Optimal credit investment and risk control for an insurer with regime-switching

Abstract

This paper studies an optimal investment and risk control problem for an insurer with default contagion and regime-switching. The insurer in our model allocates his/her wealth across multi-name defaultable stocks and a riskless bond under regime-switching risk. Default events have an impact on the distress state of the surviving stocks in the portfolio. The aim of the insurer is to maximize the expected utility of the terminal wealth by selecting optimal investment and risk control strategies. We characterize the optimal trading strategy of defaultable stocks and risk control for the insurer. By developing a truncation technique, we analyze the existence and uniqueness of global (classical) solutions to the recursive HJB system. We prove the verification theorem based on the (classical) solutions of the recursive HJB system.

Technical proofs

Proof of Lemma 4.1

Define $f(x)=Bx$ for $x\in \mathbb {R}^m$. By virtue of Proposition 1.1 of Charter 3 in Smith [19], it suffices to verify that $f:\mathbb {R}^m\rightarrow \mathbb {R}^m$ is of type K, i.e., for any $x,y\in \mathbb {R}^m$ satisfying $x\le y$ and $x_i=y_i$ for some $i=1,\ldots ,m$, then $f_i(x)\le f_i(y)$. Notice that $b_{ij}\ge 0$ for all $i\ne j$. Then, it holds that

$$\begin{aligned} f_i(x)&=(Bx)_i=\sum _{j=1}^mb_{ij}x_j=b_{ii}x_i+\sum _{j=1,j\ne i}^mb_{ij}x_j\nonumber \\&=b_{ii}y_i+\sum _{j=1,j\ne i}^mb_{ij}x_j \le b_{ii}y_i+\sum _{j=1,j\ne i}^mb_{ij}y_j=f_i(y), \end{aligned}$$

(A.1)

and hence f is of type K. Thus, we complete the proof of the lemma. $\square$

Proof of Lemma 4.2

The expression of the solution $\varphi (t,e_n)$ given by (24) is obvious. Notice that $e_m\gg 0$ and $q_{ij}\ge 0$ for all $i\ne j$ since $Q=(q_{ij})_{m\times m}$ is the generator of the Markov chain. Then, in order to prove $\varphi (t,e_n)\gg 0$ for all $t\in [0,T]$, using Lemma 4.1, it suffices to verify $[A^{(n)}]_{ij}\ge 0$ for all $i\ne j$, however, $[A^{(n)}]_{ij}=q_{ij}$ for all $i\ne j$ using (21). Thus, we have verified the condition given in Lemma 4.1, and hence $\varphi (t,e_n)\gg 0$ for all $t\in [0,T]$. $\square$

Proof of Lemma 4.3

It suffices to prove that, for any $x,y\in \mathbb {R}^m$ satisfying $x,y\ge \varepsilon e_m^{\top }$ with $\varepsilon >0$, there exists a constant $C=C(\varepsilon )>0$ depending on $\varepsilon >0$ only such that $|G_i^{(k)}(t,x)-G_i^{(k)}(t,y)|\le C\Vert x-y\Vert$ for each $i=1,\ldots ,m$. Since $\sigma (i)\sigma (i)^{\top }$ is also positive definite, $\sigma ^{(k)}(i)\sigma ^{(k)}(i)^{\top }$ is positive definite. Hence, there exists a constant $\delta >0$ such that $(\pi ^{(k)})^{\top }\sigma ^{(k)}(i)\sigma ^{(k)}(i)^{\top }\pi ^{(k)}\ge \delta \Vert \pi ^{(k)}\Vert ^2$. Then, for any $(\pi ^{(k)},l)\in \mathcal{U}^{(k)}$, there exists a positive constant $C_1>0$ such that

$$\begin{aligned}&\frac{\gamma (\gamma -1)}{2}\left\{ (\pi ^{(k)})^{\top }\sigma ^{(k)}(i)\sigma ^{(k)}(i)^{\top }\pi ^{(k)}+l^2\big (\phi (i)\phi (i)^{\top }+\bar{\phi }(i)\bar{\phi }(i)^{\top }\big )-2l(\pi ^{(k)})^{\top }\sigma ^{(k)}(i)\phi (i)\right\} \nonumber \\&\quad =\frac{\gamma (\gamma -1)}{2}\left\{ l^2\bar{\phi }(i)\bar{\phi }(i)^{\top }+\Vert \sigma ^{(k)}(i)^\top \pi ^{(k)}-l\phi (i)\Vert ^2\right\} \nonumber \\&\quad \le \frac{\gamma (\gamma -1)}{2}\left\{ \alpha l^2\bar{\phi }(i)\bar{\phi }(i)^{\top }+\frac{1}{2}(1-\alpha )\Vert \sigma ^{(k)}(i)^\top \pi ^{(k)}\Vert ^2-(1-\alpha )\Vert l\phi (i)\Vert ^2\right\} \nonumber \\&\quad =\frac{\gamma (\gamma -1)}{2}\left\{ l^2(\alpha \bar{\phi }(i)\bar{\phi }(i)^{\top }-(1-\alpha )\Vert \phi (i)\Vert ^2)+\frac{1}{2}(1-\alpha )\Vert \sigma ^{(k)}(i)^\top \pi ^{(k)}\Vert ^2\right\} \nonumber \\&\quad \le -C_1(\Vert \pi ^{(k)}\Vert ^2+l^2), \end{aligned}$$

