2. Model and problem formulation

We consider a continuous-time and right-continuous Markov chain $\{\alpha (t)\mid t\in [0,T]\}$ on a complete filtered probability space $(\Omega , \mathcal {F},\mathcal {F}(t),\mathbb {P})$ , where $\mathcal {F}(t)$ is generated by the information up to time t. The Markov chain $\alpha (t)$ is assumed to take values in a finite state space $\mathcal {M}=\{e_1,e_2,\ldots ,e_l\}$ , where $e_i\in \mathbb {R}^l$ and the jth component of $e_i$ is the Kronecker delta $\delta _{ij}.$ It is also assumed that the chain is homogeneous and has a generator $Q=(q_{ij})_{l\times l},$ in which $\sum \nolimits _{j=1}^lq_{ij}=0$ for any $ e_i\in \mathcal {M},$ and $q_{ij}>0$ if $i\neq j.$

Suppose that a financial market consists of a risk-free asset (bond) and n risky assets (stocks). Assume that risky assets can be traded continuously over $[0,T]$ and that there are no transaction costs and taxes in trading. The risk-free asset’s price process $S_0(t)$ is given by

$$ \begin{align*} dS_0(t)=r(t, \alpha(t))S_0(t)\,dt, \quad t\in[0, T], \end{align*} $$

where $r(t, e_i) (>0)$ representing the risk-free interest rate at state $e_i$ is a bounded and deterministic function on $[0,T]$ . For $k=1,2,\ldots ,n,$ the price process of the kth risky asset denoted by $S_k(t)$ is described by the following stochastic differential equation:

(2.1) $$ \begin{align} \begin{cases} dS_k(t)=S_k(t-)\bigg[b_k(t, \alpha(t-))\,dt+\displaystyle\sum\limits_{j=1}^{N}\sigma_{kj}(t, \alpha(t-))\,dW_j(t)\\ \qquad \qquad+\displaystyle\sum\limits_{j=1}^{M}\displaystyle\int_{\mathbb{R}_0}\eta_{kj}(t, z, \alpha(t-))N_{\alpha}^j(dt, dz)\bigg],\\ S_k(0)=S_{k0}, \end{cases} \end{align} $$

where the symbol “ $t-$ ” stands for the time before a jump occurring; $S_{k0}$ is the deterministic initial price; $b_k(t, \cdot )$ and $\sigma _{kj}(t, \cdot )$ represent the appreciation rate and volatility coefficient, respectively; $\eta _{kj}(t, z, \cdot )\,(>-1)$ is the jump amplitude with $\mathbb {R}_0=\mathbb {R}\setminus \{0\}$ ; $b_k(t, \cdot )$ , $\sigma _{kj}(t, \cdot )$ , $\eta _{kj}(t, z, \cdot )$ are continuous and bounded; $W(t)=(W_1(t),W_2(t),\ldots ,W_N(t))^\top $ is an N-dimensional standard Brownian motion, where the superscript $\text {``}^{\top }\text {''}$ denotes the transpose of a matrix or vector and $N_{\alpha }(dt, dz)=(N_{\alpha }^1(dt, dz),\ldots , N_{\alpha }^M(dt, dz))^\top $ is an M-dimensional Poisson random measure with compensated Poisson random measures defined as

$$ \begin{align*}\begin{array}{@{}ll} \tilde{N}_{\alpha}^j(dt, dz)=N_{\alpha}^j(dt, dz)-\nu_{\alpha}^j(dz)\,dt,\quad j=1, 2,\ldots, M. \end{array}\end{align*} $$

The diffusion component in equation (2.1) characterizes the normal fluctuation in a stock’s price, due to gradual changes in economic conditions or arrivals of new information which causes marginal changes in the stock’s price. The jump component describes sudden changes in a stock’s price, due to arrivals of important new information which has large effects on the stock’s price. It is well known from the SDE theory that a unique solution exists for the SDE (2.1).

Let $\pi _k(t)$ be the proportion of an investor’s wealth invested into the kth risky asset at time t and $\pi (t)=(\pi _1(t), \pi _2(t), \ldots , \pi _n(t))^\top , ~t\in [0,T],$ be the investor’s portfolio strategy. The remaining proportion of the investor’s wealth invested in the risk-free asset is then given by $1-\sum \nolimits _{k=1}^n \pi _k(t).$ Assume that the investor consumes wealth with nonnegative rate $c(t)$ at time t. Denote $u(t)=(\pi (t),c(t)).$ Given a consumption process c and a portfolio process $\pi $ , the wealth process $X(t)=X^u(t)$ of the investor evolves as

(2.2) $$ \begin{align} dX(t)&=\sum_{k=1}^n\dfrac{\pi_k(t)X(t)}{S_k(t-)}\,dS_k(t)+\frac{\big(1-\sum_{k=1}^n\pi_k(t)\big)X(t)}{S_0(t)}\,dS_0(t) -c(t)\,dt \nonumber \\ &=\bigg[\bigg\{r(t, \alpha(t-))+\sum_{k=1}^n\pi_k(t)\{b_k(t,\alpha(t-))-r(t, \alpha(t-))\}\bigg\}X(t) -c(t)\bigg]\,dt \nonumber \\ &\quad+\sum_{k=1}^n\pi_k(t)X(t)\sum_{j=1}^{N}\sigma_{kj}(t, \alpha(t-))\,dW_j(t) \nonumber\\ &\quad +\sum_{k=1}^n\pi_k(t)X(t)\sum_{j=1}^{M}\int_{\mathbb{R}_0}\eta_{kj}(t, z, \alpha(t-))N_{\alpha}^j(dt, dz). \end{align} $$

