Optimal renewable resource harvesting model using price and biomass stochastic variations: a utility based approach

Abstract

In this paper, we provide a general framework for analyzing the optimal harvest of a renewable resource (i.e. fish, shrimp) assuming that the price and biomass evolve stochastically and harvesters have a constant relative risk aversion. In order to take into account the impact of a sudden change in the environment linked to the ecosystem, we assume that the biomass are governed by a stochastic differential equation of the Gilpin–Ayala type, with regime change in the parameters of the drift and variance. Under the above assumptions, we find the optimal effort to be deployed by the collector (fishery for example) in order to maximize the expected utility of its profit function. To do this, we give the proof of the existence and uniqueness of the value function, which is derived from the Hamilton–Jacobi–Bellman equations associated with this problem, by resorting to a definition of the viscosity solution.

Appendices

A Proof of Lemma 3.1

• 1. Let $k\in [0,2]$ and $h=s-t$.

  According to Hölder inequality $\mathrm {E} \eta ^k \le \big [\mathrm {E} \eta ^2 \big ]^{k/2}$ for $\forall k \ge 0$,

  $$\begin{aligned} \mathrm {E}\Vert X^{t,x}_h\Vert ^k \le \Big [\mathrm {E}\Vert X^{t,x}_h\Vert ^2 \Big ]^{k/2}. \end{aligned}$$

  (16)

  Given that

  $$\begin{aligned}&dX_t=f(t, X_{t-}, E_i(t),X_t)dt+g(t, X_{t-})dw(t)\\&\quad \qquad +\int _{\mathbb {R}\backslash \{0\} }\eta (t,X_{t-},i,z)\tilde{N}_i(dt, dz);i=1, 2. \end{aligned}$$

  According to the elementary inequality $\left\| \sum _{i=1}^{M}a_i\right\| ^2 \le M \sum _{i=1}^{M} \Vert a_i\Vert ^2$,

  $$\begin{aligned}&\Vert X^{t,x}_h\Vert ^2 \le 4 \bigg [ \Vert x\Vert ^2+\Big \Vert \int _{0}^{h}f(u+t,X^{t,x}_u, i)du\Big \Vert ^2\\&\qquad \qquad \quad + \Big \Vert \int _{0}^{h}g(u+t,X^{t,x}_u,i)dw_u\Big \Vert ^2 +\Big \Vert \int _{0}^{h}\int _{\mathbb {R}\backslash \{0\}} \eta (t,X_{t-},i,z)\tilde{N}_i(dt, dz)\Big \Vert ^2 \bigg ]. \end{aligned}$$

  Using Itô-isometry and Fubini’s theorem

  $$\begin{aligned} \mathrm {E}\Vert X^{t,x}_h\Vert ^2&\le 4 \bigg [\Vert x\Vert ^2+\int _{0}^{h} \mathrm {E}\Big \Vert f(u+t,X^{t,x}_u, i)\Big \Vert ^2du\\&\quad + \int _{0}^{h}\mathrm {E}\Big \Vert g(u+t,X^{t,x}_u, i) \Big \Vert ^2du+\int _{0}^{h}\int _{\mathbb {R}\backslash \{0\} } \mathrm {E}\Big \Vert \eta (t,X_{t-},i,z)\Big \Vert ^2\nu _i(dz)du \bigg ], \end{aligned}$$

  Using the growth condition on f, g and $\eta$, there exists $C_1\in \mathbb {R}$ such that

  $$\begin{aligned} \mathrm {E}\Vert X^{t,x}_h\Vert ^2 \le C_1 \left\{ 1+\Vert x\Vert ^2+\int _{0}^{h}\mathrm {E}\Big \Vert X^{t,x}_u\Big \Vert ^2du \right\} . \end{aligned}$$

  Applying Gronwall’s inequality we obtain

  $$\begin{aligned} \mathrm {E}\Vert X^{t,x}_h\Vert ^2 \le C_1 e^{C_1h}\big [ 1+\Vert x\Vert ^2 \big ] \end{aligned}$$

  i.e

  $$\begin{aligned} \mathrm {E}\Vert X^{t,x}_h\Vert ^2 \le C\big [ 1+\Vert x\Vert ^2 \big ]. \end{aligned}$$

  (17)

  Using (16) and elementary inequalities $(a_1+a_2)^k\le 2^{k-1}(\Vert a_1\Vert ^k+\Vert a_2\Vert ^k)$ and $(\sqrt{\Vert a_1+a_2\Vert }\le \sqrt{\Vert a_1\Vert }+\sqrt{\Vert a_2\Vert })$ we deduce

  $$\begin{aligned} \mathrm {E}\Vert X^{t,x}_h\Vert ^k \le C \big [ 1+\Vert x\Vert ^k \big ]. \end{aligned}$$

  The same reasoning gives us

  $$\begin{aligned} \mathrm {E}\Vert Y^{t,y}_h\Vert ^k \le C \big [ 1+\Vert y\Vert ^k \big ]. \end{aligned}$$

• 2. We have

  $$\begin{aligned} \Vert X^{t,x}_h-x\Vert ^2&\le 3 \bigg [\Big \Vert \int _{0}^{h}f(u+t,P^{t,x}_u, i)du\Big \Vert ^2+ \Big \Vert \int _{0}^{h}g(u+t,X^{t,x}_u, i)dw_u\Big \Vert ^2\\&\quad +\Big \Vert \int _{0}^{h}\int _{\mathbb {R}\backslash \{0\} }\eta (t,X_{t-},i,z)\tilde{N}_i(dt, dz)\Big \Vert ^2 \bigg ]. \end{aligned}$$

  Similar arguments as above we obtain

  $$\begin{aligned} \mathrm {E}\Vert X^{t,x}_h-x\Vert ^2 \le C_1 \int _{0}^{h} \big [ 1+\mathrm {E}\Big \Vert X^{t,x}_u\Big \Vert ^2 \big ]du, \end{aligned}$$

  using (17) we deduce

  $$\begin{aligned} \mathrm {E}\Vert X^{t,x}_h-x\Vert ^2 \le C\big ( 1+\Vert x\Vert ^2 \big )h. \end{aligned}$$

  Hence

  $$\begin{aligned} \mathrm {E}\Vert X^{t,x}_h-x\Vert ^k \le C \big ( 1+\Vert x\Vert ^k \big )h^{k/2}. \end{aligned}$$

• 3. Let us define the process $X^{t,x}_s-X^{t,x'}_s$. Put $\bar{f}(u+t,X^{t,x}_u,X^{t,x'}_u, i)=f(u+t,X^{t,x}_u, i)-f(u+t,X^{t,x'}_u, i)$, $\bar{g}(u+t,X^{t,x}_u,X^{t,x'}_u, i)=g(u+t,X^{t,x}_u, i)-g(u+t,X^{t,x'}_u, i)$ and $\bar{\eta }(t,X^{t,x}_u,X^{t,x'}_u,i,z)=\eta (t,X^{t,x}_u,i,z)-\eta (t,X^{t,x'}_u,i,z)$. Then Applying Itô’s formula, we have

  $$\begin{aligned}&\mathrm {E}\Vert X^{t,x}_h-X^{t,x'}_h\Vert ^2 \le 4 \bigg ( \Vert x-x'\Vert ^2+\mathrm {E}\Big \Vert \int _{0}^{h}\bar{f}(u+t,X^{t,x}_u,X^{t,x'}_u, i)du\Big \Vert ^2\\&\quad +\mathrm {E}\Big \Vert \int _{0}^{h}\bar{g}(u+t,X^{t,x}_u,X^{t,x'}_u,i)dw_u\Big \Vert ^2\\&\quad +\mathrm {E}\Big \Vert \int _{0}^{h}\int _{\mathbb {R}\backslash \{0\} }\bar{\eta }(t,X^{t,x}_u,X^{t,x'}_u,i,z)\tilde{N}_i(dt, dz)\Big \Vert ^2 \bigg ). \end{aligned}$$

  Using Jensen’s inequality, we have

  $$\begin{aligned} \Big \Vert \int _{0}^{h}\bar{f}(u+t,X^{t,x}_u,X^{t,x'}_u, i)du\Big \Vert ^2\le (t-s)\int _{0}^{h}\Big [\bar{f}(u+t,X^{t,x}_u,X^{t,x'}_u, i)\Big ]^2du, \end{aligned}$$

  and using the Itô–Lévy isometry

  $$\begin{aligned}&\mathrm {E}\Big \Vert \int _{0}^{h}\bar{g}(u+t,X^{t,x}_u,X^{t,x'}_u,i)dw_u\Big \Vert ^2\\&\qquad +\mathrm {E}\Big \Vert \int _{0}^{h}\int _{\mathbb {R}\backslash \{0\} }\bar{\eta }(t,X^{t,x}_u,X^{t,x'}_u,i,z)\tilde{N}_i(dt, dz)\Big \Vert ^2\\&\quad \le \mathrm {E}\int _{0}^{h}\Big [\bar{g}(u+t,X^{t,x}_u,X^{t,x'}_u,i)\Big ]^2du\\&\qquad +\mathrm {E}\int _{0}^{h}\int _{\mathbb {R}\backslash \{0\} }\Big [\bar{\eta }(t,X^{t,x}_u,X^{t,x'}_u,i,z)\Big ]^2du \end{aligned}$$

  Therefore, the Lipschitz condition implies that there exists a constant $C_1 > 0$ such that

  $$\begin{aligned} \mathrm {E}\Vert X^{t,x}_h-X^{t,x'}_h\Vert ^2 \le 4 \Vert x-x'\Vert ^2+C_1 (T + 1) \int _{0}^{h}\mathrm {E}\Big \Vert X^{t,x}_u-X^{t,x'}_u \Big \Vert ^2du . \end{aligned}$$

  Applying Gronwall’s inequality, it follows

  $$\begin{aligned} \mathrm {E}\Vert X^{t,x}_h-X^{t,x'}_h\Vert ^2 \le 4 \Vert x-x'\Vert ^2e^{C_1 (T + 1)}. \end{aligned}$$

  Hence

  $$\begin{aligned} \mathrm {E}\Vert X^{t,x}_h-X^{t,x'}_h\Vert ^2 \le C \Vert x-x'\Vert ^2. \end{aligned}$$

  Similar arguments as above we deduce

  $$\begin{aligned} \mathrm {E}\Vert X^{t,x}_h-X^{t,x'}_h\Vert ^2 \le C \Vert x-x'\Vert ^2;~~ \mathrm {E}\Vert Y^{t,y}_h-Y^{t,y'}_h\Vert ^k \le C\Vert y-y'\Vert ^2. \end{aligned}$$

