Liquidity risk and optimal dividend/investment strategies

Abstract

In this paper, we study the problem of determining an optimal control on the dividend and investment policy of a firm operating under uncertain environment and risk constraints. We allow the company to make investment decisions by acquiring or selling producing assets whose value is governed by a stochastic process. The firm may face liquidity costs when it decides to buy or sell assets. We formulate this problem as a multi-dimensional mixed singular and multi-switching control problem and use a viscosity solution approach. We numerically compute our optimal strategies and enrich our studies with numerical results and illustrations.

Appendix

Proof of Lemma 3.2:

Consider any ${\bar{y}}\,\,{:=}\,\,({\bar{x}},{\bar{s}},{\bar{q}}) \in (0,+\infty )^2\times \mathbb {N}$ and let $\varphi (.,.,{\bar{q}})$ a $\mathcal{C}^{2}$ function on $(0,+\infty )^2$ such that $v_N^l({\bar{y}})=\varphi ({\bar{y}})$ and $v^l_N-\varphi \ge 0$ in a neighborhood of ${\bar{y}}$ denoted by ${\bar{B}}_{\varepsilon }({\bar{y}})\,\,{:=}\,\,({\bar{x}}-\varepsilon ,{\bar{x}}+\varepsilon )\times ({\bar{s}}-\varepsilon ,{\bar{s}}+\varepsilon )\times \{{\bar{q}}\}$ where $0<\varepsilon <\min ({\bar{x}},{\bar{s}})$.

On the one hand, we obviously have $v_N\ge G_{N-1}$ and as $G_{N-1}$ is continuous, it implies that $v_N^l\ge G_{N-1}$. Therefore, we

$$\begin{aligned} \varphi ({\bar{y}})=v^l_N({\bar{y}})\ge G_{N-1}({\bar{y}}). \end{aligned}$$

(4.1)

On the other hand, let us consider the admissible control ${\hat{\alpha }}=(({\hat{\tau }}_i,{\hat{q}}_i)_{i\ge 1},{\hat{Z}})$ where we decide to never make an impulse, i.e. ${\hat{\tau }}_1=+\infty$, while the dividend policy is defined by ${\hat{Z}}=\delta$ for $t \ge 0$, with $0 \le \delta \le \varepsilon$. We know that there exists a sequence $({\bar{x}}_m,{\bar{s}}_m)_{m\in \mathbb {N}}$ such that

$$\begin{aligned} \lim _{m\rightarrow +\infty }({\bar{x}}_m,{\bar{s}}_m)=({\bar{x}},{\bar{s}})\quad \text {and}\quad \lim _{m\rightarrow +\infty }v_N({\bar{x}}_m,{\bar{s}}_m, {\bar{q}})=v_{N}^l({\bar{x}},{\bar{s}},{\bar{q}}). \end{aligned}$$

With the same notation, we set ${\bar{y}}_m = ({\bar{x}}_m, {\bar{s}}_m, {\bar{q}})$.

We define the exit time $\tau ^m_{\varepsilon } \,\,{:=}\,\, \inf \{t \ge 0, Y^{{\bar{y}}_m}_{t} \not \in {\bar{B}}_{\varepsilon }({\bar{y}})\}$. We notice that $\tau ^m_{\varepsilon } < T$.

From the dynamic programming principle (see (3.16)), if we set $\gamma _m \,\,{:=}\,\, v_N({\bar{y}}_m)-\varphi ({\bar{y}}_m)\ge 0$ and $\nu _m =\tau ^m_{\varepsilon }\wedge h_m$ where $(h_m)_{m\ge 0}$ is a positive sequence such that $\lim _{m\rightarrow +\infty } h_m =0$ and $\lim _{m\rightarrow +\infty }\gamma _m/\! h_m=0$, then we have

$$\begin{aligned} \varphi ({\bar{y}}_m)= & {} v_N({\bar{y}}_m)-\gamma _m\nonumber \\\ge & {} \mathbb {E}\Big [\int _0^{\nu _m^{-}}e^{-\rho t}\, d{\hat{Z}}_t+e^{-\rho \nu _m}v_N\left( Y_{\nu _m}^{{\bar{y}}_m}\right) \Big ]-\gamma _m\nonumber \\\ge & {} \mathbb {E}\Big [ \int _0^{\nu _m^{-}}e^{-\rho t}\, d{\hat{Z}}_t+e^{-\rho \nu _m}v_N^l\left( Y_{\nu _m}^{{\bar{y}}_m}\right) \Big ]-\gamma _m\nonumber \\\ge & {} \mathbb {E}\Big [ \int _0^{\nu _m^{-}}e^{-\rho t}\, d{\hat{Z}}_t+e^{-\rho \nu _m}\varphi \left( Y_{\nu _m}^{{\bar{y}}_m}\right) \Big ]-\gamma _m. \end{aligned}$$

(4.2)

Applying Itô’s formula to the process $e^{-\rho t}\varphi (Y_{t}^{{\bar{y}}_m})$ between 0 and $\nu _m$ and taking the expectation, we obtain

$$\begin{aligned} \mathbb {E}\Big [e^{-\rho \nu _m}\varphi (Y_{\nu _m}^{{\bar{y}}_m})\Big ]= & {} \varphi ({\bar{y}}_m)+ \mathbb {E}\Big [ \int _0^{\nu _m^{-}}e^{-\rho t}\,(-\rho \varphi + \mathcal{L}\varphi )(Y_{t}^{{\bar{y}}_m})dt\Big ]\nonumber \\&+\, \mathbb {E}\left[ \sum _{0 \le t < \nu _m} e^{-\rho t}[\varphi (Y_{t}^{{\bar{y}}_m})-\varphi (Y_{t^{-}}^{{\bar{y}}_m})]\right] . \end{aligned}$$

(4.3)

Combining relations (4.2) and (4.3), we have

$$\begin{aligned}&\mathbb {E}\Big [ \int _0^{\nu _m^{-}}e^{-\rho t}\,(\rho \varphi - \mathcal{L}\varphi )(Y_{t}^{{\bar{y}}_m})dt\Big ] - \mathbb {E}\Big [ \int _0^{\nu _m^{-}}e^{-\rho t}\,d{\hat{Z}}_t\Big ]\nonumber \\&\quad -\mathbb {E}\left[ \sum _{0 \le t < \nu _m} e^{-\rho t}[\varphi (Y_{t}^{{\bar{y}}_m})-\varphi (Y_{t^{-}}^{{\bar{y}}_m})]\right] \ge -\gamma _m. \end{aligned}$$

