
Lemma 7.1
Fix \({\mathbf {x}}^{0}\) in the interior of \({\mathbf {R}}_{+}^{n}\) and let \({\overline{u}}\) be a supersolution of (3.1) satisfying the growth condition (2.14). We can find a sequence of functions \({\overline{u}} _{m}:{\mathbf {R}}_{+}^{n}\rightarrow {\mathbf {R}}\) such that:
(a) \({\overline{u}}_{m}\) is continuously differentiable and \({\overline{u}} _{m}\ge {\overline{u}}\ge f.\)
(b) \({\overline{u}}_{m}\) satisfies the growth condition (2.14).
(c) \({\mathbf {p}}{\mathbf {\cdot }}\nabla {\overline{u}}_{m}\) \(\le \left( c+\lambda \right) {\overline{u}}_{m}+\lambda \left| {\overline{u}}({\mathbf {0}})\right| +\lambda {\mathbb {E}}\left( \left| \upsilon ({\mathbf {0}},{\mathbf {U}} _{1})\right| \right) \) in \({\mathbf {R}}_{+}^{n}\) and \({\mathbf {a}} -\nabla {\overline{u}}_{m}\le {\mathbf {0}}\).
(d) \({\overline{u}}_{m}\) \(\searrow \) \({\overline{u}}\) uniformly on compact sets in \({\mathbf {R}}_{+}^{n}\) and \(\nabla {\overline{u}}_{m}\) converges to \(\nabla {\overline{u}}\) a.e. in \({\mathbf {R}}_{+}^{n}\).
(e) There exists a sequence \(c_{m}\) with \(\lim \limits _{m\rightarrow \infty }c_{m}=0\) such that
$$\begin{aligned} \sup \nolimits _{{\mathbf {x}}\in [{\mathbf {0}},{\mathbf {x}}^{0}]}{\mathcal {L}} ({\overline{u}}_{m})\left( {\mathbf {x}}\right) \le c_{m}. \end{aligned}$$
Proof of Lemma 3.6
Consider the processes \({\mathbf {z}}_{s}={\mathbf {X}}_{s}^{{\mathbf {L}}}{}\) defined in (2.3), let us call \(\tau =\tau ^{{\mathbf {L}}}\) and take \(\widetilde{\tau }={\overline{\tau }}\wedge \tau \). Let us consider the functions \({\overline{u}}_{m}\) defined in Lemma 7.1 in \({\mathbf {R}}_{+}^{n}\) . Using Lemma 3.3 for \({\widetilde{\tau }}\wedge t\), we get from Lemma 7.1 (a) and (c) that
$$\begin{aligned}{}\begin{array}[c]{l} {\overline{u}}_{m}({\mathbf {z}}_{t})e^{-ct}I_{\{t<{\widetilde{\tau }}\}} +e^{-c{\overline{\tau }}}f({\mathbf {z}}_{{\overline{\tau }}})I_{\{t\wedge {\widetilde{\tau }}={\overline{\tau }},{\overline{\tau }}<\tau \}}-e^{-c{\overline{\tau }} }\upsilon \left( {\mathbf {z}}_{\tau \mathbf {-}},{\mathbf {z}}_{\tau \mathbf {-} }-{\mathbf {z}}_{\tau }\right) I_{\{t\wedge {\widetilde{\tau }}=\tau \}}-\overline{u}_{m}({\mathbf {x}})\\ \begin{array} [c]{ll} &{}\le {\overline{u}}_{m}({\mathbf {z}}_{t})e^{-ct}I_{\{t<{\widetilde{\tau }} \}}+e^{-c{\overline{\tau }}}{\overline{u}}_{m}({\mathbf {z}}_{{\overline{\tau }} })I_{\{t\wedge {\widetilde{\tau }}={\overline{\tau }},{\overline{\tau }}<\tau \}}-e^{-c{\overline{\tau }}}\upsilon \left( {\mathbf {z}}_{\tau \mathbf {-} },{\mathbf {z}}_{\tau \mathbf {-}}-{\mathbf {z}}_{\tau }\right) I_{\{t\wedge {\widetilde{\tau }}=\tau \}}-{\overline{u}}_{m}({\mathbf {x}})\\ &{}\le \int \nolimits _{0}^{t\wedge {\widetilde{\tau }}}{\mathcal {L}}({\overline{u}} _{m})({\mathbf {z}}_{s-})e^{-cs}ds-\int _{0-}^{t\wedge {\widetilde{\tau }}} e^{-cs}{\mathbf {a}}\cdot d{\mathbf {L}}_{s}+M(t\wedge {\widetilde{\tau }}), \end{array} \end{array}\nonumber \\ \end{aligned}$$
(7.5)
where M(t) is a zero-expectation martingale. Since \({\mathbf {L}}_{s}\) is non-decreasing we get, using the monotone convergence theorem, that
$$\begin{aligned} \begin{array} [c]{l} \lim \limits _{t\rightarrow \infty }{\mathbb {E}}_{{\mathbf {x}}}\left( \int _{0-}^{t\wedge {\widetilde{\tau }}}e^{-cs}{\mathbf {a}}\cdot d{\mathbf {L}} _{s}+e^{-c{\overline{\tau }}}f({\mathbf {z}}_{{\overline{\tau }}})I_{\{t\wedge {\widetilde{\tau }}={\overline{\tau }},{\overline{\tau }}<\tau \}}-e^{-c{\overline{\tau }} }\upsilon \left( {\mathbf {z}}_{\tau \mathbf {-}},{\mathbf {z}}_{\tau \mathbf {-} }-{\mathbf {z}}_{\tau }\right) I_{\{t\wedge {\widetilde{\tau }}=\tau \}}\right) \\ =J(\pi ;{\mathbf {x}}). \end{array} \end{aligned}$$
From Lemma 7.1(c), we have