• 4. Using Doob’s inequality for martingale. We get

  $$\begin{aligned} \mathrm {E}\big [\sup _{0\le s\le h}\Vert X^{t,x}_h\Vert \big ]^k \le C(1+\Vert x\Vert ^k)h^{\frac{k}{2}};~ \mathrm {E}\big [\sup _{0\le s\le h}\Vert Y^{t,y}_h\Vert \big ]^k \le C(1+\Vert y\Vert ^k)h^{\frac{k}{2}}. \end{aligned}$$

Proof of Proposition 3.1

We first show that v is Lipschitz in (x, y), uniformly in t and its linear growth condition.

$$\begin{aligned} v_i(s, x_s, y_s)&=\sup _{E\in \mathcal {A}_i}\mathrm {E} \bigg [ \int _{s}^{T}e^{-\beta (u-s)}l(i,u, X_u^{s,x_s}, Y_u^{s,y_s}, E_u)du\\&\quad +e^{-\beta (T-s)}m(X_T^{s,x_s}, Y_T^{s,y_s}) \bigg ]. \end{aligned}$$

• 1. Using elementary inequality $\Vert \sup A-\sup B\Vert \le \sup \Vert A-B \Vert$, from Lipschitz condition (11) on l and from estimate (3.1), with k=1, we have

  $$\begin{aligned}&\Vert v_i(s, x_s, y_s)-v_i(s, x'_s, y'_s) \Vert \\&\quad \le \sup _{E\in \mathcal {A}_i}\Bigg \Vert \mathrm {E}\bigg [ \int _{s}^{T}e^{-\beta (u-s)}\Big (l(i,u, X_u^{s,x_s}, Y_u^{s,y_s}, E_u)\\&\qquad -l(i,u, X_u^{s,x'_s}, Y_u^{s,y'_s}, E_u)\Big )du\\&\qquad +e^{-\beta (T-s)}\Big (m(X_T^{s,x_s}, Y_T^{s,y_s})-m(X_T^{s,x'_s}, Y_T^{s,y'_s})\Big ) \bigg ] \Bigg \Vert \\&\quad \le \sup _{E\in \mathcal {A}_i}\mathrm {E}\bigg [ \int _{s}^{T} \bigg \Vert \Big (l(i,u, X_u^{s,x_s}, Y_u^{s,y_s}, E_u)-l(i,u, X_u^{s,x'_s}, Y_u^{s,y'_s}, E_u)\Big )\bigg \Vert du\\&\qquad +\bigg \Vert \Big (m(X_T^{s,x_s}, Y_T^{s,y_s})-m(X_T^{s,x'_s}, Y_T^{s,y'_s})\Big )\bigg \Vert \bigg ]\\&\quad \le \sup _{E\in \mathcal {A}_i}\mathrm {E}\bigg [ \int _{s}^{T} \Big (\Vert X_u^{s,x_s}-X_u^{s,x'_s}\Vert + \Vert Y_u^{s,y_s}- Y_u^{s,y'_s}\Vert \Big )du\\&\qquad +\Big (\Vert X_T^{s,b_s}-X_T^{s,x'_s}\Vert + \Vert Y_T^{s,y_s}- Y_T^{s,y'_s}\Vert \Big ) \bigg ]\\&\quad \le \sup _{E\in \mathcal {A}_i}\bigg [ \int _{s}^{T} \mathrm {E}\Big (\Vert X_u^{s,x_s}-X_u^{s,x'_s}\Vert + \mathrm {E}\Vert Y_u^{s,y_s}- Y_u^{s,y'_s}\Vert \Big )du\\&\qquad +\Big (\mathrm {E}\Vert X_T^{s,x_s}-X_T^{s,x'_s}\Vert + \mathrm {E}\Vert Y_T^{s,y_s}- Y_T^{s,y'_s}\Vert \Big ) \bigg ]\\&\qquad \Vert v_i(s, x_s, y_s)-v_i(s, x'_s, y'_s) \Vert \le C \Big (\Vert x_s - x'_s\Vert +\Vert y_s - y'_s\Vert \Big ). \end{aligned}$$

• 2. From linear growth condition (12) on l, m, and from estimates (3.1), with $k = 1$, we obtain

  $$\begin{aligned}&\Vert v_i(s, x_s, y_s)\Vert \le \sup _{E\in \mathcal {A}_i}\mathrm {E}\bigg [\int _{s}^{T} \Big \Vert l(i,u, X_u^{s,x_s}, Y_u^{s,y_s}, E_u) \Big \Vert du + \Big \Vert m(X_T^{s,x_s}, Y_T^{s,y_s})\Big \Vert \bigg ]\\&\quad \Vert v_i(s, x_s, y_s)\Vert \le \rho \sup _{E\in \mathcal {A}_i}\mathrm {E}\bigg [\int _{s}^{T} \Big ( 1+\Vert X_u^{s,x_s}\Vert +\Vert Y_u^{s,y_s}\Vert \Big )du + \bigg ( 1+\Vert X_T^{s,x_s}\Vert +\Vert Y_T\Vert \bigg )\bigg ]\\&\Vert v_i(s, x_s, y_s)\Vert \le \rho \sup _{E\in \mathcal {A}_i}\bigg [\int _{s}^{T} \Big ( 1+\mathrm {E}\Vert X_u^{s,x_s}\Vert +\mathrm {E}\Vert Y_u^{s,y_s}\Vert \Big )du\\&\qquad \qquad + \Big ( 1+\mathrm {E}\Vert X_T^{s,x_s}\Vert +\mathrm {E}\Vert Y_T^{s,y_s}\Vert \Big )\bigg ] \Vert v_i(s, x_s, y_s)\Vert \le C \Big ( 1+\Vert x_s\Vert +\Vert y_s\Vert \Big ). \end{aligned}$$

Proof of Proposition 3.2

Let $0\le t < s\le T$. To prove continuity property in time t, we use the dynamic programming principle.

$$\begin{aligned} v_i(t, x, y)&= \sup _{E\in \mathcal {A}_i}\mathrm {E}\bigg [\int _{t}^{T}e^{-\beta (u-t)}l(i,u, X_u^{t,x}, Y_u^{t,y}, E_u)du\\&\quad +e^{-\beta (T-t)}m(X_T^{t,x}, Y_T^{t,y})\bigg ]\\&= \sup _{E\in \mathcal {A}_i}\mathrm {E}\bigg [\int _{t}^{s}e^{-\beta (u-t)}l(i,u, X_u^{t,x}, Y_u^{t,y}, E_u)du+e^{-\beta (s-t)}v(s, X_s^{t,x}, Y_s^{t,y}, i) \bigg ] \\&= \sup _{E\in \mathcal {A}_i}\mathrm {E}\bigg [\int _{o}^{s-t}e^{-\beta u}l(i,t+u, X_{t+u}^{t,x}, Y_{t+u}^{t,y}, E_{t+u})du\\&\quad + e^{-\beta (s-t)}v(s, X_{s-t}^{t,x}, Y_{s-t}^{t,y}, i) \bigg ].\\&0\le v_i(t, x, y)- v_i(s, x_s, y_s)= \sup _{E\in \mathcal {A}_i}\mathrm {E}\bigg [ \int _{0}^{s-t} e^{-\beta (u)}l(i, u, X_u^{t,x}, Y_u^{t,p_t}, E_u)du \\&\qquad +e^{-\beta (s-t)}\left( v(s, X_{s-t}^{t,x}, Y_{s-t}^{t,y}, i)- v(s, x_s, y_s, i)\right) \\&\qquad +\left( e^{-\beta (s-t)}-1 \right) v(s, x_s, y_s, i) \bigg ]. \end{aligned}$$

Applying linear growth condition (12) on l, noting that $0\le 1-e^{-\beta (s-t)}\le \beta (s-t)$ and v satisfies proposition (3.1) of the article, we deduce that:

$$\begin{aligned} \Vert v_i(t, x, y)- v_i(s, x_s, y_s)\Vert&\le \sup _{E\in \mathcal {A}_i}\mathrm {E} \bigg [ \int _{0}^{s-t} \big \Vert l(i, u, X_u^{t,x}, Y_u^{t,y}, E_u)\big \Vert du\\&\quad +\Big \Vert e^{-\beta (s-t)}\big (v(s, X_{s-t}^{t,x}, Y_{s-t}^{t,x}, i)- v(s, x_s, y_s, i)\big )\Big \Vert \\&\quad +\Big \Vert \Big (e^{-\beta (s-t)}-1\Big )v(s, x_s, y_s, i) \Big \Vert \bigg ] \\&\le (s-t)^{\frac{1}{2}} \left( \int _{0}^{s-t} \sup _{E\in \mathcal {A}_i}\mathrm {E} \left\| l(i, u, X_u^{t,x}, Y_u^{t,y}, E_u)\right\| ^2du\right) ^{\frac{1}{2}} \\&\quad +\sup _{E\in \mathcal {A}_i}\mathrm {E}\Big \Vert e^{-\beta (s-t)}\left( v(s, X_{s-t}^{t,x}, Y_{s-t}^{t,y}, i)- v(s, x_s, y_s, i)\right) \Big \Vert \\&\quad +\sup _{E\in \mathcal {A}_i}\mathrm {E}\left\| \left( e^{-\beta (s-t)}- 1\right) v(s, x_s, y_s, i) \right\| \\&\le (s-t)^{\frac{1}{2}} \left( \int _{0}^{s-t} \rho ^2\sup _{E\in \mathcal {A}_i} \left( 1+\mathrm {E}\Vert X_u^{t,x}\Vert +\mathrm {E}\Vert p_u^{t,y}\Vert \right) ^2 du\right) ^{\frac{1}{2}}\\&\quad + \sup _{E\in \mathcal {A}_i}\mathrm {E}\Big \Vert \left( v(s, X_{s-t}^{t,x}, Y_{s-t}^{t,y}, i)- v(s, x_s, y_s, i)\right) \Big \Vert \\&\quad + \beta \Vert s-t\Vert \sup _{E\in \mathcal {A}_i} \mathrm {E}\big \Vert v(s, x_s, y_s, i) \big \Vert \\&\le \Vert s-t\Vert ^{\frac{1}{2}} \left( \int _{0}^{s-t} \rho \sup _{E\in \mathcal {A}_i} \left( 1+\mathrm {E}\Vert X_u^{t,x}\Vert +\mathrm {E}\Vert Y_u^{t,y}\Vert \right) du \right) \\&\quad + \sup _{E\in \mathcal {A}_i}\mathrm {E}\Big \Vert \big (v(s, X_{s-t}^{t,x}, Y_{s-t}^{t,y}, i)- v(s, x_s, y_s, i)\big )\Big \Vert \\&\quad + \beta \Vert s-t\Vert \sup _{E\in \mathcal {A}_i}\mathrm {E}\Big \Vert v(s, x_s, y_s, i) \Big \Vert \\&\le C' \bigg ( \Vert s-t\Vert ^{\frac{1}{2}} \int _{0}^{s-t} (1+ \mathrm {E}\Vert B_u^{t,x}\Vert + \mathrm {E}\Vert P_u^{t,y}\Vert )du+(\Vert x - x_s\Vert +\Vert y - y_s\Vert ) \\&\quad + \beta (1+\Vert x_s\Vert +\Vert y_s\Vert )\Vert s-t\Vert \bigg ) \\&\le C \left[ (1+\Vert x\Vert +\Vert y\Vert )\Vert s-t\Vert ^{\frac{1}{2}}+ \Vert x - x_s\Vert +\Vert y - y_s\Vert \right] . \end{aligned}$$

Proof of Theorem 3.1

We establish the viscosity super- and sub-solution properties, respectively in the following two steps.