(A.2)

where the constant $\alpha \in (\max _{i=1,\ldots ,m}\big \{\frac{\Vert \phi (i)\Vert ^2}{\bar{\phi }(i)\bar{\phi }(i)^{\top }+\Vert \phi (i)\Vert ^2}\big \},1)$. On the other hand, for any $(\pi ^{(k)},l)\in {{\mathcal {U}}}^{(k)}$, it holds that

$$\begin{aligned} \gamma \big \{(\pi ^{(k)})^{\top }\theta ^{(k)}(i)+(p^{(k)}(i)-c(i))l\big \}&\le \gamma \Vert \theta ^{(k)}(i)\Vert \Vert \pi ^{(k)}\Vert +\gamma |p^{(k)}(i)-c(i)|l\nonumber \\&\le C_2\sqrt{\Vert \pi ^{(k)}\Vert ^2+l^2}, \end{aligned}$$

(A.3)

where the constant $C_2:=\max _{i=1,\ldots ,m}\big \{\gamma \sqrt{\Vert \theta ^{(k)}(i)\Vert ^2+|p^{(k)}(i)-c(i)|^2}\big \}>0$. Finally, for any $(\pi ^{(k)},l)\in {{\mathcal {U}}}^{(k)}$, we have that $\{(1-lg(i))^\gamma -1\}\nu ^{(k)}(i)\le C_3l$, where the constant $C_3:=\max _{i=1,\ldots ,m}\{\gamma g(i)\nu ^{(k)}(i)\}>0$. Then, by virtue of (29), it follows that, for any $(\pi ^{(k)},l)\in {{\mathcal {U}}}^{(k)}$ and $i=1,\ldots ,m$,

$$\begin{aligned} H^{(k)}((\pi ^{(k)},l),i)\le -C_1\big (\Vert \pi ^{(k)}\Vert ^2+l^2\big )+C_4\sqrt{\Vert \pi ^{(k)}\Vert ^2+l^2}. \end{aligned}$$

(A.4)

Here $C_4=C_2+C_3$. This yields that there exists a constant $C_5>0$ such that when $(\pi ^{(k)},l)\in {{\mathcal {U}}}^{(k)}$ and $\Vert \pi ^{(k)}\Vert ^2+l^2>C_5$, we have $H^{(k)}((\pi ^{(k)},l),i)<0$ for all $i=1,\ldots ,m$, and meanwhile, for $x\ge \varepsilon e_{m}^{\top }$ with $\varepsilon >0$,

$$\begin{aligned}&\sum _{j\notin \{j_1,\ldots ,j_k\}}(1-\pi _j^{(k)})^\gamma h_{j}^{(k)}(i)\varphi ^{(l+1),j}(t,i)+H^{(k)}((\pi ^{(k)},l),i)x_i\nonumber \\&\quad \le \big (1+\Vert \pi ^{(k)}\Vert \big )^\gamma \sum _{j\notin \{j_1,\ldots ,j_k\}}h_{j}^{(k)}(i)\varphi ^{(l+1),j}(t,i)+\varepsilon H^{(k)}((\pi ^{(k)},l),i)\nonumber \\&\quad \le C_6\big (1+\Vert \pi ^{(k)}\Vert ^\gamma \big )\sum _{j\notin \{j_1,\ldots ,j_k\}}h^{(k)}_{j}(i)+\varepsilon \left\{ -C_1(\Vert \pi ^{(k)}\Vert ^2+l^2)+C_4\sqrt{\Vert \pi ^{(k)}\Vert ^2+l^2}\right\} \nonumber \\&\quad \le -\varepsilon C_1\big (\Vert \pi ^{(k)}\Vert ^2+l^2\big )+\varepsilon C_4\sqrt{\Vert \pi ^{(k)}\Vert ^2+l^2}+C_7\big (\sqrt{\Vert \pi ^{(k)}\Vert ^2+l^2}\big )^{\gamma }+C_8, \end{aligned}$$