$$\begin{aligned}&-\left( c+\lambda \right) {\overline{u}}_{m}({\mathbf {x}})+{\overline{u}} _{m}(0)\lambda F({\mathbf {x}})-\lambda {\mathbb {E}}\left( \left| \upsilon ({\mathbf {0}},{\mathbf {U}}_{1})\right| \right) \nonumber \\&\quad \le {\mathcal {L}} ({\overline{u}}_{m})({\mathbf {x}})\le \lambda {\overline{u}}_{m}({\mathbf {x}} )+\lambda \left| {\overline{u}}({\mathbf {0}})\right| +\lambda {\mathbb {E}}\left( \left| \upsilon ({\mathbf {0}},{\mathbf {U}}_{1})\right| \right) -{\mathcal {R}}({\mathbf {x}}). \end{aligned}$$
(7.6)
By Lemma 7.1(b), (c) and the inequality \({\mathbf {z}}_{s}\le {\mathbf {x}}+{\mathbf {p}}s,\) there exists \(d_{0}\) large enough such that
$$\begin{aligned} {\overline{u}}_{m}({\mathbf {z}}_{s})\le {\overline{u}}_{m}({\mathbf {x}}+{\mathbf {p}} s)\le d_{0}e^{\frac{c}{2n}\sum _{i=1}^{n}\frac{x_{i}+p_{i}s}{p_{i}}} =d_{0}h_{0}({\mathbf {x}})e^{\frac{c}{2}s} \end{aligned}$$
(7.7)
and
$$\begin{aligned} -\upsilon ({\mathbf {z}}_{s-},\varvec{\alpha })\le S({\mathbf {z}}_{s-})\le d_{0}h_{0}({\mathbf {x}})e^{\frac{c}{2}s}\text { for }\left( {\mathbf {z}} _{s-}-\varvec{\alpha }\right) \notin {\mathbf {R}}_{+}^{n}, \end{aligned}$$
(7.8)
where \(h_{0}\) and S are defined in (2.15) and Proposition 2.4 respectively. Therefore, from (7.6), we obtain that there exists \(d_{1}\) large enough such that,
$$\begin{aligned} e^{-cs}\left| {\mathcal {L}}({\overline{u}}_{m})\left( {\mathbf {z}}_{s-}\right) \right| \le d_{1}e^{-\frac{c}{2}s}. \end{aligned}$$
(7.9)
And using the bounded convergence theorem,
$$\begin{aligned} \lim \limits _{t\rightarrow \infty }{\mathbb {E}}_{{\mathbf {x}}}\left( \int \nolimits _{0}^{t\wedge {\widetilde{\tau }}}{\mathcal {L}}({\overline{u}}_{m} )({\mathbf {z}}_{s-})e^{-cs}ds\right) ={\mathbb {E}}_{{\mathbf {x}}}\left( \int \nolimits _{0}^{{\widetilde{\tau }}}{\mathcal {L}}({\overline{u}}_{m} )({\mathbf {z}}_{s-})e^{-cs}ds\right) .\qquad \end{aligned}$$
(7.10)
From (7.5) and (7.10), we get
$$\begin{aligned} \lim \limits _{t\rightarrow \infty }{\mathbb {E}}_{{\mathbf {x}}}\left( \overline{u}_{m}({\mathbf {z}}_{t})e^{-ct}I_{\{t<{\widetilde{\tau }}\}}\right) -\overline{u}_{m}({\mathbf {x}})\le {\mathbb {E}}_{{\mathbf {x}}}\left( \int \nolimits _{0} ^{{\widetilde{\tau }}}{\mathcal {L}}({\overline{u}}_{m})({\mathbf {z}}_{s-} )e^{-cs}ds\right) -J(\pi ;{\mathbf {x}}).\nonumber \\ \end{aligned}$$
(7.11)
By (7.7),
$$\begin{aligned} \lim \limits _{t\rightarrow \infty }{\mathbb {E}}_{{\mathbf {x}}}\left( \overline{u}_{m}({\mathbf {z}}_{t})e^{-ct}I_{\{t<{\widetilde{\tau }}\}}\right) =0. \end{aligned}$$
(7.12)
Let us prove now that
$$\begin{aligned} \limsup \limits _{m\rightarrow \infty }{\mathbb {E}}_{{\mathbf {x}}}\left( \int \nolimits _{0}^{{\widetilde{\tau }}}{\mathcal {L}}({\overline{u}}_{m} )({\mathbf {z}}_{s-})e^{-cs}ds\right) \le 0. \end{aligned}$$
(7.13)
Given any \(\varepsilon >0\), from (7.9), we can find T large enough such that
$$\begin{aligned} {\mathbb {E}}_{{\mathbf {x}}}\left( \int \nolimits _{T\wedge {\widetilde{\tau }} }^{{\widetilde{\tau }}}\left| {\mathcal {L}}({\overline{u}}_{m})({\mathbf {z}} _{s-})\right| e^{-cs}ds\right) \le \frac{2d_{1}}{c}\left( e^{-\frac{c}{2} T}\right) <\frac{\varepsilon }{2}. \end{aligned}$$
(7.14)
For \(s\le T\), we get \({\mathbf {z}}_{s-}\in [{\mathbf {0}},\) \({\mathbf {x}} +{\mathbf {p}}T\) ] , then from Lemma 7.1(e) we can find \(m_{0}\) large enough such that for any \(m\ge m_{0}\)
$$\begin{aligned} \int \nolimits _{0}^{T}{\mathcal {L}}({\overline{u}}_{m})({\mathbf {z}}_{s-} )e^{-cs}ds\le c_{m}\int \nolimits _{0}^{T}e^{-cs}ds\le \frac{c_{m}}{c}\le \frac{\varepsilon }{2} \end{aligned}$$
and so we have (7.13). Thus, from (7.11) and using (7.12) and (7.13), we obtain
$$\begin{aligned} {\overline{u}}({\mathbf {x}})=\lim \nolimits _{m\rightarrow \infty }{\overline{u}} _{m}({\mathbf {x}})\ge J(\pi ;{\mathbf {x}})\text {. } \end{aligned}$$
(7.15)
\(\square \)
Proof of Lemma 4.2
Suppose that \(\tilde{k}=\infty \), calling