(4.4)

• If we take $\delta =0$, we notice that Y is continuous on $[0,\nu _m]$ and only the first term of relation (4.4) is non zero. By dividing the above inequality by $h_m$ and letting m going to infinity, it follows from the smoothness of $\varphi$ and the continuity of the coefficients that

  $$\begin{aligned} (\rho \varphi - \mathcal{L}\varphi )({\bar{y}}) \ge 0. \end{aligned}$$

  (4.5)

• If we take now $\delta >0$ in (4.4), we notice that ${\hat{Z}}$ jumps only at $t=0$ with size $\delta$, hence

  $$\begin{aligned} \mathbb {E}\Big [ \int _0^{\nu _m^{-}}e^{-\rho t}\,(\rho \varphi - \mathcal{L}\varphi )(Y_{t}^{{\bar{y}}_m})dt\Big ]- \delta -(\varphi ({\bar{x}}_m-\delta ,{\bar{s}}_m,{\bar{q}})- \varphi ({\bar{x}}_m,{\bar{s}}_m,{\bar{q}}))\ge -\gamma _m.\nonumber \\ \end{aligned}$$

  (4.6)

By sending m to infinity, and then dividing by $\delta$ and letting $\delta \rightarrow 0$, we obtain

$$\begin{aligned} \frac{\partial \varphi }{\partial x}({\bar{x}},{\bar{s}},{\bar{q}})-1 \ge 0. \end{aligned}$$

(4.7)

We conclude by combining (4.1), (4.5) and (4.7) to obtain the required supersolution property

$$\begin{aligned} \min \left\{ \rho \varphi ({\bar{x}},{\bar{s}},{\bar{q}}) - \mathcal{L}\varphi ({\bar{x}},{\bar{s}},{\bar{q}}); \frac{\partial \varphi }{\partial x}({\bar{x}},{\bar{s}},{\bar{q}})- 1; v_N^l({\bar{x}},{\bar{s}},{\bar{q}})-G_{N-1}({\bar{x}},{\bar{s}},{\bar{q}}) \right\} \ge 0.\nonumber \\ \end{aligned}$$

(4.8)

Proof of Lemma 3.3:

Consider any ${\bar{y}}\,\,{:=}\,\,({\bar{x}},{\bar{s}},{\bar{q}}) \in (0,+\infty )^2\times \mathbb {N}$ and let $\varphi (.,.,{\bar{q}})$ a $\mathcal{C}^{2}$ function on $(0,+\infty )^2$ such that $v_N^u({\bar{y}})=\varphi ({\bar{y}})$ and $v_N^u-\varphi \le 0$ in a neighborhood of ${\bar{y}}$, denoted by ${\bar{B}}_{\varepsilon }({\bar{y}})\,\,{:=}\,\,({\bar{x}}-\varepsilon ,{\bar{x}}+\varepsilon )\times ({\bar{s}}-\varepsilon ,{\bar{s}}+\varepsilon )\times \{{\bar{q}}\}$ where $0<\varepsilon <\min ({\bar{x}},{\bar{s}})$.

Let us argue by contradiction by assuming on the contrary that $\exists \, \delta > 0$ s.t. $\forall y \in {\bar{B}}_{\varepsilon }({\bar{y}})$ we have

$$\begin{aligned}&\displaystyle \rho \varphi (y) - \mathcal{L}\varphi (y) > \delta , \end{aligned}$$

(4.9)

$$\begin{aligned}&\displaystyle \frac{\partial \varphi }{\partial x}(y)- 1 > \delta , \end{aligned}$$

(4.10)

$$\begin{aligned}&\displaystyle v_N^u(y) - G_{N-1}(y) > \delta . \end{aligned}$$

(4.11)

We know that there exists a sequence $({\bar{x}}_m,{\bar{s}}_m)_{m\in \mathbb {N}}$ such that

$$\begin{aligned} \lim _{m\rightarrow +\infty }({\bar{x}}_m,{\bar{s}}_m)=({\bar{x}},{\bar{s}}) \quad \text {and}\quad \lim _{m\rightarrow +\infty }v_N ({\bar{x}}_m,{\bar{s}}_m, {\bar{q}}) = v_N^u({\bar{x}},{\bar{s}}, {\bar{q}}). \end{aligned}$$

Let ${\bar{y}}_m \,\,{:=}\,\, ({\bar{x}}_m, {\bar{s}}_m, {\bar{q}}) \in {B}_{\varepsilon }({\bar{y}}).$ For any admissible control $\alpha =((\tau _i,q_i)_{i\ge 1},Z)$, consider the exit time $\tau _{\varepsilon }^m=\inf \{t \ge 0, Y^{{\bar{y}}_m}_{t} \not \in {\bar{B}}_{\varepsilon }({\bar{y}})\}$. We notice that $\tau _{\varepsilon }^m< T$. Applying Itô’s formula to the process $e^{-\rho t}\varphi (Y_{t}^{{\bar{y}}_m})$ between 0 and $(\tau _{\varepsilon }^m\wedge \tau _1)^{-}$ and by noting that before $(\tau _{\varepsilon }^m\wedge \tau _1)^{-}$, $Y_{t}^{{\bar{y}}_m}$ stays in the ball ${\bar{B}}_{\varepsilon }({\bar{y}})$, we obtain

$$\begin{aligned} \mathbb {E}\Big [e^{-\rho (\tau _{\varepsilon }^m\wedge \tau _1)^{-}}\varphi (Y_{(\tau _{\varepsilon }^m\wedge \tau _1)^{-}}^{{\bar{y}}_m})\Big ]= & {} \varphi ({\bar{y}}_m)+ \mathbb {E}\Big [ \int _0^{(\tau _{\varepsilon }^m\wedge \tau _1)^{-}}e^{-\rho t}\,(-\rho \varphi + \mathcal{L}\varphi )(Y_{t}^{{\bar{y}}_m})dt\Big ]\nonumber \\&-\;\mathbb {E}\Big [ \int _0^{(\tau _{\varepsilon }^m\wedge \tau _1)^{-}}e^{-\rho t}\,\frac{\partial \varphi }{\partial x}(Y_{t}^{{\bar{y}}_m})dZ^{c}_{t}\Big ]\nonumber \\&+\;\mathbb {E}\left[ \sum _{0 \le t < \tau _{\varepsilon }^m \wedge \tau _1} e^{-\rho t}[\varphi (Y_{t}^{{\bar{y}}_m})-\varphi (Y_{t^{-}}^{{\bar{y}}_m})]\right] . \end{aligned}$$