(A.5)

for some constants $C_6,C_7,C_8>0$. Notice that we used the recursive assumption that the HJB system (18) admits a positive unique (classical) solution $\varphi ^{(k+1),j}(t)$ on $t\in [0,T]$ for $j\notin \{j_1,\ldots ,j_k\}$. Then $\varphi ^{(k+1),j}(t)$ is continuous on [0, T], and hence $\varphi ^{(k+1),j}(t)$ is bounded on [0, T]. From the estimate (A.5), it follows that, for any $x\ge \varepsilon e_m^{\top }$, there exists a positive constant $C_9=C_9(\varepsilon )$ such that when $(\pi ^{(k)},l)\in {{\mathcal {U}}}^{(k)}$, $\Vert \pi ^{(k)}\Vert ^2+l^2>C_9$ and $x\ge \varepsilon e_m^{\top }$, it holds that, for each $i=1,\ldots ,m$,

$$\begin{aligned}&\sum _{j\notin \{j_1,\ldots ,j_k\}}(1-\pi _j^{(k)})^\gamma h^{(k)}_{j}(i)\varphi ^{(l+1),j}(t,i)+H^{(k)}((\pi ^{(k)},l),i)x_i<0. \end{aligned}$$

(A.6)

On the other hand, for $i=1,\ldots ,m$, it holds that

$$\begin{aligned} G^{(k)}_i(t,x)=&\sup _{(\pi ^{(k)},l)\in {{\mathcal {U}}}^{(k)}}\left\{ \sum _{j\notin \{j_1,\ldots ,j_k\}}(1-\pi _{j}^{(k)})^\gamma h_{j}^{(k)}(i)\varphi ^{(k+1),j}(t,i)+H^{(k)}((\pi ^{(k)},l),i)x_i\right\} \nonumber \\ \ge&\sum _{j\notin \{j_1,\ldots ,j_k\}} h^{(k)}_{j}(i)\varphi ^{(k+1),j}(t,i)+H^{(k)}((0e_{n-k}^{\top },0),i)x_i\nonumber \\ =&\sum _{j\notin \{j_1,\ldots ,j_k\}} h_{j}^{(k)}(i)\varphi ^{(k+1),j}(t,i)>0. \end{aligned}$$

(A.7)

Thus, using the estimate (A.6), we have that, for all $x\ge \varepsilon e_m^{\top }$,

$$\begin{aligned} G^{(k)}_i(t,x)=&\sup _{\begin{array}{c} (\pi ^{(k)},l)\in \mathcal{U}^{(k)}\\ \Vert \pi ^{(k)}\Vert ^2+l^2\le C_9(\varepsilon ) \end{array}}\left\{ \sum _{j\notin \{j_1,\ldots ,j_k\}}(1-\pi _{j}^{(k)})^\gamma h_{j}^{(k)}(i)\varphi ^{(k+1),j}(t,i)+H^{(k)}((\pi ^{(k)},l),i)x_i\right\} . \end{aligned}$$

(A.8)

It follows from (A.8) and (29) that, for all $x,y\ge \varepsilon e_m^{\top }$,

$$\begin{aligned} G^{(k)}_i(t,x)=&\sup _{\begin{array}{c} (\pi ^{(k)},l)\in {{\mathcal {U}}}^{(k)}\\ \Vert \pi ^{(k)}\Vert ^2+l^2\le C_9(\varepsilon ) \end{array}}\Bigg \{\sum _{j\notin \{j_1,\ldots ,j_k\}}(1-\pi _{j}^{(k)})^\gamma h_{j}^{(k)}(i)\varphi ^{(k+1),j}(t,i)+H^{(k)}((\pi ^{(k)},l),i)y_i\nonumber \\&\qquad \qquad \qquad +H^{(k)}((\pi ^{(k)},l),i)(x_i-y_i)\Bigg \}\nonumber \\ \le&\sup _{\begin{array}{c} (\pi ^{(k)},l)\in {{\mathcal {U}}}^{(k)}\\ \Vert \pi ^{(k)}\Vert ^2+l^2\le C_9(\varepsilon ) \end{array}}\Bigg \{\sum _{j\notin \{j_1,\ldots ,j_k\}}(1-\pi _{j}^{(k)})^\gamma h_{j}^{(k)}(i)\varphi ^{(k+1),j}(t,i)+H^{(k)}((\pi ^{(k)},l),i)y_i\nonumber \\&\qquad \qquad \qquad +\left| H^{(k)}((\pi ^{(k)},l),i)\right| \left| x_i-y_i\right| \Bigg \}\nonumber \\ \le&G^{(k)}_i(t,y)+\left| x_i-y_i\right| \sup _{\begin{array}{c} (\pi ^{(k)},l)\in {{\mathcal {U}}}^{(k)}\\ \Vert \pi ^{(k)}\Vert ^2+l^2\le C_9(\varepsilon ) \end{array}}\left\{ \left| H^{(k)}((\pi ^{(k)},l),i)\right| \right\} \nonumber \\ \le&G^{(k)}_i(t,y)+C(\varepsilon )\left| x_i-y_i\right| , \end{aligned}$$