$$\begin{aligned} k_{l}:={\mathbf {m}}{\mathbf {\cdot }}{\mathbf {1}}+(l-1)n+1, \end{aligned}$$
there are at least \(i_{l}\ge l\) control actions \({\mathbf {E}}_{0}\) in \(\left( s_{1},s_{2},\ldots ,s_{k_{l}}\right) \). Let us consider the non-decreasing sequence \((j_{l})_{l}\) defined as
$$\begin{aligned} j_{l}:=\max \{j:\tau _{j}\le t_{k_{l}}\}, \end{aligned}$$
we have that \(t_{k_{l}}\ge \tau _{j_{l}}+(i_{l}-j_{l})\delta \). If \(\lim _{l\rightarrow \infty }i_{l}-j_{l}=\infty \), then
$$\begin{aligned} \lim \nolimits _{l\rightarrow \infty }t_{k_{l}}\ge \lim \nolimits _{l\rightarrow \infty }\tau _{j_{l}}+(i_{l}-j_{l})\delta \ge \lim \nolimits _{l\rightarrow \infty }(i_{l}-j_{l})\delta =\infty ; \end{aligned}$$
if not, \(\lim _{l\rightarrow \infty }j_{l}=\infty \) and so
$$\begin{aligned} \lim \nolimits _{l\rightarrow \infty }t_{k_{l}}\ge \lim \nolimits _{l\rightarrow \infty }\tau _{j_{l}}+(i_{l}-j_{l})\delta \ge \lim \nolimits _{l\rightarrow \infty }\tau _{j_{l}} \end{aligned}$$
and since \(\lim _{l\rightarrow \infty }\tau _{j_{l}}=\) \(\lim _{i\rightarrow \infty }\tau _{i}=\) \(\infty \) a.s., we have the result. \(\square \)
Proof of Lemma 4.4
It is straightforward that \(T_{0}\), \(T_{i},\) \(T_{s}\) and T are non-decreasing and that
$$\begin{aligned} \sup \nolimits _{{\mathbf {m}}\in {\mathbf {N}}_{0}^{n}}\left| T(w_{1} )({\mathbf {m}})-T(w_{2})({\mathbf {m}})\right| \le \sup \nolimits _{{\mathbf {m}} \in {\mathbf {N}}_{0}^{n}}\left| w_{1}({\mathbf {m}})-w_{2}({\mathbf {m}} )\right| . \end{aligned}$$
Also, given a function \(w:{\mathbf {N}}_{0}^{n}\rightarrow {\mathbf {R}}\) it is immediate to see that \(T_{i}(w)\) and \(T_{s}(w)\) can be written as a linear combination of the values of \(w({\mathbf {m}})\) plus a constant. Let us prove now that
$$\begin{aligned} T_{0}(w)({\mathbf {m}})=e^{-(c+\lambda )\delta }w({\mathbf {m}}+{\mathbf {1}} )+\sum \limits _{0\le {\mathbf {k}}\le {\mathbf {m}}}a_{1}({\mathbf {k}},{\mathbf {m}} )w({\mathbf {k}})+a_{2}({\mathbf {m}})\text {,} \end{aligned}$$
Lemma 7.2
where
$$\begin{aligned} \begin{array} [c]{lll} a_{1}({\mathbf {k}},{\mathbf {m}}) &{} = &{} I_{\{{\mathbf {k}}\le {\mathbf {m}}{-}{\mathbf {1}} \}}\int \limits _{0}^{\delta }\lambda e^{-(c+\lambda )t}(F(g^{\delta }\left( {\mathbf {m}}-{\mathbf {k}}\right) {+}t{\mathbf {p}})-F(g^{\delta }\left( {\mathbf {m}} -{\mathbf {k}}-{\mathbf {1}}\right) +t{\mathbf {p}}))dt \\ &{} &{} \quad +I_{\{{\mathbf {k}}\le {\mathbf {m}},{\mathbf {k}}\nleqslant {\mathbf {m}} -{\mathbf {1}}\}}\int \limits _{0}^{\delta }\lambda e^{-(c+\lambda )t}(F(g^{\delta }\left( {\mathbf {m}}-{\mathbf {k}}\right) \\ &{}&{}\quad +t{\mathbf {p}})-F({\mathbf {0}}\vee \left( g^{\delta }\left( {\mathbf {m}}-{\mathbf {k}}\right) +t{\mathbf {p}}\right) ))dt~ \end{array} \end{aligned}$$
and
$$\begin{aligned} \begin{array} [c]{lll} a_{2}({\mathbf {m}}) &{} = &{} \sum \limits _{0\le {\mathbf {k}}<{\mathbf {m}}-{\mathbf {1}} }\int \limits _{0}^{\delta }\left( \lambda e^{-(c+\lambda )t} {\textstyle \int \limits _{g^{\delta }\left( {\mathbf {m}}-{\mathbf {k}}-{\mathbf {1}} \right) +t{\mathbf {p}}}^{g^{\delta }\left( {\mathbf {m}}-{\mathbf {k}}\right) +t{\mathbf {p}}}} {\mathbf {a}}\cdot (g^{\delta }\left( {\mathbf {m}}-{\mathbf {k}}\right) +t{\mathbf {p}} -\varvec{\alpha })dF(\varvec{\alpha })\right) dt\\ &{} &{} +\sum \limits _{{\mathbf {k}}\le {\mathbf {m}},{\mathbf {k}}\nleqslant {\mathbf {m}}-{\mathbf {1}}}\int \limits _{0}^{\delta }\left( \lambda e^{-(c+\lambda )t} {\textstyle \int \limits _{{\mathbf {0}}\vee \left( g^{\delta }\left( {\mathbf {m}} -{\mathbf {k}}\right) +t{\mathbf {p}}\right) }^{g^{\delta }\left( {\mathbf {m}} -{\mathbf {k}}\right) +t{\mathbf {p}}}} {\mathbf {a}}\cdot (g^{\delta }\left( {\mathbf {m}}-{\mathbf {k}}\right) +t{\mathbf {p}} -\varvec{\alpha })dF(\varvec{\alpha })\right) dt\\ &{} &{} -\int \limits _{0}^{\delta }e^{-(c+\lambda )t}{\mathcal {R}}(g^{\delta }({\mathbf {m}})+t{\mathbf {p}})dt. \end{array} \end{aligned}$$