(4.12)

From Taylor’s formula and (4.10), and noting that $\Delta X_{t}^{{\bar{x}}}=-\Delta Z_t$ for all $0 \le t < \tau _{\varepsilon }^m\wedge \tau _1$, we have

$$\begin{aligned} \varphi (Y_{t}^{{\bar{y}}_m})-\varphi (Y_{t^{-}}^{{\bar{y}}_m}) = \Delta X_{t}^{{\bar{x}}}\,\frac{\partial \varphi }{\partial x}(Y_{t^{-}}^{{\bar{y}}_m}) \le -(1+\delta )\Delta Z_t. \end{aligned}$$

(4.13)

Plugging the relations (4.9), (4.10) and (4.13) into (4.12), we obtain

$$\begin{aligned} \varphi ({\bar{y}}_m)\ge & {} \mathbb {E}\left[ \int _0^{(\tau _{\varepsilon }^m\wedge \tau _1)^{-}}e^{-\rho t}\,dZ_t + e^{-\rho (\tau _{\varepsilon }^m\wedge \tau _1)^{-}}\varphi (Y_{(\tau _{\varepsilon }^m\wedge \tau _1)^{-}}^{{\bar{y}}_m})\right] \nonumber \\&+\, \delta \left( \mathbb {E}\left[ \int _0^{(\tau _{\varepsilon }^m\wedge \tau _1)^{-}}e^{-\rho t}\,dt\right] +\mathbb {E}\left[ \int _0^{(\tau _{\varepsilon }^m\wedge \tau _1)^{-}}e^{-\rho t}\,dZ_t\right] \right) \nonumber \\\ge & {} \mathbb {E}\left[ \int _0^{(\tau _{\varepsilon }^m\wedge \tau _1)^{-}}e^{-\rho t}\,dZ_t + e^{-\rho \tau _{\varepsilon }^{m-}}\varphi (Y_{\tau _{\varepsilon }^{m-}}^{{\bar{y}}_m})1_{\tau _{\varepsilon }^m< \tau _1}+e^{-\rho \tau _{1}^{-}}\varphi (Y_{\tau _{1}^{-}}^{{\bar{y}}_m})1_{\tau _{1}\le \tau _{\varepsilon }^m}\right] \nonumber \\&+\, \delta \left( \mathbb {E}\left[ \int _0^{(\tau _{\varepsilon }^m\wedge \tau _1)^{-}}e^{-\rho t}\,dt\right] +\mathbb {E}\left[ \int _0^{(\tau _{\varepsilon }^m\wedge \tau _1)^{-}}e^{-\rho t}\,dZ_t\right] \right) . \end{aligned}$$

(4.14)

First step On $\{\tau _{\varepsilon }^m < \tau _1\}$, we notice that while $Y_{\tau _{\varepsilon }^{m-}}^{{\bar{y}}_m} \in {\bar{B}}_{\varepsilon }({\bar{y}})$, $Y_{\tau _{\varepsilon }^m}^{{\bar{y}}_m}$ is either on the boundary $\partial {\bar{B}}_{\varepsilon }({\bar{y}})$ or out of ${\bar{B}}_{\varepsilon }({\bar{y}})$. However, there is some random variable $\gamma$ valued in [0, 1] s.t.

$$\begin{aligned} X^{(\gamma )} \,\,{:=}\,\, X_{\tau _{\varepsilon }^{m-}}^{{\bar{x}}_m} + \gamma \Delta X_{\tau _{\varepsilon }^m}^{{\bar{x}}_m},= X_{\tau _{\varepsilon }^{m-}}^{{\bar{x}}_m}- \gamma \Delta Z_{\tau _{\varepsilon }^m} \in \{{\bar{x}}-\varepsilon ,{\bar{x}}+\varepsilon \}, \end{aligned}$$

hence, $Y^{(\gamma )}\,\,{:=}\,\,(X^{(\gamma )},S_{\tau _{\varepsilon }^m}^{{\bar{s}}_m},Q_{\tau _{\varepsilon }^m}^{{\bar{q}}})$ is on the boundary $\partial {\bar{B}}_{\varepsilon }({\bar{y}})$.

Following the same arguments as in (4.13), we have

$$\begin{aligned} \varphi (Y^{(\gamma )})-\varphi (Y_{\tau _{\varepsilon }^{m-}}^{{\bar{y}}_m}) \le -\gamma (1+\delta ) \Delta Z_{\tau _{\varepsilon }^m}. \end{aligned}$$

(4.15)

Noting that $X^{(\gamma )} = X_{\tau _{\varepsilon }^m}^{{\bar{x}}_m}+(1-\gamma ) \Delta Z_{\tau _{\varepsilon }}$, we have

$$\begin{aligned} v_N^u(Y^{(\gamma )}) \ge v_N^u(Y_{\tau _{\varepsilon }^m}^{{\bar{y}}_m}) +(1-\gamma ) \Delta Z_{\tau _{\varepsilon }^m}. \end{aligned}$$

(4.16)

Recalling that $\varphi (Y^{(\gamma )}) \ge v_N^u(Y^{(\gamma )})$, inequalities (4.15) and (4.16) imply

$$\begin{aligned} \varphi (Y_{\tau _{\varepsilon }^{m-}}^{{\bar{y}}_m}) \ge v_N^u(Y_{\tau _{\varepsilon }^m}^{{\bar{y}}_m}) +(1+\gamma \delta ) \Delta Z_{\tau _{\varepsilon }^m}. \end{aligned}$$

(4.17)