Step 1.:

$v_i(t, x, y)$, $i=1,2$ is a viscosity super-solution of Eq. (13).

We already know that $v \in \mathcal {C}^0( [0,T]\times \mathbb {R}_+\times \mathbb {R}_+)$. We first note that $v_i(T,x,y)=\kappa ^{\gamma }\frac{x^{1-\gamma }}{1-\gamma }$ so, the boundary condition at time $t = T$ is clearly satisfied. Let $(s, x_s, y_s)\in [0,T]\times \mathbb {R}_+\times \mathbb {R}_+$, $i\in \mathcal {S}$ and $\phi \in \mathcal {C}^{1,2,2}([0,T]\times \mathbb {R}_+\times \mathbb {R}_+)\cap \mathcal {C}_2([0,T]\times \mathbb {R}_+\times \mathbb {R}_+)$ such that $v_i(.,.,.)-\phi (.,.,.)$ has a local minimum at $(s,x_s,y_s)$. Let $\mathtt {N}(x_s,y_s)$ to be a neighborhood of $(s, x_s,y_s)$ where $v_i(.,.,.)-\phi (.,.,.)$ take its minimum, we introduce a new test-function $\psi$ as follows:

$$\begin{aligned} \psi (.,.,.,j)={\left\{ \begin{array}{ll} \phi (.,.,.)+[v_i(s,x_s,y_s)- \phi (s,x_s,y_s)],~ \text {if}~ j=i,\\ v_i(., ., .),\qquad \qquad \qquad \qquad \qquad \qquad ~\text {if}~ j\not = i. \end{array}\right. } \end{aligned}$$

(18)

This helps us to suppose without any loss of generality that this minimum is equal to 0. Let $\tau _{\alpha }$ be the first jump time of $\alpha (t) \big ( =\alpha (t)^{x_s,y_s,i} \big )$, i.e. $\tau _{\alpha } = \min \{t \ge s : \alpha (t)\not = i\}$. Then $\tau _{\alpha } > s$, a.s. Let $\theta _s\in (s, \tau _{\alpha })$ be such that the state $(X_t^{x_s,i},Y_t^{y_s,i})$ starts at $(x_s,y_s)$ and stays in $\mathtt {N}(x_s,y_s)$ for $s \le t \le \theta _s$. Applying the generalized Itô’s formula to the switching process $e^{-\beta t}\psi (t,X_t,Y_t,\alpha (t) )$, taking integral from $t = s$ to $t = \theta _s \wedge h$, where $h > 0$ is a positive constant, and then taking expectation we have

$$\begin{aligned}&\mathrm {E}_{x_s, y_s, i}\bigg [e^{-\beta \theta _s \wedge h }\psi (\theta _s \wedge h, X_{\theta _s \wedge h},Y_{\theta _s \wedge h},\alpha (\theta _s \wedge h))\bigg ]\nonumber \\&\quad =e^{-\beta s}\psi (s,X_s,Y_s,i)+\mathrm {E}_{x_s, y_s, i}\Bigg [\int _{s}^{\theta _s \wedge h}e^{-\beta t}\bigg \lbrace -\beta \psi (t, X_t, Y_t, \alpha (t) )\nonumber \\&\qquad +\dfrac{\partial \psi (t, X_t, Y_t, \alpha (t))}{\partial t}+X_{t}\Big [r_i-a_iX^{\lambda }_{t}-qE_i(t) \Big ] \dfrac{\partial \psi (t, X_t, Y_t, \alpha (t))}{\partial x}\nonumber \\&\qquad +\theta (\bar{Y}_0-Y_t)\dfrac{\partial \psi (t,X_t,Y_t,\alpha (t))}{\partial y} +\dfrac{1}{2}\sigma ^2X_t^2\dfrac{\partial ^2 \psi (t, X_t, Y_t, \alpha (t))}{\partial x^2}\nonumber \\&\qquad +\dfrac{1}{2}\sigma _Y^2\dfrac{\partial ^2 \psi (t, X_t, Y_t,\alpha (t))}{\partial y^2} + q_{\alpha (t)j}(\psi (t,X_t,Y_t,j)-\psi (t,X_t,Y_t,\alpha (t)))\nonumber \\&\qquad +\int _{\mathbb {R}\backslash \{0\} }\bigg (\psi (t,X_t+\eta (t,X_t,Y_t,\alpha (t),z),Y_t)-\psi (t, X_t, Y_t, \alpha (t))\nonumber \\&\qquad -\eta (t,X_t,Y_t,\alpha (t),z) \dfrac{\partial \psi (t, X_t, Y_t, \alpha (t))}{\partial x}\bigg )\nu _i(dz)\bigg \rbrace dt\Bigg ], ~~\alpha (t)\not =j. \end{aligned}$$

(19)

From hypothesis, for any $t \in [s,\theta _s \wedge h ]$

$$\begin{aligned}&v_i(t,X_t^{x_s},Y_t^{y_s})\ge \nonumber \\&\quad \phi (t,X_t^{x_s},Y_t^{y_s}) +v_i(s,x_s,y_s)- \phi (s,x_s,y_s)\ge \psi (t,X_t^{x_s},Y_t^{y_s},i). \end{aligned}$$

(20)

Moreover, we have $q_{ij}(v_j(t,X_t,Y_t)-v_i(t,X_t,Y_t))\le q_{ij}(\psi (t,X_t,Y_t,j)-\psi (t,X_t,Y_t,i))$, by definition of $\psi$. Recalling that $(X_s^{y_s},Y_s^{y_s})= (x_s,y_s)$ and using Eq. (18) and Eq. (20) we have

$$\begin{aligned}&\mathrm {E}_{x_s, y_s, i}\bigg [e^{-\beta \theta _s \wedge h }\psi (\theta _s \wedge h, X_{\theta _s \wedge h},Y_{\theta _s \wedge h},\alpha (\theta _s \wedge h))\bigg ]\ge \nonumber \\&\quad + e^{-\beta s}v_i(s,x_s,y_s)+ \mathrm {E}_{x_s, y_s, i}\Bigg [\int _{s}^{\theta _s \wedge h}e^{-\beta t}\bigg \lbrace -\beta v_i(t,X_t,Y_t)\nonumber \\&\quad +\dfrac{\partial \psi (t,X_t,Y_t,\alpha (t))}{\partial t}\nonumber \\&\quad +X_{t}\Big [r_i-a_iX^{\lambda }_{t}-qE_i(t) \Big ] \dfrac{\partial \psi (t, X_t, Y_t, \alpha (t))}{\partial x} +\theta (\bar{Y}_0-Y_t)\dfrac{\partial \psi (t,X_t,Y_t,\alpha (t))}{\partial y}\nonumber \\&\quad +\dfrac{1}{2}\sigma ^2X_t^2\dfrac{\partial ^2 \psi (t,X_t,Y_t,\alpha (t))}{\partial x^2} +\dfrac{1}{2}\sigma _Y^2 \dfrac{\partial ^2 \psi (t,X_t,Y_t,\alpha (t))}{\partial x^2} \nonumber \\&\quad + q_{ij}(v_j(t,X_t,Y_t)-v_i(t,X_t,Y_t))\nonumber \\&\quad +\int _{\mathbb {R} \backslash \{0\} }\bigg (\psi (t,X_t+\eta (t,X_t,Y_t,\alpha (t),z),Y_t) -\psi (t, X_t, Y_t, \alpha (t))\nonumber \\&\quad -\eta (t,X_t,Y_t,\alpha (t),z) \dfrac{\partial \psi (t, X_t, Y_t, \alpha (t))}{\partial x}\bigg ) \nu _i(dz)\bigg \rbrace dt\Bigg ]. \end{aligned}$$

(21)

By Bellman’s principle

$$\begin{aligned} e^{-\beta s}\psi (s,x_s,y_s,i)&=e^{-\beta s}v_i(s,x_s,y_s)\nonumber \\&\quad =\sup _{E\in \mathcal {A}_i} \mathrm {E}_{x_s, y_s, i}\bigg [\int _{s}^{\theta _s \wedge h}e^{-\beta t}l(i,t, X_{t}^{s,x_s}, Y_{t}^{s,y_s}, E_{t})dt\nonumber \\&\qquad + e^{-\beta (\theta _s \wedge h)}v_i( \theta _s \wedge h, X_{\theta _s \wedge h}^{s,x_s}, Y_{\theta _s \wedge h}^{s,y_s})\bigg ]\nonumber \\&\quad \ge \sup _{E\in \mathcal {A}_i}\mathrm {E}_{x_s, y_s, i}\bigg [\int _{s}^{\theta _s \wedge h} e^{-\beta t}l(i,t, X_{t}^{s,x_s}, Y_{t}^{s,y_s}, E_{t})dt\nonumber \\&\qquad + e^{-\beta (\theta _s \wedge h)}\psi (\theta _s \wedge h , X_{\theta _s \wedge h}^{s,x_s}, Y_{\theta _s \wedge h}^{s,y_s}, i)\bigg ]. \end{aligned}$$

(22)