Given \({\mathbf {m}}\in {\mathbf {N}}_{0}^{n}\), \(\varvec{\alpha } \in {\mathbf {R}}_{+}^{n}\) and \(0<t\le \delta \) such that \({\mathbf {0}}\le g^{\delta }({\mathbf {m}})+t{\mathbf {p}}-\varvec{\alpha }\), let us define
$$\begin{aligned} {\mathbf {k}}:=\rho ^{\delta }(g^{\delta }({\mathbf {m}})+t{\mathbf {p}}-\varvec{\alpha }), \end{aligned}$$
and so \({\mathbf {k}}\le {\mathbf {m}}.\)
If \({\mathbf {k}}\le {\mathbf {m}}-{\mathbf {1}}\),
$$\begin{aligned} g^{\delta }({\mathbf {k}})\le g^{\delta }({\mathbf {m}})+t{\mathbf {p}}-\varvec{\alpha }<g^{\delta }\left( {\mathbf {k}}+{\mathbf {1}}\right) \le g^{\delta }({\mathbf {m}}) \end{aligned}$$
that implies
$$\begin{aligned} {\mathbf {0}}<g^{\delta }\left( {\mathbf {m}}-{\mathbf {k}}-{\mathbf {1}}\right) +t{\mathbf {p}}<\varvec{\alpha }\le g^{\delta }\left( {\mathbf {m}}-{\mathbf {k}} \right) +t{\mathbf {p}}. \end{aligned}$$
If \({\mathbf {k}}\le {\mathbf {m}}\) with \({\mathbf {k}}\nleqslant {\mathbf {m}}-{\mathbf {1}} \),
$$\begin{aligned} g^{\delta }({\mathbf {k}})\le g^{\delta }({\mathbf {m}})+t{\mathbf {p}}-\varvec{\alpha }<g^{\delta }\left( {\mathbf {k}}+{\mathbf {1}}\right) \wedge \left( g^{\delta }({\mathbf {m}})+t{\mathbf {p}}\right) \end{aligned}$$
and so
$$\begin{aligned} \left( g^{\delta }\left( {\mathbf {m}}-{\mathbf {k}}-{\mathbf {1}}\right) +t{\mathbf {p}}\right) \vee {\mathbf {0}}<\varvec{\alpha }\le g^{\delta }\left( {\mathbf {m}}-{\mathbf {k}}\right) +t{\mathbf {p}}. \end{aligned}$$
Then, we can write
$$\begin{aligned} \begin{array} [c]{l} {\mathcal {I}}^{\delta }(w)({\mathbf {m}})\\ \begin{array} [c]{ll} &{}= \sum \limits _{0\le {\mathbf {k}}\le {\mathbf {m}}-{\mathbf {1}}}w({\mathbf {k}})\int _{0}^{\delta }\lambda e^{-(c+\lambda )t}\left( {\textstyle \int \nolimits _{g^{\delta }\left( {\mathbf {m}}-{\mathbf {k}} -{\mathbf {1}}\right) +t{\mathbf {p}}}^{g^{\delta }\left( {\mathbf {m}}-{\mathbf {k}} \right) +t{\mathbf {p}}}} dF(\varvec{\alpha })\right) dt\\ &{}\quad +\sum \limits _{0\le {\mathbf {k}}\le {\mathbf {m}}-{\mathbf {1}}}\int _{0}^{\delta }\lambda e^{-(c+\lambda )t}\left( {\textstyle \int \nolimits _{g^{\delta }\left( {\mathbf {m}}-{\mathbf {k}} -{\mathbf {1}}\right) +t{\mathbf {p}}}^{g^{\delta }\left( {\mathbf {m}}-{\mathbf {k}} \right) +t{\mathbf {p}}}} {\mathbf {a}}\cdot \left( g^{\delta }({\mathbf {m}}-{\mathbf {k}})+t{\mathbf {p}} -\varvec{\alpha }\right) dF(\varvec{\alpha })\right) dt\\ &{}\quad +\sum \limits _{{\mathbf {k}}\le {\mathbf {m}},{\mathbf {k}}\nleqslant {\mathbf {m}}-{\mathbf {1}} }w({\mathbf {k}})\int _{0}^{\delta }\lambda e^{-(c+\lambda )t}\left( {\textstyle \int \nolimits _{\left( g^{\delta }\left( {\mathbf {m}}-{\mathbf {k}} -{\mathbf {1}}\right) +t{\mathbf {p}}\right) \vee {\mathbf {0}}}^{g^{\delta }\left( {\mathbf {m}}-{\mathbf {k}}\right) +t{\mathbf {p}}}} dF(\varvec{\alpha })\right) dt\\ &{}\quad +\sum \limits _{{\mathbf {k}}\le {\mathbf {m}},{\mathbf {k}}\nleqslant {\mathbf {m}}-{\mathbf {1}} }\int _{0}^{\delta }\lambda e^{-(c+\lambda )t}\left( {\textstyle \int \nolimits _{\left( g^{\delta }\left( {\mathbf {m}}-{\mathbf {k}} -{\mathbf {1}}\right) +t{\mathbf {p}}\right) \vee {\mathbf {0}}}^{g^{\delta }\left( {\mathbf {m}}-{\mathbf {k}}\right) +t{\mathbf {p}}}} {\mathbf {a}}\cdot \left( g^{\delta }({\mathbf {m}}-{\mathbf {k}})+t{\mathbf {p}} -\varvec{\alpha }\right) dF(\varvec{\alpha })\right) dt. \end{array} \end{array} \end{aligned}$$
Therefore, from (4.2), we have the result. \(\square \)
Proof of Lemma 4.6