Second step On $\{\tau _1 \le \tau _{\varepsilon }^m\}$, we notice that $Y_{\tau _{1}^{-}}^{{\bar{y}}_m} \in {\bar{B}}_{\varepsilon }({\bar{y}})$, thus $v_N^u(Y_{\tau _{1}^{-}}^{{\bar{y}}_m}) \le \varphi (Y_{\tau _{1}^{-}}^{{\bar{y}}_m})$. From the assumption (4.11) we obtain

$$\begin{aligned} \varphi (Y_{\tau _{1}^{-}}^{{\bar{y}}_m})\ge & {} G_{N-1}(Y_{\tau _{1}^{-}}^{{\bar{y}}_m})+ \delta . \end{aligned}$$

(4.18)

Plugging (4.17) and (4.18) into (4.14) we have

$$\begin{aligned} \varphi ({\bar{y}}_m)\ge & {} \mathbb {E}\left[ \int _0^{(\tau _{\varepsilon }^m\wedge \tau _1)^{-}}e^{-\rho t}\,dZ_t + e^{-\rho \tau _{\varepsilon }^m}v_N^u(Y_{\tau _{\varepsilon }^m}^{{\bar{y}}_m})1_{\tau _{\varepsilon }< \tau _1}+ e^{-\rho \tau _{1}}G_{N-1}(Y_{\tau _{1}^{-}}^{{\bar{y}}_m})1_{\tau _{1}\le \tau _{\varepsilon }^m}\right] \nonumber \\&+\, \delta \, \mathbb {E}\left[ \int _0^{(\tau _{\varepsilon }^m\wedge \tau _1)^{-}}e^{-\rho t}\,dt+\, \int _0^{(\tau _{\varepsilon }^m\wedge \tau _1)^{-}}e^{-\rho t}\,dZ_t+\gamma e^{-\rho \tau _{\varepsilon }^m}\Delta Z_{\tau _{\varepsilon }^m}1_{\tau _{\varepsilon }< \tau _1}\right. \nonumber \\&\left. +\, e^{-\rho \tau _{1}}1_{\tau _{1}\le \tau _{\varepsilon }^m}\right] + \mathbb {E}\Big [e^{-\rho \tau _{\varepsilon }^m}\Delta Z_{\tau _{\varepsilon }^m}1_{\tau _{\varepsilon }^m< \tau _1}\Big ]. \end{aligned}$$

(4.19)

We now claim that there exists a constant $c_0 >0$ such that for any admissible control

$$\begin{aligned} c_0\le & {} \mathbb {E}\left[ \int _0^{(\tau _{\varepsilon }^m\wedge \tau _1)^{-}}e^{-\rho t}\,dt+ \int _0^{(\tau _{\varepsilon }^m\wedge \tau _1)^{-}}e^{-\rho t}\,dZ_t\right] \nonumber \\&+\,\mathbb {E}\Big [\gamma e^{-\rho \tau _{\varepsilon }^m}\Delta Z_{\tau _{\varepsilon }^m}1_{\tau _{\varepsilon }^m< \tau _1}+e^{-\rho \tau _{1}}1_{\tau _{1}\le \tau _{\varepsilon }^m}\Big ]. \end{aligned}$$

(4.20)

The $C^2$ function $\psi (x,s,q)=c_0[1-\frac{(x-{\bar{x}}_m)^{2}}{\varepsilon ^2}]$, with

$$\begin{aligned} 0 < c_0 \le \min \left\{ \left( \rho +\frac{4}{\varepsilon }(r({\bar{x}}_m+\varepsilon )+bH)+\frac{\eta ^2}{\varepsilon ^2}H^2\right) ^{-1},\frac{\varepsilon }{2} \right\} \end{aligned}$$

satisfies

$$\begin{aligned} \left\{ \begin{array}{l} \min \left\{ -\rho \psi (x,s,q) + \mathcal{L}\psi (x,s,q)+1; -\frac{\partial \psi }{\partial x}(x,s,q)+ 1;\right. \\ \qquad \quad \left. -\psi (x,s,q) +1 \right\} \ge 0\text { on }{\bar{B}}_{\varepsilon }({\bar{y}}),\\ \psi (x,s,q) = 0\;\text {on}\; \partial {\bar{B}}_{\varepsilon }({\bar{y}}). \end{array}\right. \end{aligned}$$

(4.21)

Applying Itô’s formula, we then obtain

$$\begin{aligned} 0< c_0 =\psi ({\bar{y}}_m)\le & {} \mathbb {E}\left[ e^{-\rho (\tau _{\varepsilon }^m\wedge \tau _1)^{-}}\psi (Y_{(\tau _{\varepsilon }^m\wedge \tau _1)^{-}}^{{\bar{y}}_m})\right] \nonumber \\&+\,\mathbb {E}\left[ \int _0^{(\tau _{\varepsilon }^m\wedge \tau _1)^{-}}e^{-\rho t}\,dt+ \int _0^{(\tau _{\varepsilon }^m\wedge \tau _1)^{-}}e^{-\rho t}\,dZ_t\right] . \end{aligned}$$

(4.22)

Noting that $\frac{\partial \psi }{\partial x}(x,s,q)\le 1$, we have

$$\begin{aligned} \psi (Y_{\tau _{\varepsilon }^{m-}}^{{\bar{y}}_m})-\psi (Y^{(\gamma )}) \le X_{\tau _{\varepsilon }^{m-}}^{{\bar{x}}_m}-X^{(\gamma )}=\gamma \Delta Z_{\tau _{\varepsilon }^m}. \end{aligned}$$

Plugging into (4.22), we obtain

$$\begin{aligned} 0< c_0\le & {} \mathbb {E}\left[ e^{-\rho \tau _{1}}\psi (Y_{\tau _1^{-}}^{{\bar{y}}_m})1_{\tau _1 \le \tau _{\varepsilon }^m }\right] +\mathbb {E}\left[ \int _0^{(\tau _{\varepsilon }^m\wedge \tau _1)^{-}}e^{-\rho t}\,dt\right] \nonumber \\&+\, \mathbb {E}\left[ \int _0^{(\tau _{\varepsilon }^m\wedge \tau _1)^{-}}e^{-\rho t}\,dZ_t \right] + \mathbb {E}\left[ e^{-\rho \tau _{\varepsilon }^m}\gamma \Delta Z_{\tau _{\varepsilon }^m}1_{ \tau _{\varepsilon }^m < \tau _1 }\right] . \end{aligned}$$

(4.23)

Since $\psi \le 1$ for all $y \in {\bar{B}}_{\varepsilon }({\bar{y}})$, this proves the claim (4.20).