Setting $\tau =\mathrm {E}(\theta _s \wedge h)$ combining (21) and (22) and multiplying both sides by $1/(\tau -s) > 0$, we obtain

$$\begin{aligned}&\sup _{E\in \mathcal {A}_i}\mathrm {E}_{x_s, y_s, i}\Bigg [\dfrac{1}{\tau -s}\int _{s}^{\theta _s \wedge h}e^{-\beta t}\bigg \lbrace \beta v_i(t,X_t,Y_t)-\dfrac{\partial \psi (t,X_t,Y_t,\alpha (t))}{\partial t}\nonumber \\&\qquad -X_{t}\Big [r_i-a_iX^{\lambda }_{t}-qE_i(t) \Big ] \dfrac{\partial \psi (t, X_t, Y_t, \alpha (t))}{\partial x} -\theta (\bar{Y}_0-Y_t)\dfrac{\partial \psi (t,X_t,Y_t,\alpha (t))}{\partial y}\nonumber \\&\qquad -\dfrac{1}{2}\sigma ^2X_t^2\dfrac{\partial ^2 \psi (t,X_t,Y_t,\alpha (t))}{\partial x^2} -\dfrac{1}{2}\sigma _Y^2\dfrac{\partial ^2 \psi (t,X_t,Y_t,\alpha (t))}{\partial x^2} \nonumber \\&\qquad - q_{ij}[v_j(t,X_t,Y_t)-v_i(t,X_t,Y_t)]\nonumber \\&\qquad -\int _{\mathbb {R}\backslash \{0\} }\bigg (\psi (t,X_t+\eta (t,X_t,Y_t,\alpha (t),z),Y_t)-\psi (t, X_t, Y_t, \alpha (t))\nonumber \\&\qquad -\eta (t,X_t,Y_t,\alpha (t),z) \dfrac{\partial \psi (t, X_t, Y_t, \alpha (t))}{\partial x}\bigg )\nu _i(dz)\nonumber \\&\qquad -l(i,t,X_t,Y_t, E_{t})\bigg \rbrace dt\Bigg ] \ge 0. \end{aligned}$$

(23)

Letting $\tau \downarrow s$ and using the dominated convergence theorem, it turns out that

$$\begin{aligned}&e^{-\beta s}\Bigg [ -\dfrac{\partial \psi (s,x_s,y_s,i)}{\partial t}+\inf _{E\in \mathcal {A}_i}\bigg \lbrace \beta v_i(s,x_s,y_s)\nonumber \\&\qquad -x_s\Big [r_i-a_ix^{\lambda }_{s}-qE_i(s) \Big ] \dfrac{\partial \psi (s,x_s,y_s,i)}{\partial x} -\theta (\bar{Y}_0-y_s)\dfrac{\partial \psi (s,x_s,y_s,i)}{\partial y}\nonumber \\&\qquad -\dfrac{1}{2}\sigma ^2x_s^2\dfrac{\partial ^2 \psi (s,x_s,y_s,i)}{\partial b^2} -\dfrac{1}{2}\sigma _Y^2\dfrac{\partial ^2 \psi (s,x_s,y_s,i)}{\partial y^2} \nonumber \\&\qquad - q_{ij}[v_j(s,x_s,y_s)-v_i(s,x_s,y_s)]\nonumber \\&\qquad -\int _{\mathbb {R}\backslash \{0\} }\bigg (\psi (s,x_s+\eta (s,x_s,y_s,i,z),y_s)-\psi (s,x_s,y_s, i)\nonumber \\&\qquad -\eta (s,x_s,y_s,i,z) \dfrac{\partial \psi (s,x_s,y_s, i)}{\partial x}\bigg )\nu _i(dz)-l(i,s,x_s,y_s, E_{s})\bigg \rbrace \Bigg ] \ge 0. \end{aligned}$$

(24)

This shows that the value function $v_i(t, x, y)$, $i=1, 2$, satisfies the viscosity super-solution property (15).

Step 2.:

$v_i(t, x, y)$, $i=1, 2$, is a viscosity sub-solution of (13).

We argue by contradiction. Assume that there exist an $i_0\in \mathcal {S}$, a point $(s,x_s,y_s)\in [0,T]\times \mathbb {R}_+^*\times \mathbb {R}_+^*$ and a testing function $\phi _{i_0}\in \mathcal {C}^{1,2,2}([0,T]\times \mathbb {R}_+^*\times \mathbb {R}_+^*)\cap \mathcal {C}_2([0,T]\times \mathbb {R}_+^*\times \mathbb {R}_+^*)$ such that $v_{i_0}(.,.,.)-\phi _{i_0}(.,.,.)$ has a local maximum at $(s, x_s, y_s)$ in a bounded neighborhood $\mathtt {N}(x_s,y_s)$, $v_{i_0}(s, x_s,y_s)=\phi _{i_0}(s, x_s,y_s)$, and

$$\begin{aligned}&\min \Bigg [ -\dfrac{\partial \phi _{i_0}(s,x_s,y_s)}{\partial t}+\inf _{E\in \mathcal {A}_{i_0}}\bigg \lbrace \beta v_{i_0}(s,x_s,y_s)\nonumber \\&\qquad - x_s\Big [r_{i_0}-a_ix^{\lambda }_{s}-qE_{i_0}(s) \Big ] \dfrac{\partial \phi _{i_0}(s,x_s,y_s)}{\partial x}\nonumber \\&\qquad -\theta (\bar{Y}_0-y_s)\dfrac{\partial \phi _{i_0}(s,x_s,y_s)}{\partial y}\nonumber \\&\qquad -\dfrac{1}{2}\sigma ^2x_s^2\dfrac{\partial ^2 \phi _{i_0}(s,x_s,y_s)}{\partial x^2} -\dfrac{1}{2}\sigma _Y^2\dfrac{\partial ^2 \phi _{i_0}(s,x_s,y_s)}{\partial y^2} \nonumber \\&\qquad - q_{{i_0}j}[v_j(s,x_s,y_s)-v_{i_0}(s,x_s,y_s)] \nonumber \\&\qquad -\int _{\mathbb {R}\backslash \{0\} }\bigg (\phi _{i_0}(s,x_s+\eta (s,x_s,y_s,{i_0},z),y_s)-\phi _{i_0}(s,x_s,y_s)\nonumber \\&\qquad -\eta (s,x_s,y_s,{i_0},z) \dfrac{\partial \phi _{i_0}(s,x_s,y_s)}{\partial x}\bigg )\nu _{i_0}(dz)\nonumber \\&\qquad -l(_{i_0},s,x_s,y_s, E_{s})\bigg \rbrace , v_{i_0}(T,x_s,y_s)-\kappa ^{\gamma }\frac{x_s^{1-\gamma }}{1-\gamma }\Bigg ] > 0, ~ i_0\not = j. \end{aligned}$$

(25)

By the continuity of all functions involved in (25) $(v_{i_0}, \frac{\partial \phi _{i_0}}{\partial x}, \frac{\partial ^2 \phi _{i_0}}{\partial x^2}, q_{ij}, l,...)$, there exists a $\delta > 0$ and an open ball $\mathtt {B}_{\delta }(x_s,y_s) \subset \mathtt {N}(x_s,y_s)$ such that,

$$\begin{aligned}&-\dfrac{\partial \phi _{i_0}(t,x,y)}{\partial t}+ \inf _{E\in \mathcal {A}_{i_0}}\bigg \lbrace \beta v_{i_0}(t,x,y)\nonumber \\&\qquad -x\Big [r_{i_0}-a_ix^{\lambda }-qE_{i_0}(t) \Big ] \dfrac{\partial \phi _{i_0}(t,x,y)}{\partial x} -\theta (\bar{Y}_0-y)\dfrac{\partial \phi _{i_0}(t,x,y)}{\partial y}\nonumber \\&\qquad -\dfrac{1}{2}\sigma ^2x^2\dfrac{\partial ^2 \phi _{i_0}(t,x,y)}{\partial x^2} -\dfrac{1}{2}\sigma _Y^2\dfrac{\partial ^2 \phi _{i_0}(t,x,y)}{\partial y^2} - q_{{i_0}j}[v_j(t,x,y)-v_{i_0}(t,x,y)]\nonumber \\&\qquad -\int _{\mathbb {R}\backslash \{0\} }\bigg (\phi _{i_0}(t,x+\eta (t,x,y,{i_0},z),y_s)-\phi _{i_0}(t,x,y)\nonumber \\&\qquad -\eta (t,x,y,{i_0},z) \dfrac{\partial \phi _{i_0}(t,x,y)}{\partial x}\bigg )\nu _{i_0}(dz) -l(_{i_0},t,x,y, E_{t})\bigg \rbrace > \delta , \nonumber \\&\qquad i_0\not = j,~~(t,x,y)\in \mathtt {B}_{\delta }(x_s,y_s) \end{aligned}$$

(26)

and

$$\begin{aligned} v_{i_0}(T,x,y)-\kappa ^{\gamma }\frac{x^{1-\gamma }}{1-\gamma } > \delta ~~(t,x,y)\in \mathtt {B}_{\delta }(x_s,y_s). \end{aligned}$$

Let $\theta _{\delta }=\min \{t\ge s: (t, X_{t}, Y_{t})\not \in \mathtt {B}_{\delta }(x_s,y_s) \}$ be the first exit time of $(t, X_{t}, Y_{t})$ $(=(t, X_{t}^{s,x_s}, Y_{t}^{s,y_s}))$ from $\mathtt {B}_{\delta }(x_s,y_s)$. Let $\theta _s= \theta _{\delta } \wedge \tau _{\alpha }$ where $\tau _{\alpha }$ is the first stopping time of $\alpha (t)^{x_s,y_s,i_0}$. Then $\theta _s > 0$, a.s.. For $0 \le t \le \theta _s$, we have

$$\begin{aligned}&\beta v_{i_0}(t,X_t,Y_t)-\dfrac{\partial \phi _{i_0}(t,X_t,Y_t)}{\partial t}\nonumber \\&\quad -X_t\Big [r_{i_0}-a_iX_t^{\lambda }-qE_{i_0}(t) \Big ] \dfrac{\partial \phi _{i_0}(t,X_t,Y_t)}{\partial x} -\theta (\bar{Y}_0-Y_t)\dfrac{\partial \phi _{i_0}(t,X_t,Y_t)}{\partial y}\nonumber \\&\quad -\dfrac{1}{2}\sigma ^2x_s^2\dfrac{\partial ^2 \phi _{i_0}(t,X_t,Y_t)}{\partial x^2} -\dfrac{1}{2}\sigma _Y^2\dfrac{\partial ^2 \phi _{i_0}(t,X_t,Y_t)}{\partial y^2}\nonumber \\&\quad - q_{{i_0}j}[v_j(t,X_t,Y_t)-v_{i_0}(t,X_t,Y_t)]\nonumber \\&\quad -\int _{\mathbb {R}\backslash \{0\} }\bigg (\phi _{i_0}(t,X_t+\eta (t,X_t,Y_t,{i_0},z),y_s) -\phi _{i_0}(t,X_t,Y_t)\nonumber \\&\quad -\eta (t,X_t,Y_t,{i_0},z) \dfrac{\partial \phi _{i_0}(t,X_t,Y_t)}{\partial x}\bigg )\nu _{i_0}(dz) -l(_{i_0},t,X_t,Y_t, E_{t}) > \delta ,\nonumber \\&\quad i_0\not = j, (t,X_t,Y_t)\in \mathtt {B}_{\delta }(x_s,y_s) \end{aligned}$$