Setting

$$ \begin{align*} \begin{cases} B(t,\alpha(t-))=(b_1(t,\alpha(t-))-r(t, \alpha(t-)),\ldots,b_n(t,\alpha(t-))-r(t, \alpha(t-)))^\top,\\ \sigma(t,\alpha(t-))=(\sigma_{kj}(t, \alpha(t-)))_{n\times N},\\ \eta(t,z,\alpha(t-))=(\eta_{kj}(t, z, \alpha(t-)))_{n\times M}, \end{cases} \end{align*} $$

we can rewrite the wealth equation (2.2) as

(2.3) $$ \begin{align} dX(t)&=[\{r(t, \alpha(t-))+\pi^\top(t)B(t,\alpha(t-))\}X(t)-c(t)]\,dt \nonumber\\ &\quad+\pi^\top(t)X(t)\sigma(t,\alpha(t-))\,dW(t) +\int_{\mathbb{R}_0}\pi^\top(t)X(t)\eta(t,z,\alpha(t-))N_{\alpha}(dt, dz). \end{align} $$

In the rest of the paper, we write $r(t,\alpha (t-))$ , $B(t,\alpha (t-))$ , $\sigma (t,\alpha (t-))$ and $\eta (t,z,\alpha (t-))$ as r, B, $\sigma $ and $\eta $ , respectively, for notational convenience.

For a real-valued continuous and concave function g (with respect to $\pi $ and c) and a concave function h (with respect to x), we define the reward function $J(t,x,u(\cdot ),e_i)$ for a finite-time horizon $[0,T]$ by

(2.4) $$ \begin{align} J(t,x,u(\cdot),e_i) =E_{t,x,e_i}\bigg[\displaystyle\int_t^Te^{-\gamma s}g(s,X(s),u(s),\alpha(s))\,ds+e^{-\gamma T}h(X(T), \alpha(T))\bigg], \end{align} $$

where $E_{t,x,e_i}[\cdot ]=E[\cdot \mid X(t)=x, \alpha (t)=e_i],$ which leads to the value function

(2.5) $$ \begin{align} \begin{array}{@{}ll} V(t,x,e_i)=\displaystyle\mathop{\mathrm{sup}}_{u(\cdot)\in\mathcal{U}}J(t,x,u(\cdot),e_i). \end{array} \end{align} $$

Without the nonnegativity constraints on the portfolio strategy $\pi $ and the consumption rate c, we denote the corresponding admissible set by $\mathcal {\tilde {U}}.$ It is obvious that $\mathcal {U}\mathcal {\subseteq \tilde {U}}.$ In the next section, we will focus on the corresponding optimization problem with $\mathcal {\tilde {U}}$ , that is,

(2.6) $$ \begin{align} \begin{array}{@{}ll} V(t,x,e_i)=\mathop{\mathrm{sup}}\limits_{u(\cdot)\in\mathcal{\tilde{U}}}J(t,x,u(\cdot),e_i), \end{array} \end{align} $$

where $J(t,x,u(\cdot ),e_i)$ is defined in (2.4) and $X(t)$ satisfies the SDE (2.3). We will return to the problem (2.5) with power utility in Section 4.

3. Some results for the optimization problem

In this section, we aim at investigating the problem (2.6) by two kinds of methods, namely, the dynamic programming principle and the stochastic maximum principle.

3.1. Dynamic programming principle

For any $e_i\in \mathcal {M}$ , let $C^{1,2}([0,T]\times \mathbb {R})$ denote the space of $\varphi (t,x,e_i)$ such that $\varphi (t,x,e_i)$ and its derivatives $\varphi _t(t,x,e_i)$ , $\varphi _x(t,x,e_i)$ , $\varphi _{xx}(t,x,e_i)$ are continuous on $[0,T]\times \mathbb {R}$ . For any function $\varphi (t,x,e_i)\in C^{1,2}([0,T]\times \mathbb {R})$ , the usual infinitesimal generator $\mathcal {A}^{u}$ under u for the jump-diffusion process (2.3) is given by

$$ \begin{align*} \mathcal{A}^{u}\varphi(t,x,e_i)&=\varphi_t(t,x,e_i)+(rx+\pi^\top Bx-c)\varphi_x(t,x,e_i) +\dfrac{1}{2}\pi^\top x\sigma\sigma^\top\pi x\varphi_{xx}(t,x,e_i)\\ &\quad+\sum\limits_{j=1}^{l}q_{ij}\varphi(t,x,e_j) +\sum\limits_{j=1}^{M}\int_{\mathbb{R}_0}[\varphi(t,x+\pi^\top x\eta^{(j)},e_i)-\varphi(t,x,e_i)]\nu_{e_i}^j(dz), \end{align*} $$

where $\eta ^{(j)}$ denotes the jth column of the matrix $\eta $ .