Finally, by taking the supremum over all admissible control $\alpha$, and using the dynamic programming principle (3.16), Eq. (4.19) implies that $\varphi ({\bar{y}}_m) \ge v_N^u({\bar{y}}_m) + \delta c_0$, which leads to a contradiction when m goes to infinity. Thus we obtain the required viscosity subsolution property:

$$\begin{aligned} \min \left\{ \rho \varphi ({\bar{x}},{\bar{s}},{\bar{q}}) - \mathcal{L}\varphi ({\bar{x}},{\bar{s}},{\bar{q}}); \frac{\partial \varphi }{\partial x}({\bar{x}},{\bar{s}},{\bar{q}})- 1; v_N^u({\bar{x}},{\bar{s}},{\bar{q}})-G_{N-1}({\bar{x}},{\bar{s}},{\bar{q}}) \right\} \le 0.\nonumber \\ \end{aligned}$$

(4.24)

$\square$

Proof of Lemma 3.4:

Step 1 We first construct a strict supersolution to the system with suitable perturbation of w. We set

$$\begin{aligned} g(y)=A + B(x+sq+1) + D(x+sq+1)^p,\quad p\in (1,2), \quad y\in \mathcal{S}, \end{aligned}$$

where

$$\begin{aligned} A=\frac{BHb+bH+1 }{\rho } + C_1 + C_3 \kappa , \quad B=2\quad \hbox {and}\quad D=B\Big (\frac{\rho -\max (r,\mu )}{2Hb}\Big ). \end{aligned}$$

We then define for all $\gamma \in (0,1)$, the lower semi-continuous function on $(0,+\infty )^2\times \mathbb {N}$ by:

$$\begin{aligned} w^{\gamma }=(1-\gamma )w+\gamma g. \end{aligned}$$

Let $y\in (0,+\infty )^2\times \mathbb {N}$. For all $\gamma \in (0,1)$, we then see that :

$$\begin{aligned} w^{\gamma }(y)-G_{N-1}(y)\ge & {} (1-\gamma )\Big (w(y)-G_{N-1}(y)\Big ) + \gamma \Big (g(y)-G_{N-1}(y)\Big )\nonumber \\\ge & {} \gamma \kappa , \end{aligned}$$

(4.25)

where the last inequality comes from the facts that as a supersolution $w(y)\ge G_{N-1}(y)$ and that for all $n \in \mathcal{A}(y)$, Corollary 3.1 implies that:

$$\begin{aligned} v_{N-1}\Big (\Gamma (y,n)\Big )\le & {} x - \kappa -nsf(n)+s(q+n)+\frac{bH}{\rho }\nonumber \\\le & {} \frac{bH}{\rho }+(x+sq+1)+sn(1-f(n))-\kappa \nonumber \\\le & {} g(y)-\kappa . \end{aligned}$$

(4.26)

Furthermore, we also easily obtain

$$\begin{aligned} \frac{\partial g}{\partial x}(y)- 1=B + p(x + sq +1)^{p-1}-1\ge 1. \end{aligned}$$

(4.27)

Recalling that $h(q)\le H$ for all $q\in \mathbb {N}$, a straight calculation gives

$$\begin{aligned} \rho g(y)-\mathcal{L}g(y)= & {} \rho \Big (A +B (x+sq+1) + D(x + sq +1)^{p}\Big )\\&-\,Dc \sigma h(q)\eta q s p(p-1)(x + sq +1)^{p-2}\\&-\,\frac{\eta ^2 h^2(q)}{2}D p(p-1)(x + sq +1)^{p-2}\\&-\,\frac{s^2 q^2\sigma ^2}{2}D p(p-1)(x + sq +1)^{p-2}\\&-\,\Big (rx+bh(q)\Big )\Big (B+Dp(x+sq+1)^{p-1}\Big )\\&-\,\mu s\Big (Bq+Dqp(x+sq+1)^{p-1}\Big )\\\ge & {} \rho A -B b H + B\Big (\rho -\max (r,\mu )\Big )(x+sq+1)\\&-\,Dc \sigma H\eta p(p-1)(x + sq +1)^{p}\\&-\,D\frac{\eta ^2 H^2}{2}p(p-1)(x + sq +1)^{p}\\&-\,D\frac{\sigma ^2}{2}p(p-1)(x + sq +1)^{p}\\&-\,D b H p (x + sq +1)^{p-1}\\&-\,D\max (r,\mu )p(x + sq +1)^{p}+D\rho (x + sq +1)^{p}\\\ge & {} \rho A -B b H +(x+sq+1)\\&\times \Big (B\Big (\rho -\max (r,\mu )\Big )-D b H p(x+sq+1)^{p-2}\Big )\\&+\,D\left( \rho -\max (r,\mu )p-\left( \frac{\eta ^2 H^2}{2}+\frac{\sigma ^2}{2}+c \sigma H\eta \right) p(p-1)\right) (x+sq+1)^{p}. \end{aligned}$$

Using that $p(x + sq +1)^{p-2}\le 2$ for all $p\in (1,2)$ and replacing A by its value, we obtain

$$\begin{aligned} \rho g(y)-\mathcal{L}g(y)\ge & {} 1 + (x+sq+1)\Big (B(\rho -\max (r,\mu ))-2DbH\Big )\nonumber \\&+\,D\left( \rho -\max (r,\mu )p-\left( \frac{\eta ^2 H^2}{2}+\frac{\sigma ^2}{2}+c \sigma H\eta \right) p(p-1)\right) (x+sq+1)^{p}.\nonumber \end{aligned}$$

Recalling that $\rho > \max (r,\mu )$, we can choose $p\in (1,2)$ s.t.