(27)

and

$$\begin{aligned} v_{i_0}(T,x,y)-\kappa ^{\gamma }\frac{x^{1-\gamma }}{1-\gamma } > \delta ~~(t,x,y)\in \mathtt {B}_{\delta }(x_s,y_s). \end{aligned}$$

(28)

As previously, we can replace $\phi _{i_0}$ by a new test-function $\psi$ defined as follows:

$$\begin{aligned} \psi (.,.,.,j)={\left\{ \begin{array}{ll} \phi _{i_0}(.,.,.),\qquad \text {if}~ j={i_0}, \\ v_{i_0}(., ., .),\qquad \text {if}~~j\not = {i_0}. \end{array}\right. } \end{aligned}$$

(29)

For any first exit time $\tau \in [s,T]$. Applying Itô’s formula to the switching process $e^{-\beta t}\psi (t,X_t,Y_t,\alpha (t) )$, taking integral from $t = s$ to $t = (\theta _s \wedge \tau )-$ and then taking expectation yield

$$\begin{aligned}&\mathrm {E_{x_s, y_s, i}}\bigg [e^{-\beta \theta \wedge \tau }\psi (\theta _s \wedge \tau , X_{ \theta _s \wedge \tau },Y_{\theta _s \wedge \tau },\alpha ( \theta _s \wedge \tau ))\bigg ]\nonumber \\&= e^{-\beta s}v_{i}(s,x_s,y_s) +\mathrm {E}_{x_s, y_s, i}\Bigg [\int _{s}^{(\theta _s \wedge \tau )- }e^{-\beta t}\bigg \lbrace -\beta \psi (t,X_t,Y_t,\alpha (t) )\nonumber \\&\quad +\dfrac{\partial \psi (t,X_t,Y_t,\alpha (t))}{\partial t}\nonumber \\&\quad +X_t\Big [r_{i}-a_iX_t^{\lambda }-qE_{i}(t) \Big ] \dfrac{\partial \psi (t,X_t,Y_t,\alpha (t))}{\partial x} +\theta (\bar{Y}_0-Y_t)\dfrac{\partial \psi (t,X_t,Y_t,\alpha (t))}{\partial y}\nonumber \\&\quad +\dfrac{1}{2}\sigma ^2X_t^2\dfrac{\partial ^2 \psi (t,X_t,Y_t,\alpha (t))}{\partial x^2} +\dfrac{1}{2}\sigma _Y^2\dfrac{\partial ^2 \psi (t,X_t,Y_t,\alpha (t))}{\partial y^2} \nonumber \\&\quad + q_{\alpha (t)j}(\psi (t,X_t,Y_t,j)-\psi (t,X_t,Y_t,\alpha (t)))\nonumber \\&\quad +\int _{\mathbb {R}\backslash \{0\} }\bigg (\psi (t,X_t+\eta (t,X_t,Y_t,\alpha (t),z),Y_t)-\psi (t, X_t, Y_t, \alpha (t))\nonumber \\&\quad -\eta (t,X_t,Y_t,\alpha (t),z) \dfrac{\partial \psi (t, X_t, Y_t, \alpha (t))}{\partial x}\bigg )\nu _i(dz)\bigg \rbrace dt\Bigg ], ~~\alpha (t)\not =j. \end{aligned}$$

(30)

in which we used $\mathrm {E}_{x_s, y_s, i}\bigg [e^{-\beta \theta _s \wedge \tau }\psi (\theta _s \wedge \tau , X_{ \theta _s \wedge \tau },Y_{\theta _s \wedge \tau },\alpha ( \theta _s \wedge \tau ))\bigg ]=\mathrm {E}_{x_s, y_s, i}\bigg [e^{-\beta \theta _s \wedge \tau }\psi (\theta _s \wedge \tau , X_{ \theta _s \wedge \tau },Y_{\theta _s \wedge \tau },\alpha ( \theta _s \wedge \tau )-)\bigg ]$ due to continuity. Noting that the integrand in the RHS of (30) is continuous in t. Using (27), (28) and that $v_{i_0}(t,X_t,Y_t) \le \phi _{i_0}(t,X_t,Y_t)$ in (30). Also noting that $\alpha (t) = i_0$ for $0 \le t \le \theta _s$, it follows

$$\begin{aligned}&e^{-\beta s}v_{i_0}(s,x_s,y_s)\nonumber \\&\quad \ge \mathrm {E}_{x_s, y_s, i_0}\bigg [e^{-\beta \theta _s \wedge \tau }\phi _{i_0}(\theta _s \wedge \tau , X_{ \theta _s \wedge \tau },Y_{\theta _s \wedge \tau },\alpha ( \theta _s \wedge \tau ))\nonumber \\&\qquad +\int _{s}^{(\theta _s \wedge \tau ) }e^{-\beta t}\bigg \lbrace \beta v_{i_0}(t,X_t,Y_t)-\dfrac{\partial \phi _{i_0}(t,X_t,Y_t)}{\partial t}\nonumber \\&\qquad -X_t\Big [r_{i_0}-a_iX_t^{\lambda }-qE_{i_0}(t) \Big ] \dfrac{\partial \phi _{i_0}(t,X_t,Y_t)}{\partial x} -\theta (\bar{Y}_0-Y_t)\dfrac{\partial \phi _{i_0}(t,X_t,Y_t)}{\partial y}\nonumber \\&\qquad -\dfrac{1}{2}\sigma ^2X_t^2\dfrac{\partial ^2 \phi _{i_0}(t,X_t,Y_t)}{\partial x^2} -\dfrac{1}{2}\sigma _Y^2\dfrac{\partial ^2 \phi _{i_0}(t,X_t,Y_t)}{\partial y^2} \nonumber \\&\qquad - q_{i_0j}[v_j(t,X_t,Y_t)-v_{i_0}(t,X_t,Y_t)]\nonumber \\&\qquad -\int _{\mathbb {R}\backslash \{0\} }\bigg (\psi (t,X_t+\eta (t,X_t,Y_t,i_0,z),Y_t)-\psi (t, X_t, Y_t, i_0)\nonumber \\&\qquad -\eta (t,X_t,Y_t,i_0,z) \dfrac{\partial \psi (t, X_t, Y_t, i_0)}{\partial x}\bigg )\nu _i(dz)\bigg \rbrace dt\Bigg ],~~i_0\not =j \end{aligned}$$

(31)

i.e

$$\begin{aligned}&e^{-\beta s}v_{i_0}(s,x_s,y_s)\nonumber \\&\quad \ge \mathrm {E_{x_s, y_s, i_0}}\bigg [e^{-\beta \tau }v_{i_0}(\tau , X_{ \tau },Y_{\tau },\alpha (\tau )) \mathbf {1}_{\{ \tau<\theta _s\}}\nonumber \\&\qquad +e^{-\beta \theta _s}v_{i_0}(\theta , X_{ \theta _s},Y_{\theta _s},\alpha (\theta _s)) \mathbf {1}_{\{ \tau \ge \theta _s\}}\nonumber \\&\qquad +\int _{s}^{(\theta _s \wedge \tau ) }e^{-\beta t}\big \lbrace l(_{i_0},t,X_t,Y_t, E_{t}) +\delta \big \rbrace dt\bigg ]\nonumber \\&\quad \ge \mathrm {E}_{x_s, y_s, i_0}\bigg [e^{-\beta \tau }[ \kappa ^{\gamma }\frac{X_{\tau }^{1-\gamma }}{1-\gamma }+\delta ]) \mathbf {1}_{\{ \tau<\theta _s\}}\nonumber \\&\qquad +e^{-\beta \theta _s}v_{i_0}(\theta _s , X_{ \theta _s},Y_{\theta _s},\alpha (\theta _s)) \mathbf {1}_{\{ \tau \ge \theta _s\}}\nonumber \\&\qquad +\int _{s}^{(\theta _s \wedge \tau ) }e^{-\beta t}\big \lbrace l(_{i_0},t,X_t,Y_t, E_{t}) +\delta \big \rbrace dt\bigg ]\nonumber \\&\quad \ge \mathrm {E}_{x_s, y_s, i_0}\bigg [ +\int _{s}^{(\theta _s \wedge \tau )}e^{-\beta t}\big \lbrace l(_{i_0},t,X_t,Y_t, E_{t}) \big \rbrace dt\nonumber \\&\qquad +e^{-\beta \theta }v_{i_0}(\theta _s , X_{ \theta _s},Y_{\theta _s},\alpha (\theta _s)) \mathbf {1}_{\{ \tau \ge \theta _s\}}\nonumber \\&\qquad + e^{-\beta \tau }[ \kappa ^{\gamma }\frac{X_{\tau }^{1-\gamma }}{1-\gamma } ]\mathbf {1}_{\{ \tau<\theta _s\}}\bigg ] +\delta \mathrm {E_{x_s, y_s, i_0}}\bigg [\int _{s}^{(\theta _s \wedge \tau )}e^{-\beta t} dt+ e^{-\beta \tau }\mathbf {1}_{\{ \tau <\theta _s\}}\bigg ]. \end{aligned}$$

(32)

Now considering the estimate of the term $\mathrm {E}_{x_s, y_s, i_0}\bigg [\int _{s}^{(\theta _s \wedge \tau )}e^{-\beta t} dt+ e^{-\beta \tau }\mathbf {1}_{\{ \tau <\theta _s\}}\bigg ]$, there exists a positive constant $C_0$ such that,

$$\begin{aligned} \mathrm {E_{x_s, y_s, i_0}}\bigg [\int _{s}^{(\theta _s \wedge \tau )}e^{-\beta t} dt+ e^{-\beta \tau }\mathbf {1}_{\{ \tau <\theta _s\}}\bigg ]\ge C_0\big (1-\mathrm {E_{x_s, y_s, i_0}}\big [e^{-\beta \tau _{\alpha }}\big ]\big ). \end{aligned}$$

It follows that

$$\begin{aligned}&v_{i_0}(s,x_s,y_s)\nonumber \\&\quad \ge \sup _{\tau \in [s,T], E\in \mathcal {A} }\mathrm {E_{x_s, y_s, i_0}}\bigg [ +\int _{s}^{(\theta \wedge \tau )}e^{-\beta t}\big \lbrace l(_{i_0},t,X_t,Y_t, E_{t}) \big \rbrace dt \nonumber \\&\qquad +e^{-\beta \theta }v_{i_0}(\theta , X_{ \theta },Y_{\theta },\alpha (\theta )) \mathbf {1}_{\{ \tau \ge \theta \}} + e^{-\beta \tau }[ \kappa ^{\gamma }\frac{X_{\tau }^{1-\gamma }}{1-\gamma } ]\mathbf {1}_{\{ \tau <\theta _s\}}\bigg ]\nonumber \\&\qquad +C_0\delta \big (1-\mathrm {E_{x_s, y_s, i_0}}\big [e^{-\beta \tau _{\alpha }}\big ]\big ) \end{aligned}$$

(33)

which is a contradiction to the DP principle since $\mathrm {E_{x_s, y_s, i_0}}\big [e^{-\beta \tau _{\alpha }}\big ] < 1$. Therefore the value function $v_i(t, x, y)$, $i=1,2$, is a viscosity sub-solution of the system (13).