According to standard stochastic control theory (see the book by Fleming and Soner [Reference Fleming and Soner8]), one can show that the HJB equation associated with the problem (2.6) is

(3.1) $$ \begin{align} \begin{cases} \begin{aligned} &\mathop{\mathrm{sup}}\limits_{u}\{\mathcal{A}^{u}V(t,x,e_i)+e^{-\gamma t}g(t,x,u,e_i)\}=0,\\ &V(T,x,e_i)=e^{-\gamma T}h(x,e_i). \end{aligned} \end{cases} \end{align} $$

The following theorem shows a connection between the solution to the HJB equation (3.1) and the optimal strategy as well as the value function.

Theorem 3.1 (Verification theorem). Assume that $W(t,x,e_i)\in C^{1,2}$ is a solution to the HJB equation (3.1) with $W_x(t,x,e_i)>0, W_{xx}(t,x,e_i)<0$ and $u^*=(\pi ^*, c^*)$ satisfying

$$ \begin{align*}\begin{array}{@{}ll} u^*=\arg\mathop{\mathrm{sup}}\limits_{u}\{\mathcal{A}^{u}W(t,x,e_i)+e^{-\gamma t}g(t,x,u,e_i)\}. \end{array}\end{align*} $$

Then the value function $V(t,x,e_i)$ coincides with $W(t,x,e_i)$ , that is,

$$ \begin{align*}W(t,x,e_i)=V(t,x,e_i).\end{align*} $$

Furthermore, $u^*$ is an optimal strategy for the problem (2.6).

Proof. The HJB equation (3.1) that is satisfied by $W(t,x,e_i)$ can be rewritten as

$$ \begin{align*}\begin{array}{@{}ll} \begin{aligned} W_t(t,x,e_i)+\mathop{\mathrm{sup}}\limits_{u}\bigg\{\psi(\pi,c)+\displaystyle\sum\limits_{j=1}^{l}q_{ij}W(t,x,e_j)\bigg\}=0 \end{aligned} \end{array}\end{align*} $$

with the boundary condition $W(T,X(T),\alpha (T))=e^{-\gamma T}h(X(T),\alpha (T)),$ where

$$ \begin{align*} \psi(\pi,c)&=(rx+{\pi}^\top Bx-c)W_x(t,x,e_i)+\dfrac{1}{2}{\pi}^\top x\sigma\sigma^\top{\pi} xW_{xx}(t,x,e_i)\\ &\quad+\sum\limits_{j=1}^{M}\int_{\mathbb{R}_0}[W(t,x+{\pi}^\top x\eta^{(j)},e_i)-W(t,x,e_i)]\nu_{e_i}^j(dz) +e^{-\gamma t}g(t,x,u,e_i). \end{align*} $$

For any $s\in [t, T],$ applying Itô’s formula [Reference Fleming and Soner8] to the function $W(s, X(s), \alpha (s))$ yields

(3.2) $$ \begin{align} dW(s, X(s), \alpha(s)) &=\dfrac{1}{2}\pi^\top X(s-)\sigma(s,\alpha(s-))\sigma^\top(s,\alpha(s-))\pi X(s-)W_{xx}\nonumber \\ &\quad+\bigg[W_s+\{r(s,\alpha(s-))X(s-)+\pi^\top X(s-)B(s,\alpha(s-))-c(s)\}W_x \nonumber\\ &\quad+\sum\limits_{j=1}^{M}\int_{\mathbb{R}_0}\{W(s,X(s-)+\pi^\top X(s-)\eta^{(j)},\alpha(s))\nonumber\\ &\quad-W(s,X(s-),\alpha(s-))\}\nu_{\alpha}^j(dz) \nonumber\\ &\quad+\sum\limits_{j=1}^{l}\{W(s,X(s-),e_j)-W(s,X(s-),\alpha(s-))\}q_{\alpha(s-)j}\bigg]\,ds+dM(t), \end{align} $$

where

$$ \begin{align*} M(t)=&\int_0^t\pi^\top(s)X(s)\sigma(s,\alpha(t-))W_x(s, X(s), \alpha(s-))\,dW(s)\\ &+\sum\limits_{j=1}^{M}\int_0^t\int_{\mathbb{R}_0}\{W(s,X(s-)+\pi^\top(s) X(s-)\eta^{(j)},\alpha(s))\\ &-W(s,X(s-),\alpha(s-))\}\tilde{N}_{\alpha}^j(ds,dz)\\ &+\sum\limits_{j=1}^{l}\int_0^t\{W(s,X(s-),e_j)-W(s,X(s-),\alpha(s-))\}\,d\tilde{\Phi}_j(s). \end{align*} $$

Here $\tilde {\Phi }_j(s),j=1,2,\ldots ,l$ is an $(\mathcal {F},\mathbb {P})$ martingale (see the paper by Zhang et al. [Reference Zhang, Elliott and Siu33] for more details). Then $M(t)$ is a local martingale. Since $b_k(t, \cdot )$ , $\sigma _{kj}(t, \cdot )$ , $\eta _{kj}(t, z, \cdot )$ are continuous and bounded by the parameter hypothesis and $u(\cdot )=(\pi (\cdot ),c(\cdot ))$ is an admissible strategy, the first and second terms are square-integrable. Moreover, the third term is also square-integrable due to the assumption $W(t,x,e_i)\in C^{1,2}.$ Thus, $M(t)$ is a martingale.