The proof of this lemma is a discrete version of the one of Lemma 3.6. Assume that \(\pi =({\mathbf {L}},{\overline{\tau }})\in \Pi _{g^{\delta }({\mathbf {m}})}^{\delta }\). For any \(\omega =(\tau _{i},{\mathbf {U}}_{i})_{i\ge 1}\), consider the sequence \({\mathbf {s}}=(s_{k})_{k=1,\ldots ,{\tilde{k}}}\) with \(s_{k}\in {\mathcal {E}}\) corresponding to \(\pi \) and \({\mathbf {m}}^{k}\), \({\mathbf {y}}^{k}\) and times \(t_{k}\) and \(\Delta _{k}\) as defined in Section 4. Let \(\left( \kappa _{l}\right) _{l\ge 1}\) be the indices of the sequence \({\mathbf {s}}=(s_{k})_{k=1,\ldots ,{\tilde{k}}}\) where \(s_{k}\) is either \({\mathbf {E}}_{s}\) or \({\mathbf {E}}_{0}\). If the sequence stops at \({\tilde{k}}=\kappa _{l_{0}}<\infty \), we define
$$\begin{aligned} \kappa _{l}=\kappa _{l_{0}}\text { for }l\ge l_{0},\text { }t_{\kappa _{l_{0}+j} }=t_{\kappa _{l_{0}}}+\Delta _{\kappa _{l_{0}}}\text { for }j\ge 1; \end{aligned}$$
and if \({\tilde{k}}=\infty \) we put \(l_{0}=\infty \). Consider the case in which the process goes to ruin at \(\kappa _{l}\), that is \({\mathbf {y}}^{\kappa _{l} }\notin {\mathbf {R}}_{+}^{n}\); then the surplus prior to the ruin is \({\mathbf {y}}^{\kappa _{l}}+{\mathbf {U}}\) and the penalty paid at ruin is \(\upsilon ({\mathbf {y}}^{\kappa _{l}}+{\mathbf {U}},{\mathbf {U}})\), where \({\mathbf {U}} \) is the last jump of the uncontrolled process. So we define, for \(l\ge 1\),
$$\begin{aligned} H(l)=w({\mathbf {m}}^{1+\kappa _{l}})I_{\{s_{\kappa _{l}}={\mathbf {E}}_{0} \}}I_{\{{\mathbf {y}}^{\kappa _{l}}\in {\mathbf {R}}_{+}^{n}\}}-\upsilon ({\mathbf {y}}^{\kappa _{l}}+{\mathbf {U}},{\mathbf {U}})I_{\{s_{\kappa _{l}} ={\mathbf {E}}_{0}\}}I_{\{{\mathbf {y}}^{\kappa _{l}}\notin {\mathbf {R}}_{+}^{n} \}}+f(g^{\delta }\left( {\mathbf {m}}^{\kappa _{l}}\right) )I_{\{s_{\kappa _{l} }={\mathbf {E}}_{s}\}}\text {.} \end{aligned}$$
If we put \(H(0)=w({\mathbf {m}})\), \(\kappa _{0}=0\) and \(t_{0}=0\), we have using \(\left( T_{i}(w)-w\right) _{i=1,\ldots ,n}\le 0,\)
$$\begin{aligned} \begin{array} [c]{lll} e^{-ct_{\kappa _{l+1}}}H(l)-w({\mathbf {m}}) &{} = &{} \sum \limits _{j=1}^{l}\left( e^{-ct_{\kappa _{j+1}}}H(j)-e^{-ct_{\kappa _{j}}}H(j-1)\right) \\ &{} = &{} \sum \limits _{j=1}^{l}I_{\{\kappa _{j+1}\ne \kappa _{j}\}}\left( e^{-ct_{\kappa _{j+1}} }H(j)-e^{-ct_{\kappa _{j}}}H(j-1)\right) \\ &{} = &{} \sum \limits _{j=1}^{l}I_{\{\kappa _{j+1}\ne \kappa _{j}\}}\left( e^{-ct_{1+\kappa _{j-1} }}\left( \sum \nolimits _{k=1+\kappa _{j-1}}^{\kappa _{j}-1}\left( w({\mathbf {m}}^{k+1} )-w({\mathbf {m}}^{k})\right) \right) \right) \\ &{} &{} +\sum \limits _{j=1}^{l}I_{\{\kappa _{j+1}\ne \kappa _{j}\}}\left( e^{-ct_{\kappa _{j+1}} }H(j)-e^{-ct_{\kappa _{j}}}w({\mathbf {m}}^{\kappa _{j}})\right) \\ &{} \le &{} \sum \limits _{j=1}^{l}I_{\{\kappa _{j+1}\ne \kappa _{j}\}}\left( \sum \limits _{k=1+\kappa _{j-1}}^{\kappa _{j}-1}e^{-ct_{1+\kappa _{j-1}}}\left( \sum \limits _{i=1}^{n}\left( -a_{i}p_{i}\delta \right) I_{\{s_{k}={\mathbf {E}}_{i}\}}\right) \right) \\ &{} &{} +\sum \limits _{j=1}^{l}I_{\{\kappa _{j+1}\ne \kappa _{j}\}}\left( e^{-ct_{\kappa _{j+1}} }H(j)-e^{-ct_{\kappa _{j}}}w({\mathbf {m}}^{\kappa _{j}})\right) ; \end{array} \end{aligned}$$
(7.16)
and since \(T_{0}(w)-w\le 0\) and \(T_{s}(w)-w\le 0,\) if \(\kappa _{j+1}\ne \kappa _{j}\),
$$\begin{aligned} \begin{array} [c]{l} {\mathbb {E}}\left( \left. e^{-ct_{\kappa _{j+1}}}H(j)-e^{-ct_{\kappa _{j}} }w({\mathbf {m}}^{\kappa _{j}})\right| {\mathcal {F}}_{t_{\kappa _{j}}}\right) \\ \begin{array}[c]{ll} &{}= {\mathbb {E}}\left( \left. (e^{-ct_{\kappa _{j+1}}}H(j)-e^{-ct_{\kappa _{j}} }w({\mathbf {m}}^{\kappa _{j}}))I_{\{s_{\kappa _{j}} ={\mathbf {E}}_{0}\}}\right| {\mathcal {F}}_{t_{\kappa _{j}}}\right) \\ &{}\quad +\,I_{\{s_{\kappa _{j}}={\mathbf {E}}_{s} \}}e^{-ct_{\kappa _{j}}}\left( f(g^{\delta }({\mathbf {m}}^{\kappa _{j} }))-w({\mathbf {m}}^{\kappa _{j}})\right) \\ &{}\le {\mathbb {E}}\left( \left. e^{-ct_{\kappa _{j+1}}}I_{\{s_{\kappa _{j} }={\mathbf {E}}_{0}\}}(w({\mathbf {m}}^{1+\kappa _{j}})I_{\{{\mathbf {y}}^{\kappa _{j} }\in {\mathbf {R}}_{+}^{n}\}}-\upsilon ({\mathbf {y}}^{\kappa _{j}}+{\mathbf {U}} ,{\mathbf {U}})I_{\{{\mathbf {y}}_{\kappa _{j}}\notin {\mathbf {R}}_{+}^{n} \}})\right| {\mathcal {F}}_{t_{\kappa _{j}}}\right) \\ &{}\quad -\,e^{-ct_{\kappa _{j}}}w({\mathbf {m}}^{\kappa _{j}})I_{\{s_{\kappa _{j} }={\mathbf {E}}_{0}\}}\\ &{}= e^{-ct_{\kappa _{j}}}I_{\{s_{\kappa _{j}}={\mathbf {E}}_{0}\}}\left( T_{0}(w)\left( {\mathbf {m}}^{\kappa _{j}}\right) -w({\mathbf {m}}^{\kappa _{j} })\right) \\ &{}\quad -\,e^{-ct_{\kappa _{j}}}I_{\{s_{\kappa _{j}}={\mathbf {E}}_{0}\}}\int \limits _{0}^{\delta } {\textstyle \int \limits _{\varvec{\alpha }\in [{\mathbf {0}},{\mathbf {z}} _{j}(t)]}} \lambda e^{-(c+\lambda )t}{\mathbf {a}}\cdot \left( {\mathbf {z}}_{j} (t)-\varvec{\alpha }-\left\langle {\mathbf {z}}_{j}(t)-\varvec{\alpha }\right\rangle ^{\delta }\right) dF(\varvec{\alpha })dt\\ &{}\le -e^{-ct_{\kappa _{j}}}I_{\{s_{\kappa _{j}}={\mathbf {E}}_{0}\}} \int \limits _{0}^{\delta } {\textstyle \int \limits _{\varvec{\alpha }\in [{\mathbf {0}},{\mathbf {z}} _{j}(t)]}} \lambda e^{-(c+\lambda )t}{\mathbf {a}}\cdot \left( {\mathbf {z}}_{j} (t)-\varvec{\alpha }-\left\langle {\mathbf {z}}_{j}(t)-\varvec{\alpha }\right\rangle ^{\delta }\right) dF(\varvec{\alpha })dt\text {,} \end{array} \end{array} \end{aligned}$$
(7.17)
where \({\mathbf {z}}_{j}(t)=g^{\delta }({\mathbf {m}}^{\kappa _{j}})+t{\mathbf {p}}\). From (7.16) and (7.17), and calling the initial surplus \({\mathbf {x}}=g^{\delta }({\mathbf {m}})\in {\mathcal {G}}^{\delta }\) we have,
$$\begin{aligned} \lim \sup _{l\rightarrow \infty }{\mathbb {E}}_{{\mathbf {x}}}\left( e^{-ct_{\kappa _{l+1}}}H(l)-w({\mathbf {m}})\right) \le -{\mathbb {E}}_{{\mathbf {x}}}\left( \int _{0-}^{{\overline{\tau }}\wedge \tau _{L}}e^{-cs}{\mathbf {a}}\cdot d{\mathbf {L}} _{s}\right) . \end{aligned}$$
Then,
$$\begin{aligned} w({\mathbf {m}})\ge J(\pi ;g^{\delta }({\mathbf {m}}))+\lim \sup _{l\rightarrow \infty }{\mathbb {E}}_{{\mathbf {x}}}\left( I_{\{l\le l_{0}\}}e^{-ct_{1+\kappa _{l}} }w({\mathbf {m}}^{1+\kappa _{l}})I_{\{{\mathbf {y}}^{\kappa _{l}}\in {\mathbf {R}}_{+} ^{n}\}}\right) . \end{aligned}$$
Since
$$\begin{aligned} g^{\delta }({\mathbf {m}}^{1+\kappa _{l}})\le g^{\delta }\left( {\mathbf {m}} +\rho ^{\delta }(t_{1+\kappa _{l}}{\mathbf {p}})\right) \end{aligned}$$
and w satisfies the growth condition (4.7), there exists d large enough such that
$$\begin{aligned}&\lim \sup _{l\rightarrow \infty }\left( {\mathbb {E}}_{{\mathbf {x}}}I_{\{l\le l_{0}\}}e^{-ct_{1+\kappa _{l}}}w({\mathbf {m}}^{1+\kappa _{l}})I_{\{{\mathbf {y}} ^{\kappa _{l}}\in {\mathbf {R}}_{+}^{n}\}}\right) \\&\quad \le d\lim _{l\rightarrow \infty }{\mathbb {E}}_{{\mathbf {x}}}\left( I_{\{l\le l_{0}\}}e^{-ct_{1+\kappa _{l}} }e^{c\delta {\mathbf {m}}{\mathbf {\cdot }}{\mathbf {1}}/\left( 2n\right) }e^{\frac{c}{2}t_{1+\kappa _{l} }})\right) =0; \end{aligned}$$
so we have the result. \(\square \)
Proof of Lemma 5.1
(1) Take the \({\mathcal {G}}^{\delta }\)-optimal strategy \(\pi _{g^{\delta }({\mathbf {m}})}^{\delta }\in \Pi _{g^{\delta }({\mathbf {m}})}^{\delta }\) and define \({\overline{\pi }}_{g^{\delta }({\mathbf {m}}+{\mathbf {e}}_{i})}\in \Pi _{g^{\delta }({\mathbf {m}}+{\mathbf {e}}_{i})}^{\delta }\) by applying first the control action \({\mathbf {E}}_{i}\) and then the \({\mathcal {G}}^{\delta }\)-optimal strategy \(\pi _{g^{\delta }({\mathbf {m}})}^{\delta }\). The value function of this strategy is given by
$$\begin{aligned} a_{i}p_{i}\delta +v^{\delta }({\mathbf {m}}), \end{aligned}$$
so we obtain the the first inequality of this proposition. Now, take the \({\mathcal {G}}^{\delta }\)-optimal strategy \(\pi _{g^{\delta }\left( {\mathbf {m}} +{\mathbf {1}}\right) }^{\delta }\in \Pi _{g^{\delta }\left( {\mathbf {m}} +{\mathbf {1}}\right) }^{\delta }\) and define \({\overline{\pi }}_{g^{\delta }({\mathbf {m}})}\in \Pi _{g^{\delta }({\mathbf {m}})}^{\delta }\) by applying first the control action \({\mathbf {E}}_{0}\) and then the \({\mathcal {G}}^{\delta }\)-optimal strategy \(\pi _{g^{\delta }\left( {\mathbf {m}}+{\mathbf {1}}\right) }^{\delta }\). Hence, we obtain the second inequality from