This completes the proof of Theorem 3.1.

Proof of Theorem 3.2

For $\varrho , \epsilon ,\delta , \lambda > 0$, we define the auxiliary functions $\phi$ : $(0,T]\times \mathbb {R}_+^2\times \mathbb {R}_+^2 \rightarrow \mathbb {R}$ and $\Xi$: $[0; T] \times \mathbb {R}_+^2 \times \mathbb {R}_+^2\times \mathcal {S}$ by

$$\begin{aligned} \phi (t, (x, y),(x',y'))= \frac{\varrho }{t}+\frac{1}{2\epsilon }\Vert (x, y)- (x', y') \Vert ^2+\delta e^{\lambda (T-t)}( \Vert (x, y)\Vert ^2 +\Vert (x', y')\Vert ^2 ) \end{aligned}$$

and

$$\begin{aligned} \Xi (t, (x, y),(x', y'), i)=v_i(t,x,y)-u_i(t,x',y')-\phi (t, (x, y),(x', y')). \end{aligned}$$

By using the linear growth of $v_i$ and $u_i$, we have for each $i\in \mathcal {S}$

$$\begin{aligned} \lim \limits _{ \Vert (x, y)\Vert +\Vert (x', y')\Vert \rightarrow \infty }\Xi (t, (x, y),(x', y'), i)=-\infty . \end{aligned}$$

Then, since $v_i$ and $u_i$ are uniformly continuous with respect to (t, x, y) on each compact subset of $[0,T] \times \mathbb {R}_+ \times \mathbb {R}_+\times \mathcal {S}$ and that $\mathcal {S}$ is a finite set, $\Xi$ attains its global maximum at some finite point belonging to a compact $K \subset [0,T] \times \mathbb {R}_+^2 \times \mathbb {R}_+^2\times \mathcal {S}$ say, $\left( t_{\delta \epsilon }, ( x_{1\delta \epsilon } , y_{1\delta \epsilon } ) , ( x_{2\delta \epsilon }, y_{2\delta \epsilon } ), \alpha _{\delta \epsilon }\right)$. Observing that $2 \Xi \big (t_{\delta \epsilon }, ( x_{1\delta \epsilon } , y_{1\delta \epsilon } ) , ( x_{2\delta \epsilon }, y_{2\delta \epsilon } ), \alpha _{\delta \epsilon }\big )\ge \Xi \big (t_{\delta \epsilon }, ( x_{1\delta \epsilon } , y_{1\delta \epsilon } ), ( x_{2\delta \epsilon }, y_{2\delta \epsilon } ), \alpha _{\delta \epsilon }\big ) +\Xi \big (t_{\delta \epsilon }, ( x_{1\delta \epsilon } , y_{1\delta \epsilon } ) , ( x_{2\delta \epsilon }, y_{2\delta \epsilon } ), \alpha _{\delta \epsilon }\big )$ and using the uniform continuity of $v_i$ and $u_i$ on K we have

$$\begin{aligned}&\frac{1}{\epsilon }\Vert ( x_{1\delta \epsilon } , y_{1\delta \epsilon } ) - (x_{2\delta \epsilon }, y_{2\delta \epsilon } ) \Vert ^2 \\&\quad \le v_i \left( t_{\delta \epsilon }, ( x_{1\delta \epsilon } , y_{1\delta \epsilon } )\right) -v_i\big (t_{\delta \epsilon }, (x_{2\delta \epsilon } , y_{2\delta \epsilon } )\big )\\&\qquad +u_i\left( t_{\delta \epsilon }, ( x_{1\delta \epsilon } , y_{1\delta \epsilon } )\right) -u_i\left( t_{\delta \epsilon }, ( x_{2\delta \epsilon } , y_{2\delta \epsilon } )\right) \\&\quad \le 2C \Vert ( x_{1\delta \epsilon } , y_{1\delta \epsilon })-( x_{2\delta \epsilon }, y_{2\delta \epsilon } )\Vert . \end{aligned}$$

Thus,

$$\begin{aligned} \Vert ( x_{1\delta \epsilon } , y_{1\delta \epsilon })-(x_{2\delta \epsilon }, y_{2\delta \epsilon })\Vert \le 2C\epsilon \end{aligned}$$

(34)

where C is a positive constant independent of $\varrho , \epsilon ,\delta , \lambda$. From the inequality,

$$\begin{aligned} 2 \Xi \left( T,(0 ,0 ) , (0,0 ), \alpha _{\delta \epsilon }\right) \le 2 \Xi \left( t_{\delta \epsilon }, ( x_{1\delta \epsilon } , y_{1\delta \epsilon } ) , ( x_{2\delta \epsilon }, y_{2\delta \epsilon } ), \alpha _{\delta \epsilon }\right) \end{aligned}$$

and the linear growth for $v_i$ and $u_i$, we have:

$$\begin{aligned}&\delta \left( \Vert (x_{1\delta \epsilon }, y_{1\delta \epsilon })\Vert ^2+\Vert \left( x_{2\delta \epsilon }, y_{2\delta \epsilon } \right) \Vert ^2\right) \nonumber \\&\quad \le e^{-\lambda (T-t_{\delta \epsilon })}\Big [v_i\left( t_{\delta \epsilon }, x_{1\delta \epsilon } , y_{1\delta \epsilon } \right) -v_i\left( T, 0 ,0\right) \nonumber \\&\qquad +u_i\big (T, 0 ,0 \big )-u_i\left( t_{\delta \epsilon }, x_{2\delta \epsilon } , y_{2\delta \epsilon } \right) \Big ] \nonumber \\&\quad \le e^{-\lambda (T-t_{\delta \epsilon })}C_2\left( 1+\Vert (x_{1\delta \epsilon } , y_{1\delta \epsilon })\Vert + \Vert (x_{2\delta \epsilon } , y_{2\delta \epsilon })\Vert \right) . \end{aligned}$$

(35)

It follows that

$$\begin{aligned} \dfrac{\delta \big (\Vert (x_{1\delta \epsilon } , y_{1\delta \epsilon })\big )\Vert ^2+\Vert (x_{2\delta \epsilon }, y_{2\delta \epsilon }\Vert ^2\big ) }{(1+\Vert (x_{1\delta \epsilon } , y_{1\delta \epsilon })\Vert + \Vert (x_{2\delta \epsilon } , y_{2\delta \epsilon })\Vert \big ) }\le C_2. \end{aligned}$$

Consequently, there exists $C_{\delta } > 0$ such that,

$$\begin{aligned} \Vert (x_{1\delta \epsilon } , y_{1\delta \epsilon })\Vert + \Vert (x_{2\delta \epsilon } , y_{2\delta \epsilon })\Vert \le C_{\delta }. \end{aligned}$$

(36)

This inequality implies that for any fixed $\delta \in (0, 1)$, the sets $\{ (x_{1\delta \epsilon } , y_{1\delta \epsilon }), \epsilon >0 \}$ and $\{(x_{2\delta \epsilon } , y_{2\delta \epsilon }), \epsilon >0 \}$ are bounded by $C_{\delta }$ independent of $\epsilon$. It follows from inequalities (35) and (36) that, possibly if necessary along a subsequence, named again $\big (t_{\delta \epsilon }, ( x_{1\delta \epsilon } , y_{1\delta \epsilon } ) , ( x_{2\delta \epsilon }, y_{2\delta \epsilon } ), \alpha _{\delta \epsilon }\big )$ that there exists $( x_{1\delta 0} , y_{1\delta 0} ) \in \mathbb {R}_+^2$, $t_{\delta \epsilon 0}\in (0, T]$ and $\alpha _{\delta \epsilon 0}\in \mathcal {S}$ such that: $\lim \limits _{\epsilon \downarrow 0} ( x_{1\delta \epsilon } , y_{1\delta \epsilon } )= ( x_{1\delta 0} , y_{1\delta 0} )= \lim \limits _{\epsilon \downarrow 0} ( x_{2\delta \epsilon } , y_{2\delta \epsilon } )$, $\lim \limits _{\epsilon \downarrow 0} t_{\delta \epsilon } = t_{\delta 0}$, $\lim \limits _{\epsilon \downarrow 0}\alpha _{\delta \epsilon } = \alpha _{\delta 0}$.