Adding $\int _t^Te^{-\gamma s}g(s,X(s),u(s),\alpha (s))\,ds$ and taking expectation in equation (3.2),

$$ \begin{align*} E_{t,x,e_i}\bigg[&W(T, X(T), \alpha(T))+\int_t^Te^{-\gamma s}g(s,X(s),u(s),\alpha(s))\,ds\bigg] \\ &=W(t,x,e_i) +E_{t,x,e_i}\bigg[\int_t^T\{\mathcal{A}^uW(s, X(s), \alpha(s))+e^{-\gamma s}g(s,X(s),u(s),\alpha(s))\}\,ds\bigg]. \end{align*} $$

From (3.1),

$$ \begin{align*} \mathcal{A}^uW(s, X(s), \alpha(s))+e^{-\gamma s}g(s,X(s),u(s),\alpha(s))\leq0, \end{align*} $$

where equality holds when $u=u^*.$ By the definition of the objective function (2.4) and the terminal condition $W(T, X(T), \alpha (T))=e^{-\gamma T}h(X(T), \alpha (T))$ ,

(3.3) $$ \begin{align} J(t,x,u,e_i)\leq W(t,x,e_i), \end{align} $$

where the equality in (3.3) will hold when $u=u^*.$ Therefore, $V(t,x,e_i)\leq W(t,x,e_i).$ On the other hand,

$$ \begin{align*}W(t,x,e_i)=J(t,x,u^*,e_i)\leq V(t,x,e_i).\end{align*} $$

As a result, $W(t,x,e_i)=V(t,x,e_i),$ that is, the solution $W(t,x,e_i)$ to (3.1) is the value function and $u^*$ is an optimal strategy. This completes the proof.

Furthermore, the following theorem shows that under some condition, the optimal result for the problem (2.6) is the solution to a system of partial differential equations.

Theorem 3.2. For the portfolio–consumption problem (2.6), if the inequality

(3.4) $$ \begin{align} e^{-\gamma t}&g_{cc}(t,x,u,e_i)\bigg[x\sigma\sigma^\top xV_{xx}(t,x,e_i) +e^{-\gamma t}g_{\pi\pi}(t,x,u,e_i) \nonumber \\ &+\sum\limits_{j=1}^{M}\displaystyle\int_{\mathbb{R}_0}V_{xx}(t,x+{\pi}^\top x\eta^{(j)},e_i)x^2\eta^{(j)}{\eta^{(j)}}^T\nu_{e_i}^j(dz)\bigg] -e^{-2\gamma t}g_{c\pi}^2(t,x,u,e_i)>0 \end{align} $$

holds, then the optimal strategy $(\pi ^*,c^*)$ is the solution to the following equations:

(3.5) $$ \begin{align} \begin{cases} BxV_x(t,x,e_i)+\sigma\sigma^\top\pi x^2V_{xx}(t,x,e_i)+e^{-\gamma t}g_{\pi}(t,x,u,e_i)\\ \quad+\displaystyle\sum\limits_{j=1}^{M}\displaystyle\int_{\mathbb{R}_0}V_x(t,x+{\pi}^\top x\eta^{(j)},e_i)x\eta^{(j)}\nu_{e_i}^j(dz)=0,\\ e^{-\gamma t}g_{c}(t,x,u,e_i)-V_{x}(t,x,e_i)=0. \end{cases} \end{align} $$

Proof. Differentiating $\psi (\pi ,c)$ with respect to $\pi $ and $c,$ respectively, yields

$$ \begin{align*} \begin{cases} \dfrac{\partial\psi}{\partial c}=-V_x(t,x,e_i)+e^{-\gamma t}g_c(t,x,u,e_i),\\[2mm] \dfrac{\partial\psi}{\partial \pi}= BxV_x(t,x,e_i)+x\sigma\sigma^\top{\pi} xV_{xx}(t,x,e_i)+e^{-\gamma t}g_{\pi}(t,x,u,e_i)\\ \quad\quad\ \ +\displaystyle\sum\limits_{j=1}^{M}\displaystyle\int_{\mathbb{R}_0}V_x(t,x+{\pi}^\top x\eta^{(j)},e_i)x\eta^{(j)}\nu_{e_i}^j(dz),\\ \dfrac{\partial^2\psi}{\partial c^2}=e^{-\gamma t}g_{cc}(t,x,u,e_i),\\[2mm] \dfrac{\partial^2\psi}{\partial \pi^2}=x\sigma\sigma^\top xV_{xx}(t,x,e_i)+e^{-\gamma t}g_{\pi\pi}(t,x,u,e_i)\\ \quad\quad\quad+\displaystyle\sum\limits_{j=1}^{M}\displaystyle\int_{\mathbb{R}_0}V_{xx}(t,x+{\pi}^\top x\eta^{(j)},e_i)x^2\eta^{(j)}{\eta^{(j)}}^\top\nu_{e_i}^j(dz),\\ \dfrac{\partial^2\psi}{\partial c\partial\pi}=\dfrac{\partial^2\psi}{\partial\pi\partial c}=e^{-\gamma t}g_{c\pi}(t,x,u,e_i). \end{cases} \end{align*} $$

Since g is a concave function with respect to $\pi $ and c and the Hessian matrix

$$ \begin{align*}H=\left(\begin{array}{@{}ll} \dfrac{\partial^2\psi}{\partial c^2}\quad \dfrac{\partial^2\psi}{\partial c\partial\pi}\\[8pt] \dfrac{\partial^2\psi}{\partial\pi\partial c}\quad \dfrac{\partial^2\psi}{\partial \pi^2} \end{array}\!\!\right)\end{align*} $$

is a negative-definite matrix because of inequality (3.4), the maximizer $u^*=(\pi ^*,c^*)$ to the HJB equation (3.1) satisfies equation (3.5).