$$\begin{aligned} v^{\delta }\left( {\mathbf {m}}+{\mathbf {1}}\right) e^{-(c+\lambda )\delta }\le T_{0}(v^{\delta })\left( {\mathbf {m}}\right) \le T(v^{\delta })\left( {\mathbf {m}}\right) =v^{\delta }\left( {\mathbf {m}}\right) . \end{aligned}$$
(2) In order to avoid any confusion, in the remainder of the proof we put a superindex \(\delta \) to the control actions in \({\mathcal {G}}^{\delta }\). Note first that given any surplus in \({\mathbf {R}}_{+}^{n}\), the strategy of paying dividends in such a way that the surplus goes to the nearest smaller point in \({\mathcal {G}}^{2\delta }\) corresponds to go first to the nearest smaller point in \({\mathcal {G}}^{\delta }\) and then to apply (possibly) a combination of control actions \({\mathbf {E}}_{i}^{\delta \prime }s \). Consider \(\pi _{2g^{\delta }({\mathbf {m}})}\in \Pi _{g^{2\delta }\left( {\mathbf {m}}\right) }^{2\delta }\) given by the random sequence \({\mathbf {s}}=(s_{k})_{k=1,\ldots ,{\tilde{k}}}\) with
$$\begin{aligned} s_{k}\in {\mathcal {E}}^{2\delta }=\left\{ {\mathbf {E}}_{s}^{2\delta },\left( {\mathbf {E}}_{i}^{2\delta }\right) _{i=1,\ldots ,n},{\mathbf {E}}_{0}^{2\delta }\right\} . \end{aligned}$$
We can see that \(\pi _{2g^{\delta }({\mathbf {m}})}\) also belongs to \(\Pi _{2g^{\delta }({\mathbf {m}})}^{\delta }\) rewriting the sequence as follows: If \(s_{k}={\mathbf {E}}_{i}^{2\delta }\), we replace it by the pair \({\mathbf {E}} _{i}^{\delta }\mathbf {,E}_{i}^{\delta }\); if \(s_{k}={\mathbf {E}}_{s}^{2\delta } \), we replace it by \({\mathbf {E}}_{s}^{\delta }\); and if \(s_{k} ={\mathbf {E}}_{0}^{2\delta }\), we replaces it
-
either by \({\mathbf {E}}_{0}^{\delta },{\mathbf {E}}_{0}^{\delta }\) if the next jump in the uncontrolled process arrives at time \(\tau>\) \(2\delta ;\)
-
or by \({\mathbf {E}}_{0}^{\delta },{\mathbf {E}}_{0}^{\delta }\mathbf {,}\) and a possible combination of \({\mathbf {E}}_{i}^{\delta \prime }s\), if it arrives at time \(\tau \in \) \((\delta \),\(2\delta ]\), so the surplus goes to the nearest smaller point in \({\mathcal {G}}^{2\delta }\);
-
or by \({\mathbf {E}}_{0}^{\delta }\), and a possible combination of \({\mathbf {E}}_{i}^{\delta \prime }s\), if it arrives at time \(\tau \le \) \(\delta \), so again the surplus goes to the nearest smaller point in \({\mathcal {G}} ^{2\delta }\).
So we have the result.
\(\square \)
Proof of Lemma 5.3
Let us first prove that
$$\begin{aligned} \begin{array} [c]{l} \left| V^{\delta _{k}}({\mathbf {y}})-V^{\delta _{k}}({\mathbf {x}})\right| \\ \le \frac{2}{{\hat{p}}}V^{\delta _{k}}\left( \left\langle {\mathbf {x}}\vee {\mathbf {y}} \right\rangle ^{\delta _{k}}\right) \left( \frac{e^{(c+\lambda )\delta _{k}}-1}{\delta _{k} }\right) \left\| \left\langle {\mathbf {y}}\right\rangle ^{\delta _{k}}-\left\langle {\mathbf {x}}\right\rangle ^{\delta _{k}}\right\| _{1}+2\delta _{k} {\mathbf {a}}\cdot {\mathbf {p}}, \end{array} \end{aligned}$$
(7.18)
for any \({\mathbf {x}}\) and \({\mathbf {y}}\) in \({\mathbf {R}}_{+}^{n}\). Let us assume first that \({\mathbf {y}}>{\mathbf {x}}\). We have from Lemma 5.1,
$$\begin{aligned}&V^{\delta _{k}}(g^{\delta _{k}}\left( {\mathbf {m}}+{\mathbf {e}}_{i}\right) )-V^{\delta _{k}}(g^{\delta _{k}}({\mathbf {m}}))\\&\quad \le V^{\delta _{k}}(g^{\delta _{k} }\left( {\mathbf {m}}+{\mathbf {1}}\right) )-V^{\delta _{k}}(g^{\delta _{k} }({\mathbf {m}}))\le V^{\delta _{k}}(g^{\delta _{k}}({\mathbf {m}}))(e^{(c+\lambda )\delta _{k}}-1). \end{aligned}$$
Let us call \({\mathbf {m}}_{{\mathbf {y}}}=\rho ^{\delta _{k}}({\mathbf {y}})\) and \({\mathbf {m}}_{{\mathbf {x}}}=\rho ^{\delta _{k}}({\mathbf {x}})\). Then,