$$\begin{aligned} \xi \,\,{:=}\,\,\rho -\max (r,\mu )p-\left( \frac{\eta ^2 H^2}{2}+\frac{\sigma ^2}{2}+c \sigma H\eta \right) p(p-1)>0, \end{aligned}$$

we then have

$$\begin{aligned} \rho g(y)-\mathcal{L}g(y)\ge & {} 1 + D\xi (x + sq +1)^{p}, \quad \forall y \in (0,+\infty )^2\times \mathbb {N}. \end{aligned}$$

(4.28)

Combining (4.26), (4.27) and (4.28), we obtain that $w^{\gamma }$ is a strict supersolution of Eq. (3.15): $\forall y \in (0,+\infty )^2\times \mathbb {N}$, we have

$$\begin{aligned} \min \left\{ \rho w^{\gamma }(y) - \mathcal{L}w^{\gamma }(y); \frac{\partial w^{\gamma }}{\partial x}(y){\!}{\!}- {\!}{\!}1; w^{\gamma }(y)-G_{N-1}(y) \right\} \ge \gamma \min (1,\kappa )&\,\,{:=}\,\,&\delta .\nonumber \\ \end{aligned}$$

(4.29)

Step 2 In order to prove the comparison principle, it suffices to show that for all $\gamma \in (0,1)$:

$$\begin{aligned} \sup _{y\in (0,+\infty )^2\times \mathbb {N}}{(u-w^{\gamma })}\le 0, \end{aligned}$$

since the required result is obtained by letting $\gamma$ to 0. We argue by contradiction and suppose that there exist some $\gamma \in (0,1)$ s.t.

$$\begin{aligned} \theta \,\,{:=}\,\,\sup _{y\in (0,+\infty )^2\times \mathbb {N}}{(u-w^{\gamma })}>0. \end{aligned}$$

(4.30)

Notice that $u(y)-w^{\gamma }(y)$ goes to $-\infty$ as x, s and q go to infinity. For any ${\bar{y}}\in \{0\}\times (0,+\infty )\times \mathbb {N}$, we also have

$$\begin{aligned} \limsup _{y\rightarrow {\bar{y}}}u(y)- w^{\gamma }(y)\le \gamma (\liminf _{y\rightarrow {\bar{y}}}w(y)-A)\le 0. \end{aligned}$$

Hence, by semi-continuity of the functions u and $w^{\gamma }$, there exists $y_0=(x_0,s_0,q_0)\in (0,+\infty )^2\times \mathbb {N}$ s.t.

$$\begin{aligned} \theta =u(y_0)-w^{\gamma }(y_0). \end{aligned}$$

For any $\varepsilon >0$, we consider the the functions

$$\begin{aligned} \Phi _{\varepsilon }(y,y')= & {} u(y)-w^{\gamma }(y')-\varphi _{\varepsilon }(y,y')\\ \varphi _{\varepsilon }(y,y')= & {} \frac{1}{4}|y-y_0|^4+\frac{1}{2\varepsilon }|y-y'|^2, \end{aligned}$$

for all y, $y'$ $\in \mathcal{S}$. By standard arguments in comparison principle, the function $\Phi _{\varepsilon }$ attains its maximum in $(y_{\varepsilon },y'_{\varepsilon })\in ((0,+\infty )^2\times \mathbb {N})^2$, which converges (up to a subsequence) to $(y_0,y_0)$ when $\varepsilon$ goes to zero. Moreover,

$$\begin{aligned} \lim _{\varepsilon \rightarrow 0}{\frac{|y_{\varepsilon }-y_{\varepsilon }'|^2}{\varepsilon }}=0. \end{aligned}$$

(4.31)

Applying Theorem 3.2 in [9], we get the existence of two $2\times 2$ symmetric matrices $M^{\varepsilon }$ and $N^{\varepsilon }$ s.t.:

$$\begin{aligned} (p^{\varepsilon },M^{\varepsilon })\in & {} J^{2,+}u(y_{\varepsilon }),\\ (d^{\varepsilon },N^{\varepsilon })\in & {} J^{2,-}w^{\gamma }(y'_{\varepsilon }), \end{aligned}$$

and

$$\begin{aligned} \left( \begin{array}{cc} M^{\varepsilon }&{} 0\\ 0&{}-N^{\varepsilon }\\ \end{array}\right) \le D_{y,y'}^2\varphi _{\varepsilon }(y_{\varepsilon },y'_{\varepsilon })+ \varepsilon \Big (D_{y,y'}^2\varphi _{\varepsilon }(y_{\varepsilon },y'_{\varepsilon })\Big )^2, \end{aligned}$$

(4.32)

where

$$\begin{aligned}&\displaystyle p^{\varepsilon } = D_y\varphi _{\varepsilon }(y_{\varepsilon },y'_{\varepsilon })=\left( \begin{array}{c} (x_{\varepsilon }-x_0)|y_{\varepsilon }-y_0|^2+\frac{1}{\varepsilon }(x_{\varepsilon }-x'_{\varepsilon })\\ (s_{\varepsilon }-s_0)|y_{\varepsilon }-y_0|^2+\frac{1}{\varepsilon }(s_{\varepsilon }-s'_{\varepsilon })\\ \end{array}\right) \\&\displaystyle d^{\varepsilon } = -D_{y'}\varphi _{\varepsilon }(y_{\varepsilon },y'_{\varepsilon })=\left( \begin{array}{c} \frac{1}{\varepsilon }(x_{\varepsilon }-x'_{\varepsilon })\\ \frac{1}{\varepsilon }(s_{\varepsilon }-s'_{\varepsilon })\\ \end{array}\right) ,\\&\displaystyle D_{y,y'}^2\varphi _{\varepsilon }(y_{\varepsilon },y'_{\varepsilon })= \left( \begin{array}{cccc} 2(x_{\varepsilon }-x_0)^2+|y_{\varepsilon }-y_0|^2+\frac{1}{\varepsilon }&{} 2(x_{\varepsilon }-x_{0})(s_{\varepsilon }-s_{0})&{} \frac{-1}{\varepsilon }&{} 0\\ 2(x_{\varepsilon }-x_{0})(s_{\varepsilon }-s_{0})&{} 2(s_{\varepsilon }-s_0)^2+|y_{\varepsilon }-y_0|^2+\frac{1}{\varepsilon }&{} 0&{} \frac{-1}{\varepsilon }\\ \frac{-1}{\varepsilon }&{} 0&{} \frac{1}{\varepsilon }&{} 0\\ 0&{} \frac{-1}{\varepsilon }&{} 0&{}\frac{1}{\varepsilon }\\ \end{array}\right) . \end{aligned}$$