3.2. Stochastic maximum principle

In this subsection, we discuss the problem (2.6) using the approach of the stochastic maximum principle. Define the Hamiltonian function $H: [0,T]\times \mathbb {R}\times U\times \mathcal {M}\times \mathbb {R}\times \mathbb {R}^N\times \mathbb {R}^M\longrightarrow \mathbb {R}$ by

$$ \begin{align*} H(t,x,u,e_i,p,\varphi,\psi) &=-e^{-\gamma t}g(t,x,u,e_i)+\bigg[\bigg\{r+\pi^\top\bigg(B+\int_{\mathbb{R}_0}\eta \nu_{e_i}(dz)\bigg)\bigg\}x-c\bigg]p(t)\\ &\quad+\sum\limits_{k=1}^n\sum\limits_{j=1}^N\pi_kx\sigma_{kj}\varphi_j(t) +\sum\limits_{k=1}^n\sum\limits_{j=1}^M\displaystyle\int_{\mathbb{R}_0}\pi_kx\eta_{kj}\psi_j(t,z)\nu^j_{e_i}(dz). \end{align*} $$

Assume that the Hamiltonian function H is differentiable with respect to x.

The adjoint equation corresponding to u and $X(t)$ for the unknown adapted processes $p(t), \varphi (t)$ and $\psi (t,\cdot ),$ where $p(t)\in \mathbb {R}, \varphi (t)\in \mathbb {R}^N$ and $\psi (t,\cdot )\in \mathbb {R}^M,$ is given by the following SDE:

(3.6) $$ \begin{align} \begin{cases} \begin{aligned} dp(t)=&\bigg\{e^{-\gamma t}g_x(t,X(t-),u,\alpha(t-))-\bigg[r+\pi^\top\bigg(B+\int_{\mathbb{R}_0}\eta \nu_{\alpha(t-)}(dz)\bigg)\bigg]p(t-)\\ &-\displaystyle\sum\limits_{k=1}^n \displaystyle\sum\limits_{j=1}^N\pi_k\sigma_{kj}\varphi_j(t) - \displaystyle\sum\limits_{k=1}^n \displaystyle\sum\limits_{j=1}^M\displaystyle\int_{\mathbb{R}_0}\pi_k\eta_{kj}\psi_j(t,z)\nu^j_{\alpha(t-)}(dz)\bigg\}\,dt\\ &+\displaystyle\sum\limits_{j=1}^N\varphi_j(t)\,dW_j(t)+ \displaystyle\sum\limits_{j=1}^M\displaystyle\int_{\mathbb{R}_0}\psi_j(t,z)\tilde{N}^j_{\alpha}(dt,dz),\\ p(T)=&-e^{-\gamma t}h_x(X(T),\alpha(T)). \end{aligned} \end{cases} \end{align} $$

For the existence and uniqueness of the solution to the SDE (3.6) with jumps, interested readers are referred to Zhang et al. [Reference Zhang, Elliott and Siu33].

The next theorem develops a sufficient stochastic maximum principle for the optimal problem.

Theorem 3.3 (Sufficient conditions for optimality). Let $u^*\in \mathcal {\tilde {U}}$ with the corresponding solution $X^*=X^{u^*}$ . Suppose that there exists an adapted solution $(\hat {p}(t),\hat {\varphi }(t),\hat {\psi }(t,\cdot ))$ to the corresponding adjoint equation (3.6) such that for all $u(\cdot )\in \mathcal {\tilde {U}}$ ,

$$ \begin{align*}\begin{array}{@{}ll} E\bigg[\displaystyle\int_0^T(X^*(t)-X(t))^2\bigg(\hat{\varphi}^\top(t)\hat{\varphi}(t) +\displaystyle\int_{\mathbb{R}_0}\hat{\psi}^\top(t,z) \mbox{ {Diag}}(\nu_\alpha(dz))\hat{\psi}(t,z)\bigg)\,dt\bigg]<\infty \end{array}\end{align*} $$

and

$$ \begin{align*} &E\bigg[\displaystyle\int_0^T\hat{p}(t)^2\bigg({\pi^*(t)}^\top X^*(t-)\sigma\sigma^\top\pi^*(t)X^*(t-)\\ &\quad+\displaystyle\int_{\mathbb{R}_0}{\pi^*(t)}^\top X^*(t-)\eta \mbox{ {Diag}}(\nu_\alpha(dz))\eta^\top\pi^*(t)X^*(t-)\bigg)\,dt\bigg]<\infty. \end{align*} $$

Assume further that the following three conditions hold:

1. (i) for almost all $t\in [0,T],$

   $$ \begin{align*} & H(t,X^*(t-),u^*(t),\alpha(t-),\hat{p}(t-),\hat{\varphi}(t),\hat{\psi}(t,\cdot))\\ & \quad=\inf\limits_{u}H(t,X^*(t-),u(t),\alpha(t-),\hat{p}(t-),\hat{\varphi}(t),\hat{\psi}(t,\cdot)); \end{align*} $$

2. (ii) for each fixed pair $(t,e_i)\in [0,T]\times \mathcal {M}$ ,

   $$ \begin{align*}\hat{H}(x)=\inf\limits_{u}H(t,x,u,e_i,\hat{p}(t-),\hat{\varphi}(t),\hat{\psi}(t,\cdot))\end{align*} $$
   exists and is a convex function with respect to x;

3. (iii) $h(x, e_i)$ is a concave function with respect to x for each $e_i\in \mathcal {M}.$

Then $u^*$ is an optimal strategy for the problem (2.6) and $X^*$ is the corresponding controlled state process.