(A.9)

where $C(\varepsilon )>0$ is a constant which depends on $\varepsilon >0$ only. Then, the above estimate results in the validity of the estimate (30) for all $x,y\in \mathbb {R}^m$ satisfying $x,y\ge \varepsilon e_m^{\top }$. Thus, we complete the proof of the lemma. $\square$

Proof of Lemma 4.4

For $p>0$, let $g_{1}^{(p)}(t)=(g_{1i}^{(p)}(t);\ i=1,\ldots ,m)^{\top }$ be the solution to the following dynamical system given by

$$\begin{aligned} \left\{ \begin{aligned} \frac{d}{dt}g_{1}^{(p)}(t)=&f(t,g_{1}^{(p)}(t))+\tilde{f}(t,g^{(p)}_{1}(t))+\frac{1}{p}e_m^{\top },\ \text { in }(0,T];\\ g_{1}^{(p)}(0)=&\xi _1+\frac{1}{p}e_m^{\top }. \end{aligned} \right. \end{aligned}$$

(A.10)

Then, for all $t\in (0,T]$, it holds that

$$\begin{aligned} \Vert g_{1}^{(p)}(t)-g_1(t)\Vert \le&\Vert g_{1}^{(p)}(0)-g_1(0)\Vert +\int _0^t\big \Vert f(s,g_{1}^{(p)}(s))-f(s,g_1(s))\big \Vert ds\nonumber \\&+\int _0^t\big \Vert \tilde{f}(s,g_{1}^{(p)}(s))-\tilde{f}(s,g_1(s))\big \Vert ds+\frac{1}{p}\int _0^t\Vert e_m\Vert ds\nonumber \\ \le&\frac{2}{p}\Vert e_m\Vert +(C+\tilde{C})\int _0^t\big \Vert g_{1}^{(p)}(s)-g_1(s)\big \Vert ds. \end{aligned}$$

Here $C>0$ (resp. $\tilde{C}>0$) is the Lipschitz constant of f(t, x) (resp. $\tilde{f}(t,x)$) in x. Then, the Gronwall’s lemma yields that $g_{1}^{(p)}(t)\rightarrow g_1(t)$ for all $t\in [0,T]$ as $p\rightarrow \infty$. We claim that $g_{1}^{(p)}(t)\gg g_2(t)$ for all $t\in [0,T]$. If the claim were false, notice that $g_{1}^{(p)}(0)\gg g_2(0)$, and $g_1^{(p)}(t),g_2(t)$ are continuous on [0, T], then there exists a $t_0\in (0,T]$ such that $g_{1}^{(p)}(s)\ge g_2(s)$ on $s\in [0,t_0]$ and $g_{1i}^{(p)}(t_0)=g_{2i}(t_0)$ for some $i\in \{1,\ldots ,m\}$. Since $t_0>0$, $g_1^{(p)}(t)$ and $g_2(t)$ are differentiable on (0, T], we have that

$$\begin{aligned} \frac{d}{dt}g_{1i}^{(p)}(t)\big |_{t=t_0}=\lim _{\epsilon \rightarrow 0}\frac{g_{1i}^{(p)}(t_0)-g_{1i}^{(p)}(t_0-\epsilon )}{\epsilon } \le \lim _{\epsilon \rightarrow 0}\frac{g_{2i}(t_0)-g_{2i}(t_0-\epsilon )}{\epsilon }= \frac{d}{dt}g_{2i}(t)\big |_{t=t_0}. \end{aligned}$$