If $t_{\delta \epsilon }=T$ then writing that

$\Xi \big (t, (x, y), (x, y), \alpha _{\delta \epsilon }\big )\le \Xi \big (T, ( x_{1\delta \epsilon }, y_{1\delta \epsilon } ) , ( x_{2\delta \epsilon }, y_{2\delta \epsilon } ), \alpha _{\delta \epsilon }\big )$, we have

$$\begin{aligned}&u_i(t,x,y)- v_i(t,x,y)-\frac{\varrho }{t}-2\delta e^{\lambda (T-t)}( \Vert (x, y)\Vert ^2 ) \\&\quad \le u_i(T, ( x_{1\delta \epsilon }, y_{1\delta \epsilon } ) )- v_i(T, ( x_{2\delta \epsilon }, y_{2\delta \epsilon } )) - \frac{\varrho }{T}\\&\qquad -\frac{1}{2\epsilon } \Vert ( x_{1\delta \epsilon } , y_{1\delta \epsilon } ) - ( x_{2\delta \epsilon }, y_{2\delta \epsilon } )\Vert ^2 -\delta (\Vert ( x_{1\delta \epsilon }, y_{1\delta \epsilon } )\Vert ^2 + \Vert ( x_{2\delta \epsilon }, y_{2\delta \epsilon } )\Vert ^2)\\&\quad \le u_i(T, ( x_{1\delta \epsilon }, y_{1\delta \epsilon } ) )- v_i(T, ( x_{2\delta \epsilon }, y_{2\delta \epsilon } ))\\&\quad = [u_i(T, ( x_{1\delta \epsilon }, y_{1\delta \epsilon } ) )- v_i(T, ( x_{1\delta \epsilon }, y_{1\delta \epsilon } ))]\\&\qquad + [v_i(T, ( x_{1\delta \epsilon }, y_{1\delta \epsilon } ) )- v_i(T, ( x_{2\delta \epsilon }, y_{2\delta \epsilon } ))]\\&\quad \le C_1 \Vert ( x_{1\delta \epsilon } , y_{1\delta \epsilon } ) - ( x_{2\delta \epsilon }, y_{2\delta \epsilon } )\Vert \end{aligned}$$

where the last inequality follows from the uniform continuity of $v_i$ and by assumption that

$u_i(T, ( x_{1\delta \epsilon }, y_{1\delta \epsilon } ) )=\kappa ^{\gamma }\frac{x_{1\delta \epsilon }^{1-\gamma }}{1-\gamma } = v_i(T, ( x_{1\delta \epsilon }, y_{1\delta \epsilon } )).$

Sending $\varrho , \epsilon , \delta \downarrow 0$ and using estimate (34), we have:

$$u_i(t,x,y)\le v_i(t,x,y).$$

Assume now that $t_{ \delta \epsilon } < T$.

Applying Lemma 3.2 with $u_i$, $v_i$ and $\phi (t, (x, y),(x',y'))$ at the point $(t_{\delta \epsilon }, ( x_{1\delta \epsilon }, y_{1\delta \epsilon } ), ( x_{2\delta \epsilon }, y_{2\delta \epsilon }), \alpha _{\delta \epsilon } ) \in (0,T)\times \mathbb {R}_+^2\times \mathbb {R}_+^2\times \mathbb {S}$, for any $\zeta \in (0, 1)$ there are $d\in \mathbb {R}, M_{\delta \epsilon }, N_{\delta \epsilon } \in \mathbb {S}^2$ such that:

$$\begin{aligned}&\Bigg (d-\dfrac{\varrho }{t_{\delta \epsilon }^2}-\lambda \delta e^{\lambda (T-t_{\delta \epsilon })}( \Vert (x_{\delta \epsilon }, y_{\delta \epsilon })\Vert ^2 +\Vert (x_{\delta \epsilon }', y_{\delta \epsilon }')\Vert ^2 ), \frac{1}{\epsilon }((x_{\delta \epsilon }, y_{\delta \epsilon })-(x_{\delta \epsilon }', y_{\delta \epsilon }') ) \\&\quad +2\delta e^{\lambda (T-t_{\delta \epsilon })}(x_{\delta \epsilon }, y_{\delta \epsilon }), M_{\delta \epsilon }+2\delta e^{\lambda (T-t_{\delta \epsilon })}I\Bigg )\in \mathcal {\bar{P}}^{2,+} u(t_{\delta \epsilon },x_{\delta \epsilon },y_{\delta \epsilon },i)\\&\quad \Bigg (d, \frac{1}{\epsilon }((x_{\delta \epsilon }, y_{\delta \epsilon })-(x_{\delta \epsilon }', y_{\delta \epsilon }') ) -2\delta e^{\lambda (T-t_{\delta \epsilon })}(x'_{\delta \epsilon }, y'_{\delta \epsilon }), N_{\delta \epsilon }-2\delta e^{\lambda (T-t_{\delta \epsilon })}I \Bigg )\in \\&\quad \mathcal {\bar{P}}^{2,-} v(t_{\delta \epsilon },x'_{\delta \epsilon },y'_{\delta \epsilon },i) \end{aligned}$$

and

$$\begin{aligned}&\quad -\frac{1}{\zeta }\left( \begin{array}{cc} I &{} 0 \\ 0 &{} I \end{array} \right) \le \left( \begin{array}{cc} M_{\delta \epsilon } &{} 0 \\ 0 &{} -N_{\delta \epsilon } \end{array} \right) \le D^2_{ (x, y) , (x', y')}\phi (t_{\delta \epsilon }, (x_{\delta \epsilon }, y_{\delta \epsilon }),(x_{\delta \epsilon }', y_{\delta \epsilon }'))\\&\qquad +\zeta \Big ( D^2_{ (x, y) , (x', y')}\phi (t_{\delta \epsilon }, (x_{\delta \epsilon }, y_{\delta \epsilon }),(x_{\delta \epsilon }', y_{\delta \epsilon }'))\Big )^2\\&\quad \le \dfrac{\epsilon +\zeta (2+4\delta \epsilon e^{\lambda (T-t)})}{\epsilon ^2}\left( \begin{array}{cc} I &{} -I \\ -I &{} I \end{array} \right) +(2\delta +4\zeta \delta ^2\epsilon e^{\lambda (T-t)})e^{\lambda (T-t)}\left( \begin{array}{cc} I &{} 0 \\ 0 &{} I \end{array} \right) . \end{aligned}$$

Letting $\delta \downarrow 0$ and taking $\zeta =\dfrac{\epsilon }{2}$, we obtain

$$\begin{aligned} -\frac{1}{\epsilon }\left( \begin{array}{cc} I &{} 0 \\ 0 &{} I \end{array} \right) \le \left( \begin{array}{cc} M_{\delta \epsilon } &{} 0 \\ 0 &{} -N_{\delta \epsilon } \end{array} \right) \le \frac{2}{\epsilon } \left( \begin{array}{cc} I &{} -I \\ -I &{} I \end{array} \right) . \end{aligned}$$

It follows that

$$\begin{aligned}&(x_{\delta \epsilon }, y_{\delta \epsilon }) M_{\delta \epsilon } \left( \begin{array}{c}x_{\delta \epsilon }\\ y_{\delta \epsilon }\end{array} \right) - (x'_{\delta \epsilon }, y'_{\delta \epsilon }) N_{\delta \epsilon } \left( \begin{array}{c}x'_{\delta \epsilon }\\ y'_{\delta \epsilon }\end{array} \right) \\&\quad =((x_{\delta \epsilon }, y_{\delta \epsilon }),(x_{\delta \epsilon }', y_{\delta \epsilon }'))\left( \begin{array}{cc} M_{\delta \epsilon } &{} 0 \\ 0 &{} -N_{\delta \epsilon } \end{array} \right) \left( \begin{array}{c} \left( \begin{array}{c}x_{\delta \epsilon }\\ y_{\delta \epsilon }\end{array} \right) \\ \left( \begin{array}{c} x_{\delta \epsilon }'\\ y_{\delta \epsilon }'\end{array} \right) \end{array} \right) \\&\quad \le ((x_{\delta \epsilon }, y_{\delta \epsilon }), (x_{\delta \epsilon }', y_{\delta \epsilon }'))\left[ \frac{2}{\epsilon } \left( \begin{array}{cc} I &{} -I \\ -I &{} I \end{array} \right) \right] \left( \begin{array}{c} \left( \begin{array}{c}x_{\delta \epsilon }\\ y_{\delta \epsilon }\end{array} \right) \\ \left( \begin{array}{c} x_{\delta \epsilon }'\\ y_{\delta \epsilon }'\end{array} \right) \end{array} \right) \\&\quad \le \dfrac{2}{\epsilon } \Vert ( x_{\delta \epsilon } , y_{\delta \epsilon } ) - ( x'_{\delta \epsilon }, y'_{\delta \epsilon } )\Vert ^2. \end{aligned}$$

Furthermore, the definition of the viscosity subsolution $u_i$ and supersolution $v_i$ implies that

$$\begin{aligned}&\min \Bigg [ \beta u_{i_0}(t_{\delta \epsilon },x_{\delta \epsilon } ,y_{\delta \epsilon })- \Big (d-\dfrac{\varrho }{t_{\delta \epsilon }^2}-\lambda \delta e^{\lambda (T-t_{\delta \epsilon })}( \Vert (x_{\delta \epsilon }, y_{\delta \epsilon })\Vert ^2 +\Vert (x_{\delta \epsilon }', y_{\delta \epsilon }')\Vert ^2 ) \Big ) \nonumber \\&\quad +\inf _{E\in \mathcal {A}_{i_0} }\bigg \lbrace -x_{\delta \epsilon }\Big [r_{i_0}-a_ix_{\delta \epsilon }^{\lambda }-qE_i(s) \Big ]\Big ( \frac{1}{\epsilon }(x_{\delta \epsilon }-x_{\delta \epsilon }')+2\delta e^{\lambda (T-t)}x_{\delta \epsilon }\Big )\\&\quad -\theta (\bar{Y}_0-y_{\delta \epsilon })\Big (\frac{1}{\epsilon }( x_{\delta \epsilon }-y_{\delta \epsilon }') +2\delta e^{\lambda (T-t)} y_{\delta \epsilon }\Big ) \\&\quad -\dfrac{1}{2}(\sigma x_{\delta \epsilon },\sigma _y)(M_{\delta \epsilon } +2\delta e^{\lambda (T-t)}I)\left( \begin{array}{c}\sigma x_{\delta \epsilon }\\ \sigma _y\end{array} \right) \\&\quad - q_{{i_0}j}[u_j(t_{\delta \epsilon },x_{\delta \epsilon } ,y_{\delta \epsilon }) -u_{i_0}(t_{\delta \epsilon },x_{\delta \epsilon } ,y_{\delta \epsilon }) ]\\&\quad -l(_{i_0},t_{\delta \epsilon },x_{\delta \epsilon } ,y_{\delta \epsilon }, E_{t_{\delta \epsilon } }) \bigg \rbrace ,~~ u_{i_0}(T,x_{\delta \epsilon } ,y_{\delta \epsilon })-\kappa ^{\gamma }\frac{x_{\delta \epsilon }^{1-\gamma }}{1-\gamma }\Bigg ] \le 0, ~~ i_0\not = j \end{aligned}$$