Proof. Define $\tilde {J}(t,x,u(\cdot ),e_i)$ and $\tilde {V}(t,x,e_i)$ by

$$ \begin{align*} \tilde{J}(t,x,u(\cdot),e_i)&=-J(t,x,u(\cdot),e_i)\\ &=E_{t,x,e_i}\bigg[\displaystyle\int_t^T-e^{-\gamma s}g(s,X(s),u(s),\alpha(s))\,ds-e^{-\gamma T}h(X(T), \alpha(T))\bigg],\\ \tilde{V}(t,x,e_i)&=-V(t,x,e_i), \end{align*} $$

respectively. Then the problem (2.6) is equivalent to the following problem:

(3.7) $$ \begin{align} \tilde{V}(t,x,e_i)=\inf\limits_{u(\cdot)\in\mathcal{\tilde{U}}}\tilde{J}(t,x,u(\cdot),e_i) \end{align} $$

with $X(t)$ satisfying the SDE (2.3).

For the problem (3.7) with the Hamiltonian function $H(t,x,u,e_i,p,\varphi ,\psi )$ and the adjoint equation (3.6), one can follow the proof of Theorem 3.1 in Zhang et al. [Reference Zhang, Elliott and Siu33] to obtain the optimal strategy $u^*$ for the problem (2.6). This completes the proof.

The next theorem presents the results derived by the method of the stochastic maximum principle for the problem (2.6).

Theorem 3.4. For the portfolio–consumption problem (2.6), the optimal strategy $(\pi ^*, c^*)$ is the solution to the following equations:

(3.8) $$ \begin{align} \begin{cases} e^{-\gamma t}g_{\pi}(t,x,u,e_i)+BxV_x(t,x,e_i)+\sigma\sigma^\top\pi x^2V_{xx}(t,x,e_i) \\ \quad{} +\displaystyle\sum\limits_{j=1}^M\displaystyle\int_{\mathbb{R}_0}V_{x}(t,x+\pi^\top x\eta^{(j)},e_i)x\eta^{(j)}\nu^j_{e_i}(dz)=0,\\ e^{-\gamma t}g_c(t,x,u,e_i)-V_x(t,x,e_i)=0. \end{cases} \end{align} $$

Proof. For the problem (3.7) with the adjoint equation (3.6), we mimic the proof of Theorem 4.2 in Zhang et al. [Reference Zhang, Elliott and Siu33] to obtain the optimal results. Note that

(3.9) $$ \begin{align} \begin{cases} \hat{p}(t)=\tilde{V}_x(t,x,e_i)=-V_x(t,x,e_i),\\ \hat{\varphi}_j(t)=\tilde{V}_{xx}(t,x,e_i)\displaystyle\sum\limits_{k=1}^n\pi_kx\sigma_{kj}=-V_{xx}(t,x,e_i)\displaystyle\sum\limits_{k=1}^n\pi_kx\sigma_{kj},\\ \hat{\psi}_j(t,z)=\tilde{V}_x(t,x+\displaystyle\sum\limits_{k=1}^n\pi_kx\eta_{kj},e_i)-\tilde{V}_x(t,x,e_i)\\ \qquad \quad=V_x(t,x,e_i)-V_x(t,x+\displaystyle\sum\limits_{k=1}^n\pi_kx\eta_{kj},e_i). \end{cases} \end{align} $$

Since $H(t,x,u,e_i,\hat {p}(t),\hat {\varphi }(t),\hat {\psi }(t,\cdot ))$ is linear in $\pi $ and c, the coefficients of $\pi $ and c should vanish at optimality, that is,

(3.10) $$ \begin{align} \begin{cases} -e^{-\gamma t}g_{\pi}(t,x,u,e_i)+\bigg[B+\displaystyle\int_{\mathbb{R}_0}\eta \nu_{e_i}(dz)\bigg]x\hat{p}(t) \\ \quad {} + {} x\sigma\hat{\varphi}(t) +\displaystyle\sum\limits_{j=1}^M\displaystyle\int_{\mathbb{R}_0}x\eta^{(j)}\hat{\psi}_j(t,z)\nu^j_{e_i}(dz)=0,\\ -e^{-\gamma t}g_c(t,x,u,e_i)-\hat{p}(t)=0. \end{cases} \end{align} $$

Substituting $(\hat {p}(t),\hat {\varphi }(t),\hat {\psi }(t,z))$ of (3.9) into (3.10), we obtain the equation (3.8).