On the other hand, since $f(t,\cdot )$ satisfies the type K condition for each $t\in [0,T]$ and $\tilde{f}(t,x)\ge 0$ for all $(t,x)\in [0,T]\times \mathbb {R}^m$, for the above i, we also have that

$$\begin{aligned} \frac{d}{dt}g_{1i}^{(p)}(t)\big |_{t=t_0}=&f_i(t_0,g_{1i}^{(p)}(t_0))+\tilde{f}_i(t_0,g_{1}^{(p)}(t_0))+\frac{1}{p}\nonumber \\ >&f_i(t_0,g_{1i}^{(p)}(t_0))\ge f_i(t_0,g_2(t_0))=\frac{d}{dt}g_{2i}(t)\big |_{t=t_0}. \end{aligned}$$

(A.11)

This results in a contradiction, and hence $g_{1}^{(p)}(t)\gg g_2(t)$ for all $t\in [0,T]$. Thus, it holds that $g_1(t)\ge g_2(t)$ for all $t\in [0,T]$ by letting p tend to infinity. $\square$

Proof of Theorem 4.6

For $(t,x,i,z)\in [0,T]\times \mathbb {R}_+\times \{1,\ldots ,m\}\times {{\mathcal {S}}}$, note that $\varphi (T,i,z)=\frac{1}{\gamma }$. Then, by virtue of Itô’s formula, for all $(\pi ,l)\in \tilde{{\mathcal {U}}}$, it follows that

$$\begin{aligned}&\frac{1}{\gamma }(X^{\pi ,l}(T))^\gamma =(X^{\pi ,l}(t))^\gamma \varphi (t,Y(t),Z(t))+\int _t^T(X^{\pi ,l}(s))^\gamma \frac{\partial \varphi (s,Y(s),Z(s))}{\partial s}ds\nonumber \\&\qquad +\int _t^T\gamma (X^{\pi ,l}_s)^{\gamma -1}\varphi (s,Y(s),Z(s))dX^{\pi ,l}(s)^c\\&\qquad +\frac{\gamma (\gamma -1)}{2}\int _t^T(X^{\pi ,l}(s))^{\gamma -2}\varphi (s,Y(s),Z(s))d[X^{\pi ,l},X^{\pi ,l}]^c(s)\\&\qquad +\int _t^T\varphi (s,Y(s-),Z(s-))(X^{\pi ,l}(s-))^\gamma [(1-l(s)g(Y(s-)))^\gamma -1]dN(s)\\&\qquad +\sum _{j=1}^n\int _t^T(X^{\pi ,l}(s-))^\gamma \big [(1-\pi _j(s-))^\gamma \varphi (s,Y(s-),Z^j(s-))-\varphi (s,Y(s-),Z(s-))\big ]dZ_j(s)\\&\qquad +\int _t^T\sum _{j\ne Y(s-)}(X^{\pi ,l}(s-))^\gamma \big [\varphi (s,j,Z(s-))-\varphi (s,Y(s-),Z(s-))\big ]dH_{Y(s-),j}(s)\\&\quad =(X^{\pi ,l}(t))^\gamma \varphi (t,Y(t),Z(t))+\int _t^T(X^{\pi ,l}(s))^\gamma \mathcal{A}(\pi ,l;s,Y(s),Z(s))ds+M^{\pi ,l}(T)-M^{\pi ,l}(t). \end{aligned}$$

Here for $(\pi ,l)\in (-\infty ,1]^n\times [0,\infty )$ and $(t,i,z)\in [0,T]\times \{1,\ldots ,m\}\times {{\mathcal {S}}}$, the coefficient is given by

$$\begin{aligned}&{{\mathcal {A}}}(\pi ,l;t,i,z)\nonumber \\&\quad =\frac{\partial \varphi (t,i,z)}{\partial t}+\Bigg \{\gamma \Big [r(i)+\pi ^{\top }(I-diag(z))\theta (i,z)+\pi ^{\top }(I-diag(z))h(i,z)+(p(i,z)-c(i))l\Big ]\\&\qquad +\frac{\gamma (\gamma -1)}{2}\Big [\pi ^{\top }(I-diag(z))\sigma (i)\sigma (i)^{\top }(I-diag(z))\pi +l^2\big (\phi (i)\phi (i)^{\top }+\bar{\phi (i)}\bar{\phi (i)}^{\top }\big )\\&\qquad -2l\pi ^{\top }(I-diag(z))\sigma (i)\phi (i)^{\top }\Big ]+[(1-lg(i))^\gamma -1]\nu (i,z)\bigg \}\varphi (t,i,z)\\&\qquad +\sum _{j=1}^n[(1-\pi _j)^\gamma \varphi (t,i,z^j)-\varphi (t,i,z)](1-z_j)h_j(i,z)+\sum _{j\ne i}[\varphi (t,j,z)-\varphi (t,i,z)]q_{ij}, \end{aligned}$$