By writing the viscosity subsolution property of u and the viscosity supersolution property (4.29) of $w^{\gamma }$, we have the following inequalities:

$$\begin{aligned}&\min \left\{ \rho u(y_{\varepsilon })-\Big (rx_{\varepsilon }+bh(q_{\varepsilon })\Big )p^{\varepsilon }_{1}-\mu s_{\varepsilon }p^{\varepsilon }_{2}-c\sigma \eta s_{\varepsilon }h(q_{\varepsilon })M^{\varepsilon }_{12}-\frac{\eta ^2h^2(q_{\varepsilon })}{2}M^{\varepsilon }_{11}\right. \nonumber \\&\left. \quad \qquad -\frac{\sigma ^2s_{\varepsilon }^2}{2}M^{\varepsilon }_{22};p^{\varepsilon }_{1}-1;u(y_{\varepsilon })-G_{N-1}(y_{\varepsilon })\right\} \le 0 \end{aligned}$$

(4.33)

$$\begin{aligned}&\min \left\{ \rho w^{\gamma }(y'_{\varepsilon })-\Big (rx'_{\varepsilon }+b h(q'_{\varepsilon })\Big )d^{\varepsilon }_{1}-\mu s'_{\varepsilon }d^{\varepsilon }_{2}-c\sigma \eta s'_{\varepsilon }h(q'_{\varepsilon })N^{\varepsilon }_{12}-\frac{\eta ^2h^2(q'_{\varepsilon })}{2}N^{\varepsilon }_{11}\right. \nonumber \\&\left. \quad \qquad -\frac{\sigma ^{2}(s'_{\varepsilon })^2}{2}N^{\varepsilon }_{22};d^{\varepsilon }_{1}-1;w^{\gamma }(y'_{\varepsilon })-G_{N-1}(y'_{\varepsilon })\right\} \ge \delta \end{aligned}$$

(4.34)

We then distinguish three cases:

• Case 1 $u(y_{\varepsilon })-G_{N-1} (y_{\varepsilon })\le 0$ in (4.33). From the definition of $(y_{\varepsilon },y_{\varepsilon }^\prime )$, we have

  $$\begin{aligned} \theta= & {} u(y_0)-w^\gamma (y_0)\\\le & {} u(y_\varepsilon )-w^\gamma (y_\varepsilon ^\prime )-\varphi _\varepsilon (y_\varepsilon ,y^\prime _\varepsilon )\\\le & {} G_{N-1}(y_\varepsilon )-G_{N-1}(y^\prime _\varepsilon )-\delta -\varphi _\varepsilon (y_\varepsilon ,y^\prime _\varepsilon ). \end{aligned}$$

  Now, letting $\varepsilon$ going to 0, we deduce from Eq. (4.31) and the continuity of $G_{N-1}$ that $0<\theta \le -\delta <0,$ which is obviously a contradiction.

• Case 2 $(x_{\varepsilon }-x_0)|y_{\varepsilon }-y_0|^2+\frac{1}{\varepsilon }(x_{\varepsilon }-x'_{\varepsilon }) -1=p^{\varepsilon }_{1}-1\le 0$ in (4.33). Notice by (4.34), we have

  $$\begin{aligned} \frac{1}{\varepsilon }(x_{\varepsilon }-x'_{\varepsilon })-1=d^{\varepsilon }_{1}-1\ge \delta , \end{aligned}$$

  which implies in this case

  $$\begin{aligned} (x_{\varepsilon }-x_0)|y_{\varepsilon }-y_0|^2\le -\delta . \end{aligned}$$

  By sending $\varepsilon$ to zero, we obtain again a contradiction.

• Case 3 $\rho u(y_{\varepsilon })-(rx_{\varepsilon }+bh(q_{\varepsilon }))p^{\varepsilon }_{1}-\mu s_{\varepsilon }p^{\varepsilon }_{2}-c\sigma \eta s_{\varepsilon }h(q_{\varepsilon })M^{\varepsilon }_{12}-\frac{\eta ^2h^2(q_{\varepsilon })}{2}M^{\varepsilon }_{11} -\frac{\sigma ^2s_{\varepsilon }^2}{2}M^{\varepsilon }_{22}\le 0$ in (4.33).

From (4.34), we have

$$\begin{aligned}&\rho w^{\gamma }(y'_{\varepsilon })-\Big (rx'_{\varepsilon }+b h(q'_{\varepsilon })\Big )d^{\varepsilon }_{1}-\mu s'_{\varepsilon }d^{\varepsilon }_{2}\nonumber \\&-c\sigma \eta s'_{\varepsilon }h(q'_{\varepsilon })N^{\varepsilon }_{12}-\frac{\eta ^2h^2(q'_{\varepsilon })}{2}N^{\varepsilon }_{11} -\frac{\sigma ^{2}(s'_{\varepsilon })^2}{2}N^{\varepsilon }_{22}\ge \delta , \end{aligned}$$

which implies in this case

$$\begin{aligned}&\rho \Big (u(y_{\varepsilon })-w^{\gamma }(y'_{\varepsilon })\Big ) -r(x_{\varepsilon }p^{\varepsilon }_{1}-x'_{\varepsilon }d^{\varepsilon }_{1}) -b(h(q_{\varepsilon })p^{\varepsilon }_{1}-h(q'_{\varepsilon })d^{\varepsilon }_{1}) -\mu (s_{\varepsilon }p^{\varepsilon }_{1}-s'_{\varepsilon }d^{\varepsilon }_{1})\nonumber \\&\quad -\,c\sigma \eta (s_{\varepsilon }h(q_{\varepsilon })M^{\varepsilon }_{12}-s'_{\varepsilon }h(q'_{\varepsilon })N^{\varepsilon }_{12}) -\frac{\eta ^2}{2}(h^2(q_{\varepsilon })M^{\varepsilon }_{11}-h^2(q'_{\varepsilon })N^{\varepsilon }_{11})\nonumber \\&\quad -\frac{\sigma ^2}{2}\Big ((s_{\varepsilon })^{2}M^{\varepsilon }_{22}-(s'_{\varepsilon })^{2}N^{\varepsilon }_{22}\Big )\le -\delta \end{aligned}$$

(4.35)

We have that

$$\begin{aligned} x_{\varepsilon }p^{\varepsilon }_{1}-x'_{\varepsilon }d^{\varepsilon }_{1}= x_{\varepsilon }(x_{\varepsilon }-x_0)|y_{\varepsilon }-y_0|^2+\frac{1}{\varepsilon }(x_{\varepsilon }-x'_{\varepsilon })^2. \end{aligned}$$

From (4.31) we have that this last quantity goes to zero when $\varepsilon$ goes to zero. Using the same argument, we also have that the quantity $s_{\varepsilon }p^{\varepsilon }_{1}-s'_{\varepsilon }d^{\varepsilon }_{1}$ goes to zero when $\varepsilon$ goes to zero.