Remark 3.5 For the portfolio–consumption problem (2.6), by the techniques of the dynamic programming principle and the stochastic maximum principle, we obtain the same results for the general objective function, that is, the optimal strategy without nonnegative constraints is the solution to a system of partial differential equations. In the next section, we shall focus on the optimization problem with a power utility function and derive closed-form expressions for the optimal strategy and the value function.

4. Optimal results under power utility

In this section, we work on the problem (2.6) in which the utility function of the investor is of power type, that is,

(4.1) $$ \begin{align} \begin{cases} g(t,x,\pi,c,e_i)=\dfrac{c^\delta}{\delta},\\[-1mm] h(x,e_i)=\dfrac{x^\delta}{\delta}, \end{cases} \end{align} $$

with $ 0<\delta <1.$ For illustration purposes, we consider the case with $n=N=M=1$ . The derivations of the optimal strategy and the value function are based on the HJB equation. From (4.1) and based on the terminal condition of the value function, we try to fit a solution of the form

(4.2) $$ \begin{align} V(t,x,e_i)=e^{-\gamma t}k^{1-\delta}(t,e_i)\dfrac{x^\delta}{\delta}, \end{align} $$

where $k(t,e_i)$ is a suitable function such that (4.2) is a solution to the HJB equation (3.1). The boundary condition $V(T,x,e_i)=e^{-\gamma T} x^\delta / \delta $ implies that $k(T,e_i)=1$ . Then the original HJB equation (3.1) can be rewritten as

(4.3) $$ \begin{align} \begin{cases} \dfrac{x^\delta}{\delta}[(1-\delta)k_t(t,e_i)-\gamma k(t,e_i)]+rk(t,e_i)x^{\delta}+\mathop{\mathrm{sup}}\limits_{c}\psi(c)\\ \quad +\mathop{\mathrm{sup}}\limits_{\pi}[k(t,e_i)x^{\delta}\phi(\pi)] +\displaystyle \sum\limits_{j=1}^{l}q_{ij}k^{1-\delta}(t,e_j)k^\delta(t,e_i)\dfrac{x^\delta}{\delta}=0,\\ k(T,e_i)=1, \end{cases} \end{align} $$

where $k_t$ represents the partial derivative of the function k with respect to t and

$$ \begin{align*}\left\{\begin{array}{@{}ll} \psi(c)=\dfrac{c^\delta}{\delta}k^\delta(t,e_i)-k(t,e_i)cx^{\delta-1},\\ \phi(\pi)=\pi (b-r)+\dfrac{1}{2}(\delta-1)\sigma^2\pi^2 +\dfrac{1}{\delta}\displaystyle\int_{\mathbb{R}_0}((1+\pi\eta)^\delta-1)\nu_{e_i}(dz). \end{array}\right.\end{align*} $$

Recalling Theorem 3.2, we find that the inequality

(4.4) $$ \begin{align} \begin{array}{@{}ll} (\delta-1)^2e^{-2\gamma t}k^{1-\delta}(t,e_i)x^\delta\bigg[\sigma^2+\displaystyle\int_{\mathbb{R}_0}(1+\pi\eta)^{\delta-2}\eta^2\nu_{e_i}(dz)\bigg]>0 \end{array} \end{align} $$

holds and, thus, the maximizer of the problem (2.6) without constraints is the solution to the following equations:

(4.5) $$ \begin{align} \begin{cases} e^{-\gamma t}\psi'(c)=0,\\[.2cm] e^{-\gamma t}k^{1-\delta}(t,e_i) \phi'(\pi)=0, \end{cases} \end{align} $$

where $\psi '$ ( $\phi '$ ) represents the first derivative of $\psi $ ( $\phi $ ). From (4.5),

(4.6) $$ \begin{align} c^*=k^{-1}(t,e_i)x, \end{align} $$

which is the maximizer of $\psi (c)$ .

We see later in Proposition 4.6 that $k(t,e_i)$ is positive for any state $e_i$ . Hence, if $X^*(t)\geq 0$ holds, then $c^*(t)=k^{-1}(t,e_i)X^*(t)$ is the optimal consumption strategy. Before investigating the optimal portfolio strategy, we need the following result.

Proof. Inserting $c(t)=k^{-1}(t,e_i)X(t)$ back into (2.3) yields

$$ \begin{align*}\begin{array}{@{}ll} dX(t)=X(t)\bigg\{[r+ (b-r)\pi-k^{-1}(t,\alpha(t-))]\,dt+\sigma\pi\,dW(t)+\displaystyle\int_{\mathbb{R}_0}\eta\pi N_{\alpha}(dt, dz)\bigg\}. \end{array}\end{align*} $$

Applying Itô’s formula to $\ln X(t)$ and integrating on both sides from $0$ to s for $s\in [0,T],$

$$ \begin{align*}\begin{array}{@{}ll} \begin{aligned} X(s)&=x\prod\limits_{k=1}^{N(s)}(1+\pi\eta(\tau_k,Z_{\tau_k},\alpha(\tau_k)))\\ &\quad \times\exp\bigg\{\int_0^s\bigg[r+ (b-r)\pi-\frac{1}{2}\sigma^2\pi^2-k^{-1}(u,\alpha(u-))\bigg]\,du+\int_0^s\sigma\pi\, dW(u)\bigg\}, \end{aligned} \end{array}\end{align*} $$

where $N(s)$ is the Poisson counting process (see the papers by Cont and Tankov and Framstad et al. [Reference Cont and Tankov7, Reference Framstad, Øksendal and Sulem9] for details) up to time $s.$ Since the exponent of the exponential function is positive, we have $X(s)\geq 0$ if $1+\pi \eta (u,\cdot ,\alpha (u))\geq 0$ for any $u\in [0,s].$ This completes the proof.