and the $\mathbb {P}$-(local) martingale is defined as

$$\begin{aligned} M^{\pi ,l}(t)=&\int _0^t\gamma (X^{\pi ,l}(s))^\gamma \varphi (s,Y(s),Z(s))\big [\pi (s)^{\top }(I-diag(Z(s)))\sigma (Y(s))-l(s)\phi (Y(s))\big ]dW(s)\\&+\int _0^t\gamma (X^{\pi ,l}(s))^\gamma \varphi (s,Y(s),Z(s))l(s)\bar{\phi }(Y(s))d\bar{W}(s)\\&+\int _0^t(X^{\pi ,l}(s))^\gamma \varphi (s,Y(s-),Z(s-))[(1-l(s)g(Y(s-)))^\gamma -1]d\tilde{N}(s)\\&+\sum _{j=1}^n\int _0^T(X^{\pi ,l}(s-))^\gamma [(1-\pi _j(s))^\gamma \varphi (s,Y(s-),Z(s-)^j)-\varphi (s,Y(s-),Z(s-))]dM_j(s)\\&+\int _0^t\sum _{j\ne Y(s-)}(X^{\pi ,l}(s-))^\gamma \big [\varphi (s,j,Z(s-))-\varphi (s,Y(s-),Z(s-))]d\tilde{H}_{Y(s-),j}(s), \end{aligned}$$

where we used the following $\mathbb {P}$-martingale processes given by, for $t\in [0,T]$,

$$\begin{aligned} \tilde{N}(t)&:=N(t)-\int _0^t\nu (Y(s),Z(s))ds,\quad \tilde{H}_{ij}(t):=H_{ij}(t)-\int _0^tq_{ij}{\mathbb {1}}_{Y(s)=i}ds, \end{aligned}$$

for all $i,j\in \{1,\ldots ,m\}$ and $i\ne j$. Here, we recall that the process $H_{ij}(t)$ is defined by (15). Using (18), (23) and (38), for $t\in [0,T]\times \{1,\ldots ,m\}\times {{\mathcal {S}}}$, we have that $\mathcal{A}(\pi ,l;t,i,z)\le {{\mathcal {A}}}(\pi ^*,l^*;t,i,z)=0$ for all $(\pi ,l)\in {{\mathcal {U}}}$. Moreover, define $\tau _a:=\inf \{s\ge t;\ |X^{\pi ,l}(s)|>a\}$ for $a>0$. Eq. (12) gives that, for $s\in [t,T]$,

$$\begin{aligned} X^{\pi ,l}(s\wedge \tau _a)=&X^{\pi ,l}(s\wedge \tau _a-)\nonumber \\&\times [1-\tilde{\pi }^\top (s\wedge \tau _a)\Delta M(s\wedge \tau _a)-\tilde{l}(s\wedge \tau _a)g(Y(s\wedge \tau _a-))\Delta N(s\wedge \tau _a)], \end{aligned}$$

(A.12)

where the feedback controls are given by

$$\begin{aligned} \tilde{\pi }(s\wedge \tau _a)=\,&\pi \big (s\wedge \tau _a,X^{\pi ,l}(s\wedge \tau _a-),Y(s\wedge \tau _a-),Z(s\wedge \tau _a-)\big ),\\ \tilde{l}(s\wedge \tau _a)=\,&l\big (s\wedge \tau _a,X^{\pi ,l}(s\wedge \tau _a-),Y(s\wedge \tau _a-),Z(s\wedge \tau _a-)\big ). \end{aligned}$$

Notice that $(\pi ,l)\in {{\mathcal {U}}}$ is locally bounded, and hence

$$\begin{aligned} |\tilde{\pi }(s\wedge \tau _a)|+|\tilde{l}(s\wedge \tau _a)|\le C_1(\pi ,l,a,T),\quad s\in [t,T]. \end{aligned}$$

The positive constant $C_1$ depends on $(\pi ,l)$, a and T only. Since $|\Delta M|\vee |\Delta N|\le 1$, it follows that

$$\begin{aligned} |X^{\pi ,l}(s\wedge \tau _a)|\le C_2(\pi ,l,a,T),\quad s\in [t,T], \end{aligned}$$

where $C_2$ is a positive constant which depends on $(\pi ,l)$, a and T only. This implies that $M^{\pi ,l}(\cdot \wedge \tau _a)$ is a $\mathbb {P}$-martingale. Hence, it holds that

$$\begin{aligned} \mathbb {E} _{t,x,i,z}\left[ U(X^{\pi ,l}(T\wedge \tau _a)\big )\right]&\le x^\gamma \varphi (t,i,z)+ \mathbb {E} _{t,x,i,z}\left[ {M}^{\pi ,l}(T\wedge \tau _a)-{M}^{\pi ,l}(t)\right] \nonumber \\&=x^\gamma \varphi (t,i,z), \end{aligned}$$