$$\begin{aligned} \begin{array} [c]{lll} V^{\delta _{k}}({\mathbf {y}})-V^{\delta _{k}}({\mathbf {x}}) &{} \le &{} V^{\delta _{k} }\left( g^{\delta _{k}}({\mathbf {m}}_{{\mathbf {y}}})\right) -V^{\delta _{k}}\left( g^{\delta _{k} }\left( {\mathbf {m}}_{{\mathbf {x}}}\right) \right) +{\mathbf {a}}\cdot ({\mathbf {y}} -g^{\delta _{k}}({\mathbf {m}}_{{\mathbf {y}}}))\\ &{} \le &{} \left( \frac{e^{(c+\lambda )\delta _{k}}-1}{\delta _{k}}\right) V^{\delta _{k} }({\mathbf {y}})\sum \nolimits _{i=1}^{n}\frac{g_{i}^{\delta _{k}}\left( {\mathbf {m}} _{{\mathbf {y}}}-{\mathbf {m}}_{{\mathbf {x}}}\right) }{p_{i}}+\delta _{k} {\mathbf {a}}\cdot {\mathbf {p}}\\ &{} \le &{} \left( \frac{e^{(c+\lambda )\delta _{k}}-1}{{\hat{p}}\delta _{k}}\right) V^{\delta _{k}}({\mathbf {y}})\left\| g^{\delta _{k}}\left( {\mathbf {m}} _{{\mathbf {y}}}-{\mathbf {m}}_{{\mathbf {x}}}\right) \right\| _{1}+\delta _{k}{\mathbf {a}}\cdot {\mathbf {p}}. \end{array} \end{aligned}$$
Let us consider now \({\mathbf {x}}\) and \({\mathbf {y}}\) in \({\mathbf {R}}_{+}^{n}\), consider \({\mathbf {m}}_{0}=\rho ^{\delta _{k}}({\mathbf {x}}\wedge {\mathbf {y}})\),
$$\begin{aligned} \begin{array} [c]{l} \left| V^{\delta _{k}}({\mathbf {y}})-V^{\delta _{k}} ({\mathbf {x}})\right| \\ \begin{array} [c]{ll} &{}\le V^{\delta _{k}}({\mathbf {y}})-V^{\delta _{k}}({\mathbf {x}}\wedge {\mathbf {y}})+V^{\delta _{k}}({\mathbf {x}})-V^{\delta _{k}}({\mathbf {x}} \wedge {\mathbf {y}})\\ &{}\le \frac{1}{{\hat{p}}}V^{\delta _{k}}({\mathbf {x}}\vee {\mathbf {y}})\left( \frac{e^{(c+\lambda )\delta _{k}}-1}{\delta _{k}}\right) \left( \left\| g^{\delta _{k} }\left( {\mathbf {m}}_{{\mathbf {y}}}-{\mathbf {m}}_{0}\right) \right\| _{1}+\left\| g^{\delta _{k}}\left( {\mathbf {m}}_{{\mathbf {x}}}-{\mathbf {m}} _{0}\right) \right\| _{1}\right) +2\delta _{k}{\mathbf {a}} \cdot {\mathbf {p}}\\ &{}\le \frac{2}{{\hat{p}}}V^{\delta _{k}}({\mathbf {x}}\vee {\mathbf {y}})\left( \frac{e^{(c+\lambda )\delta _{k}}-1}{\delta _{k}}\right) \left\| g^{\delta _{k}}\left( {\mathbf {m}}_{{\mathbf {y}}}-{\mathbf {m}}_{{\mathbf {x}}}\right) \right\| _{1}+2\delta _{k}{\mathbf {a}}\cdot p. \end{array} \end{array} \end{aligned}$$
Therefore we have (7.18).
By definitions (4.9) and (5.1), and since \(T_{i}\left( v^{\delta _{k}}\right) \le v^{\delta _{k}}\),
$$\begin{aligned} \begin{array} [c]{lll} {\overline{V}}({\mathbf {y}})-{\overline{V}}({\mathbf {x}}) &{} \ge &{} \overline{V}({\mathbf {y}})-V^{\delta _{k}}({\mathbf {y}})+{\mathbf {a}}\cdot g^{\delta _{k} }\left( \rho ^{\delta _{k}}({\mathbf {y}})-\rho ^{\delta _{k}}({\mathbf {x}})\right) \\ &{} &{}\quad +\,{\mathbf {a}}\cdot ({\mathbf {y}}-g^{\delta _{k}}(\rho ^{\delta _{k}} ({\mathbf {y}})-\rho ^{\delta _{k}}({\mathbf {x}}))+{\mathbf {x}})+V^{\delta _{k} }({\mathbf {x}})-{\overline{V}}({\mathbf {x}}); \end{array} \end{aligned}$$
taking the limit as k goes to infinity, we obtain the first inequality of the Lipschitz inequality.
We can write, from (7.18),
$$\begin{aligned} \begin{array} [c]{lll} {\overline{V}}({\mathbf {y}})-{\overline{V}}({\mathbf {x}}) &{} = &{} \overline{V}({\mathbf {y}})-V^{\delta _{k}}({\mathbf {y}})+V^{\delta _{k}}({\mathbf {y}} )-V^{\delta _{k}}({\mathbf {x}})+V^{\delta _{k}}({\mathbf {x}})-\overline{V}({\mathbf {x}})\\ &{} \le &{} {\overline{V}}({\mathbf {y}})-V^{\delta _{k}}({\mathbf {y}})+\frac{2}{{\hat{p}} }{\overline{V}}({\mathbf {y}})\left( \frac{e^{(c+\lambda )\delta _{k}}-1}{\delta _{k} }\right) \left\| g^{\delta _{k}}\left( \rho ^{\delta _{k}}({\mathbf {y}})-\rho ^{\delta _{k}}({\mathbf {x}})\right) \right\| _{1}\\ &{} &{}\quad +\,2\delta _{k}{\mathbf {a}}\cdot {\mathbf {p}}+V^{\delta _{k}}({\mathbf {x}} )-{\overline{V}}({\mathbf {x}}); \end{array} \end{aligned}$$
taking the limit as k goes to infinity, we obtain the second inequality of the Lipschitz inequality. \(\square \)