and

$$\begin{aligned}&\min \Bigg [ \beta v_{i_0}(t_{\delta \epsilon },x_{\delta \epsilon } ,y_{\delta \epsilon })- d +\inf _{E\in \mathcal {A}_{i_0} }\bigg \lbrace -x_{\delta \epsilon }\Big [r_{i_0}-a_ix_{\delta \epsilon }^{\lambda }-qE_i(s) \Big ]\Big ( \frac{1}{\epsilon }(x_{\delta \epsilon }-x_{\delta \epsilon }')\\&\quad +2\delta e^{\lambda (T-t)}x'_{\delta \epsilon }\Big )\\&\quad -\theta (\bar{Y}_0-y_{\delta \epsilon })\Big (\frac{1}{\epsilon }( y_{\delta \epsilon }-y_{\delta \epsilon }') +2\delta e^{\lambda (T-t)} y'_{\delta \epsilon }\Big ) \\&\quad -\dfrac{1}{2}(\sigma x'_{\delta \epsilon };\sigma _P)(N_{\delta \epsilon }-2\delta e^{\lambda (T-t)}I)\left( \begin{array}{c}\sigma x'_{\delta \epsilon }\\ \sigma _Y\end{array} \right) \\&\quad - q_{{i_0}j}[v_j(t_{\delta \epsilon },x'_{\delta \epsilon } ,y'_{\delta \epsilon }) -v_{i_0}(t_{\delta \epsilon },x'_{\delta \epsilon } ,y'_{\delta \epsilon }) ]\\&\quad -l(_{i_0},t_{\delta \epsilon },x'_{\delta \epsilon } ,y'_{\delta \epsilon }, E_{t_{\delta \epsilon } })\bigg \rbrace ,~~ v_{i_0}(T,x_{\delta \epsilon } ,y_{\delta \epsilon })-\kappa ^{\gamma }\frac{x_{\delta \epsilon }^{1-\gamma }}{1-\gamma }\Bigg ] \ge 0,~~ i_0\not = j. \end{aligned}$$

Let us define operators $A^{E}(x, v, \phi ,X, Z)$ and $B^{E}(x, v)$.

$$\begin{aligned} A^{E}(t, x, y, w, X, Y Z)= & {} x\Big [r_{i_0}-a_ix^{\lambda }-qE_i(t) \Big ]X+\theta (\bar{Y}_0-y)Y+\dfrac{1}{2}wZw'\nonumber \\ B^{E}(t,x,y, v)= & {} q_{{i_0}j}[v_j(t,x ,y) -v_{i_0}(t,x ,y)]. \end{aligned}$$

By subtracting these last two inequalities and remarking that $\min (x; y)-\min (z; t)\le 0$ implies either $x-z\le 0$ or $y-t\le 0$, we divide our consideration into two cases:

Case 1:

$$\begin{aligned}&\beta \big [u_{i_0}(t_{\delta \epsilon },x_{\delta \epsilon } ,y_{\delta \epsilon }) -v_{i_0}(t_{\delta \epsilon },x_{\delta \epsilon } ,y_{\delta \epsilon })\big ]+\dfrac{\varrho }{t_{\delta \epsilon }^2}+\lambda \delta e^{\lambda (T-t_{\delta \epsilon })}( \Vert (x_{\delta \epsilon }, y_{\delta \epsilon })\Vert ^2 \nonumber \\&\qquad +\Vert (x_{\delta \epsilon }', y_{\delta \epsilon }')\Vert ^2 )\nonumber \\&\quad \le \sup _{E\in \mathcal {A}_{i_0} }\big \lbrace l(_{i_0},t_{\delta \epsilon },x_{\delta \epsilon } ,y_{\delta \epsilon }, E_{t_{\delta \epsilon } })-l(_{i_0},t_{\delta \epsilon },x'_{\delta \epsilon } ,y'_{\delta \epsilon }, E_{t_{\delta \epsilon } })\big \rbrace \nonumber \\&\qquad +\sup _{E\in \mathcal {A}_{i_0}}\bigg \lbrace A^{E}\Big (t_{\delta \epsilon },x_{\delta \epsilon } ,y_{\delta \epsilon },(\sigma x_{\delta \epsilon };\sigma _P) , \frac{1}{\epsilon }(x_{\delta \epsilon }-x_{\delta \epsilon }')+2\delta e^{\lambda (T-t_{\delta \epsilon })}x_{\delta \epsilon },\nonumber \\&\quad \frac{1}{\epsilon }( y_{\delta \epsilon }-y_{\delta \epsilon }') +2\delta e^{\lambda (T-t_{\delta \epsilon })} y_{\delta \epsilon }, M_{\delta \epsilon }+2\delta e^{\lambda (T-t_{\delta \epsilon })}I\Big )\nonumber \\&\qquad -A^{E}\Big (t_{\delta \epsilon },x'_{\delta \epsilon } ,y'_{\delta \epsilon },(\sigma b'_{\delta \epsilon };\sigma _P) ,\nonumber \\&\quad \frac{1}{\epsilon }(x_{\delta \epsilon }-x_{\delta \epsilon }')+2\delta e^{\lambda (T-t)}x'_{\delta \epsilon }, \frac{1}{\epsilon }( y_{\delta \epsilon }-y_{\delta \epsilon }') +2\delta e^{\lambda (T-t_{\delta \epsilon })} y'_{\delta \epsilon },\nonumber \\&\qquad N_{\delta \epsilon }-2\delta e^{\lambda (T-t_{\delta \epsilon })}I\Big ) \bigg \rbrace \nonumber \\&\qquad +\sup _{E\in \mathcal {A}_{i_0}}\bigg \lbrace B^{E}(t_{\delta \epsilon },x_{\delta \epsilon } ,y_{\delta \epsilon } u)-B^{E}(t_{\delta \epsilon },x'_{\delta \epsilon } ,y'_{\delta \epsilon } v) ] \bigg \rbrace \equiv \mathcal {I}_1+ \mathcal {I}_2+\mathcal {I}_3. \end{aligned}$$

(37)

In view of condition (11) on l and from estimate (3.1), we have the classical estimates of $\mathcal {I}_1$ and $\mathcal {I}_2$:

$$\begin{aligned} \mathcal {I}_1\le & {} C\Vert ( x_{\delta \epsilon } , y_{\delta \epsilon } ) - ( x'_{\delta \epsilon }, y'_{\delta \epsilon } )\Vert \\ \mathcal {I}_2\le & {} C(\dfrac{1}{\epsilon } \Vert ( x_{\delta \epsilon } , y_{\delta \epsilon } ) - ( x'_{\delta \epsilon }, y'_{\delta \epsilon })\Vert ^2\\&+2\delta e^{\lambda (T-t_{\delta \epsilon })}(1+\Vert ( x_{\delta \epsilon } , y_{\delta \epsilon } )\Vert ^2+ \Vert ( x'_{\delta \epsilon }, y'_{\delta \epsilon } )\Vert ^2 ). \end{aligned}$$

Using the Lipschitz condition for u and v, we have

$$\begin{aligned} \mathcal {I}_3\le 2C \Vert ( x_{\delta \epsilon } , y_{\delta \epsilon } ) - ( x'_{\delta \epsilon }, y'_{\delta \epsilon })\Vert . \end{aligned}$$

Writing that $\Xi (t, (x, y),(x, y), i)\le \Xi (t_{\delta \epsilon }, (x_{\delta \epsilon }, y_{\delta \epsilon }),(x_{\delta \epsilon }, y_{\delta \epsilon }), i)$ for $i \in \mathcal {S}$ and using the inequality (37),

$$\begin{aligned}&u_i(t,x,y)-v_i(t,x,y)- \frac{\varrho }{t}-2\delta e^{\lambda (T-t)} \Vert (x, y)\Vert ^2 \le \\&\quad v_i(t_{\delta \epsilon },x_{\delta \epsilon },y_{\delta \epsilon })- u_i(t_{\delta \epsilon },x_{\delta \epsilon },y_{\delta \epsilon }) - \frac{\varrho }{t_{\delta \epsilon }}-2\delta e^{\lambda (T-t)} \Vert (x_{\delta \epsilon },y_{\delta \epsilon })\Vert ^2\le \\&\quad \dfrac{1}{\beta }\Big [\mathcal {I}_1+\mathcal {I}_2+\mathcal {I}_3\Big ]- \dfrac{\varrho }{\beta t_{\delta \epsilon }^2}-\dfrac{\lambda }{\beta }\delta e^{\lambda (T-t_{\delta \epsilon })}( \Vert (x_{\delta \epsilon }, y_{\delta \epsilon })\Vert ^2 +\Vert (x_{\delta \epsilon }', y_{\delta \epsilon }')\Vert ^2 ) \end{aligned}$$

this implies

$$\begin{aligned}&u_i(t,x,y)-v_i(t,x,y)- \frac{\varrho }{t}-2\delta e^{\lambda (T-t)} \Vert (x, y)\Vert ^2 \le \\&\quad \dfrac{1}{\beta }\Big [\mathcal {I}_1+\mathcal {I}_2+\mathcal {I}_3\Big ]- \dfrac{\lambda }{\beta }\delta e^{\lambda (T-t_{\delta \epsilon })}( \Vert (x_{\delta \epsilon }, y_{\delta \epsilon })\Vert ^2 +\Vert (x_{\delta \epsilon }', y_{\delta \epsilon }')\Vert ^2 ). \end{aligned}$$

Sending $\epsilon \downarrow 0$, with the above estimates of $(\mathcal {I}_1)-(\mathcal {I}_2)-(\mathcal {I}_3)$, we obtain:

$$\begin{aligned} u_i(t,x,y)-v_i(t,x,y)- \dfrac{\varrho }{t}-2\delta e^{\lambda (T-t)} \Vert (x,y)\Vert ^2 \\ \le \dfrac{2\delta }{\beta } e^{\lambda (T-t_0)}\Big [ C(1+2\Vert (x_0, y_0)\Vert ^2)-\lambda \Vert (x_0, y_0)\Vert ^2\Big ]. \end{aligned}$$

Choose $\lambda$ sufficiently large positive $(\lambda \ge 2C)$ and send $\varrho , \delta \rightarrow 0^+$ to conclude that $u_i(t,x,y)\le v_i(t,x,y)$.

Case 2:

The second case occurs if

$$\begin{aligned} u_{i_0}(T,x_{\delta \epsilon } ,y_{\delta \epsilon })-v_{i_0}(T,x_{\delta \epsilon } ,y_{\delta \epsilon })\le 0 \end{aligned}$$

and finally that $u_i(t,x,y)\le v_i(t,x,y)$.

This completes the proof.