(A.13)

where we set $\mathbb {E} _{t,i,z}[\cdot ]:= \mathbb {E} [\cdot \mid X^{\pi ,l}(t)=x,Y(t)=i,Z(t)=z]$ for $(t,x,i,z)\in [0,T]\times \mathbb {R}_+\times \{1,\ldots ,m\}\times {{\mathcal {S}}}$. It follows from Fatou’s lemma that

$$\begin{aligned} \mathbb {E} _{t,x,i,z}[U(X^{\pi ,l}(T))]\le \varliminf _{a\rightarrow \infty } \mathbb {E} _{t,x,i,z}[U(X^{\pi ,l}(T\wedge \tau _a))]\le x^\gamma \varphi (t,i,z). \end{aligned}$$

This verifies the validity of the conclusion (i).

We next prove the conclusion (ii). In fact, recall that the optimal feedback strategy $(\pi ^{*},l^*)=(\pi ^*(t,i,z),l^*(t,i,z))$ for $i=1,\ldots ,m$ is given by (23) for $k=n$ and given by (38) for $k=0,1,\ldots ,n-1$. Then, there exists a constant $C>0$ which is independent of (t, i, z) such that $\Vert \pi ^*(t,i,z)\Vert ^2+|l^*(t,i,z)|^{2}\le C$ for all $(t,i,z)\in [0,T]\times \{1,\ldots ,m\}\times {{\mathcal {S}}}$. We next estimate $\mathbb {E} [(X^{\pi ^*,l^*}(T\wedge \tau _a))^{2\gamma }]$. First of all, the dynamics of the wealth process can be rewritten as, for $s\in [t,T]$,

$$\begin{aligned} dX^{\pi ^*,l^*}(s)=&X^{\pi ^*,l^*}(s)[r(Y_{s})+\pi ^{*}(s,Y(s),Z(s))^{\top }\theta (Y(s),Z(s))\nonumber \\&\qquad +l^*(s,Y(s),Z(s))(p(Y(s),Z(s))-c(Y(s)))]ds\nonumber \\&\qquad +X^{\pi ^*,l^*}(s)[\pi ^{*}(s,Y(s),Z(s))^{\top }\sigma (Y(s))-l^*(s,Y(s),Z(s))\phi (Y(s))]dW(s)\nonumber \\&\qquad -X^{\pi ^*,l^*}(s)l^*(s,Y(s),Z(s))\bar{\phi }(Y(s))d\bar{W}(s)\nonumber \\&\qquad -X^{\pi ^*,l^*}(s-)\pi ^{*}(s,Y(s),Z(s))^{\top }dZ(s)\nonumber \\&\qquad \quad -l^*(s-,Y(s-),Z(s-))X^{\pi ^*,l^*}(s-)g(Y(s-))dN(s). \end{aligned}$$

Then, Itô’s formula yields that for $u\in [t,T]$,

$$\begin{aligned} (X^{\pi ^*,l^*}(u))^{2\gamma }=&(X^{\pi ^*,l^*}(t))^{2\gamma }+\tilde{M}^{\pi ^*,l^*}(u)-\tilde{M}^{\pi ^*,l^*}(t)\nonumber \\&+\int _t^u(X^{\pi ^*,l^*}(s))^{2\gamma }\tilde{\mathcal{A}}(\pi ^*(s,Y(s),Z(s)),l^*(s,Y(s),Z(s));Y(s),Z(s))ds. \end{aligned}$$

Here, for $(\pi ,l)\in (-\infty ,1]^n\times [0,\infty )$ and $(i,z)\in \{1,\ldots ,m\}\times {{\mathcal {S}}}$,

$$\begin{aligned} \tilde{{\mathcal {A}}}(\pi ,l;i,z)=&{2\gamma }\big [r(i)+\pi ^{\top }(I-diag(z))\theta (i,z)+(p(i,z)-c(i))l\big ]\\&+{\gamma }({2\gamma }-1)\big [\pi ^{\top }(I-diag(z))\sigma (i)\sigma (i)^{\top }(I-diag(z))\pi +l^2\big (\phi (i)\phi (i)^{\top }\nonumber \\&\quad +\bar{\phi }(i)\bar{\phi }(i)^{\top }\big )\\&-2l\pi ^{\top }(I-diag(z))\sigma (i)\phi (i)^{\top }\big ]+[(1-lg(i))^{2\gamma }-1]\nu (i,z)\\&+\sum _{j=1}^n[(1-\pi _j)^{2\gamma }-1](1-z_j)h_j(i,z). \end{aligned}$$