Using that $h(q)\le H$ for all $q\in \mathbb {N}$ we have

$$\begin{aligned} h(q_{\varepsilon })p^{\varepsilon }_{1}-h(q'_{\varepsilon })d^{\varepsilon }_{1}\le & {} h(q_{\varepsilon })(x_{\varepsilon }-x_0)|y_{\varepsilon }-y_0|^2 +\frac{1}{\varepsilon }(x_{\varepsilon }-x'_{\varepsilon })\Big (h(q_{\varepsilon })-h(q'_{\varepsilon })\Big )\\\le & {} h(q_{\varepsilon })(x_{\varepsilon }-x_0)|y_{\varepsilon }-y_0|^2 +H\frac{(x_{\varepsilon }-x'_{\varepsilon })^2+(q_{\varepsilon }-q'_{\varepsilon })^2}{\varepsilon }. \end{aligned}$$

Again, by (4.31), this last quantity goes to zero when $\varepsilon$ goes to zero.

Moreover, assuming that $c\ne 0$, from (4.32), we have

$$\begin{aligned}&c\sigma \eta (s_{\varepsilon }h(q_{\varepsilon })M^{\varepsilon }_{12}-s'_{\varepsilon }h(q'_{\varepsilon })N^{\varepsilon }_{12}) +\frac{\eta ^2}{2}(h^2(q_{\varepsilon })M^{\varepsilon }_{11}-h^2(q'_{\varepsilon })N^{\varepsilon }_{11})\nonumber \\&\quad +\frac{c^2\sigma ^2}{2}\Big ((s_{\varepsilon })^{2}M^{\varepsilon }_{22}-(s'_{\varepsilon })^{2}N^{\varepsilon }_{22}\Big )\le \Upsilon \end{aligned}$$

(4.36)

where

$$\begin{aligned} \Upsilon =\Lambda \Big (D_{y,y'}^2\varphi _{\varepsilon }(y_{\varepsilon },y'_{\varepsilon })+ \varepsilon \Big (D_{y,y'}^2\varphi _{\varepsilon }(y_{\varepsilon },y'_{\varepsilon })\Big )^2\Big )\Lambda ^{\intercal } \end{aligned}$$

with

$$\begin{aligned} \Lambda =\frac{1}{\sqrt{2}}\Big (\eta h(q_{\varepsilon }),c\sigma s_{\varepsilon },\eta h(q'_{\varepsilon }),c\sigma s'_{\varepsilon }\Big )^{\intercal } \end{aligned}$$

Here $\intercal$ denotes the transpose operator.

From (4.32), we have also

$$\begin{aligned} \frac{c^2\sigma ^2}{2}\Big ((s_{\varepsilon })^{2}M^{\varepsilon }_{22}-(s'_{\varepsilon })^{2}N^{\varepsilon }_{22}\Big )\le {\bar{\Upsilon }} \end{aligned}$$

(4.37)

where

$$\begin{aligned} {\bar{\Upsilon }}={\bar{\Lambda }}\Big (D_{y,y'}^2\varphi _{\varepsilon }(y_{\varepsilon },y'_{\varepsilon })+ \varepsilon \Big (D_{y,y'}^2\varphi _{\varepsilon }(y_{\varepsilon },y'_{\varepsilon })\Big )^2\Big ){\bar{\Lambda }}^{\intercal } \end{aligned}$$

with

$$\begin{aligned} {\bar{\Lambda }}=\frac{1}{\sqrt{2}}\Big (0,c\sigma s_{\varepsilon },0,c\sigma s'_{\varepsilon }\Big )^{\intercal } \end{aligned}$$

Combining (4.36) and (4.37) and the fact that $-1 \le c\le 1$, we obtain

$$\begin{aligned}&c\sigma \eta (s_{\varepsilon }h(q_{\varepsilon })M^{\varepsilon }_{12}-s'_{\varepsilon }h(q'_{\varepsilon })N^{\varepsilon }_{12}) +\frac{\eta ^2}{2}(h^2(q_{\varepsilon })M^{\varepsilon }_{11}-h^2(q'_{\varepsilon })N^{\varepsilon }_{11})\nonumber \\&+\frac{\sigma ^2}{2}\Big ((s_{\varepsilon })^{2}M^{\varepsilon }_{22}-(s'_{\varepsilon })^{2}N^{\varepsilon }_{22}\Big )\le \Upsilon + \frac{1-c^2}{c^2}{\bar{\Upsilon }}. \end{aligned}$$

After some straightforward calculations and using (4.31) and that $h(q)\le H$ for all $q\in \mathbb {N}$, we have that $\Upsilon$ and ${\bar{\Upsilon }}$ go to zero when $\varepsilon$ goes to zero.

The case of $c=0$ is treated in the same way by choosing $\Lambda =\frac{1}{\sqrt{2}}(\eta h(q_{\varepsilon }),0,\eta h(q'_{\varepsilon }),0)^{\intercal }$ and ${\bar{\Lambda }}=\frac{1}{\sqrt{2}}(0,\sigma s_{\varepsilon },0,\sigma s'_{\varepsilon })^{\intercal }$.

Finally, using all the above arguments and the continuity of u and $w^{\gamma }$ we can see that when $\varepsilon$ goes to zero in the inequality (4.35), we obtain the required contradiction: $\rho \theta \le -\delta <0$. This ends the proof.$\square$