The $\mathbb {P}$-(local) martingale is given by, for $t\in [0,T]$,

$$\begin{aligned} \tilde{M}^{\pi ^*,l^*}(t):=&\int _0^t(X^{\pi ^*,l^*}(s))^{2\gamma }[\pi ^{*}(s,Y(s),Z(s))^{\top }\sigma (Y(s))-l^*(s,Y(s),Z(s))\phi (Y(s))]dW(s)\nonumber \\&-\int _0^t(X^{\pi ^*,l^*}(s))^{2\gamma }l^*(s,Y(s),Z(s))\bar{\phi }(Y(s))d\bar{W}(s)\nonumber \\&+\sum _{j=1}^n\int _0^t(X^{\pi ^*,l^*}(s-))^{2\gamma }[(1-\pi _j^*(s,Y(s-),Z(s-)))^{2\gamma }-1]dM_j(s)\nonumber \\&+\int _0^t(X^{\pi ^*,l^*}(s-))^{2\gamma }[(1-lg(Y(s-)))^{2\gamma }-1]d\tilde{N}(s). \end{aligned}$$

As above, we have that $\Vert \pi ^*(t,i,z)\Vert ^2+|l^*(t,i,z)|^{2}\le C$ for all $(t,i,z)\in [0,T]\times \{1,\ldots ,m\}\times {{\mathcal {S}}}$, and hence

$$\begin{aligned} |(1-l^*(t,i,z)g(i))^{2\gamma }-1|\le (1+\gamma )(|l^*(t,i,z)g(i)|^2+|l^*(t,i,z)g(i)|). \end{aligned}$$

Then, there exists a constant $C>0$ such that for all $(t,i,z)\in [0,T]\times \{1,\ldots ,m\}\times {{\mathcal {S}}}$,

$$\begin{aligned} |\tilde{{\mathcal {A}}}(\pi ^*(t,i,z),l^*(t,i,z),i,z)|\le C. \end{aligned}$$

Thus, we have that for all $t\in [0,T]$,

$$\begin{aligned}&\mathbb {E} _{t,x,i,z}\big [(X^{\pi ^*,l^*}(T\wedge \tau _a))^{2\gamma }\big ] =x^{2\gamma }\nonumber \\&\qquad + \mathbb {E} _{t,x,i,z}\left[ \int _t^{T\wedge \tau _a}(X^{\pi ^*,l^*}(s))^{2\gamma }\tilde{{\mathcal {A}}}(\pi ^*(s,Y(s),Z(s)),l^*(s,Y(s),Z(s));Y(s),Z(s))ds\right] \\&\quad \le x^{2\gamma }+ \mathbb {E} _{t,x,i,z}\left[ \int _t^{T}(X^{\pi ^*,l^*}(s\wedge \tau _a))^{2\gamma }\big |\tilde{{\mathcal {A}}}(\pi ^*(s,Y(s),Z(s)),l^*(s,Y(s);Z(s)),Y(s),Z(s))\big |ds\right] \\&\quad \le x^{2\gamma }+C\int _t^{T} \mathbb {E} _{t,x,i,z}[(X^{\pi ^*,l^*}(s\wedge \tau _a))^{2\gamma }]ds. \end{aligned}$$

The Gronwall’s inequality yields that

$$\begin{aligned} \sup _{a\in \mathbb {R}_+} \mathbb {E} _{t,x,i,z}\big [(X^{\pi ^*,l^*}(T\wedge \tau _a))^{2\gamma }\big ]\le x^{2\gamma }e^{CT}, \end{aligned}$$

and hence $\{(X^{\pi ^*,l^*}(T\wedge \tau _a))^{\gamma }\}_{a\in \mathbb {R}_+}$ is uniformly integrable. This yields that

$$\begin{aligned} V(t,x,i,z)= \mathbb {E} _{t,x,i,z}[U(X^{\pi ^*,l^*}(T))]=\lim _{a\rightarrow \infty } \mathbb {E} _{t,x,i,z}[U(X^{\pi ^*,l^*}(T\wedge \tau _a))]= x^\gamma \varphi (t,i,z). \end{aligned}$$

This verifies the validity of the conclusion (ii). $\square$