Define

$$ \begin{align*}\begin{array}{@{}ll} \bar{M}=-\text{ess~inf} ~\eta(t,\cdot,\alpha(t))=-{\mathrm{sup}}\{m\mid \mathbb{P}(\{z \mid \eta(t,z,\alpha(t))<m\})=0\}, \end{array}\end{align*} $$

where the symbol “ $\text {ess~inf}$ ” is the essential infimum. Let $\pi _0=1/\bar {M}$ for $\bar {M}>0$ and $\pi _0=+\infty $ for $\bar {M}\leq 0.$ It is easy to see that $\pi _0 \geq 1$ . Then, for $\pi \in (\pi _0,+\infty ),$ it is possible that $1+\pi \eta (t,\cdot ,\alpha (t))\geq 0$ or $1+\pi \eta (t,\cdot ,\alpha (t))<0.$ If $1+\pi \eta (t,\cdot ,\alpha (t))<0,$ then it follows from Lemma 4.1 that the nonnegativity of $X(t)$ cannot be guaranteed and hence the consumption process $c(t)$ could be negative due to equation (4.6), which is meaningless and impractical. Thus, in the following context, we shall consider the optimal portfolio strategy in the interval $[0,\pi _0]$ , which guarantees that $X(t)\geq 0.$

The following proposition gives the result of the optimal portfolio strategy.

Proof. Obviously, according to the function $\phi (\pi ),$

$$ \begin{align*}\left\{\begin{array}{@{}ll} \phi'(\pi)= b-r+(\delta-1)\sigma^2\pi+\displaystyle\int_{\mathbb{R}_0}(1+\pi\eta)^{\delta-1}\eta\nu_{e_i}(dz),\\ \phi''(\pi)= (\delta-1)\bigg[\sigma^2+\displaystyle\int_{\mathbb{R}_0}(1+\pi\eta)^{\delta-2}\eta^2\nu_{e_i}(dz)\bigg]. \end{array}\right.\end{align*} $$

For the jump-diffusion risk model, the jump item in $\phi (\pi )$ is a power function with respect to $\pi $ and $\eta (t,\cdot ,\alpha (t))$ is a random jump size with values in $(-1,+\infty )$ . Therefore, it is not easy to obtain an explicit expression for the optimal portfolio strategy simply using $\phi '(\pi )$ .

By the definition of $\pi _0,$ we know that $1+\pi \eta (t,\cdot ,\alpha (t))\geq 0$ and $\phi ''(\pi )<0$ for $\pi \in [0,\pi _0].$ Hence, $\phi (\pi )$ is continuous and concave in $[0,\pi _0].$ Also, following from Assumption 2.1, we have $\phi '(0)>0$ . Thus, we claim that if

(4.8) $$ \begin{align} \phi'(\pi_0)= b-r+(\delta-1)\sigma^2\pi_0+\displaystyle\int_{\mathbb{R}_0}(1+\pi_0\eta)^{\delta-1}\eta\nu_{e_i}(dz)<0 \end{align} $$

holds, according to the zero theorem [Reference Gille and Hornbostel13], there exists a unique solution $\pi ^*\in (0,\pi _0)$ such that $\phi '(\pi ^*)=0,$ which is the maximum point of $\phi (\pi ),$ and thus $\pi ^*$ is the optimal portfolio strategy. If $\phi '(\pi _0)\geq 0,$ that is, $\phi (\pi )$ increases in $[0,\pi _0],$ then $\pi _0$ is the maximizer of $\phi (\pi )$ and hence it is the optimal portfolio strategy.

Remark 4.3 If the jump size of the risky asset is nonnegative, that is, $\eta (t,\cdot ,\alpha (t))\geq ~0$ , we have $\pi _0=+\infty $ and

$$ \begin{align*}\phi'(+\infty)= \lim\limits_{\pi\rightarrow +\infty}\phi'(\pi)=-\infty,\end{align*} $$

so that the condition (4.8) is satisfied. Therefore, the optimal portfolio strategy is the unique positive solution to the equation (4.7) in this case.

The following example illustrates the result in Proposition 4.2.

Example 4.5 Assume that the jump size of the risky asset is negative, that is, $-1<\eta (t,\cdot ,\alpha (t))<0.$ Set $\delta = q/p \in (0,1)$ with two positive integers p and q such that $p>q$ and $q/p$ is irreducible. Suppose that both p and q are odd numbers. In this case, $p-q$ and $3p-q$ are even numbers and $2p-q$ is an odd number, so that $(1+\pi \eta (t,\cdot ,\alpha (t)))^{\delta -1}\geq 0$ and $(1+\pi \eta (t,\cdot ,\alpha (t)))^{\delta -3}\geq 0$ for $\pi \in [0,+\infty ).$ Denote

$$ \begin{align*}\varphi(\pi)=\sigma^2+\displaystyle\int_{\mathbb{R}_0}(1+\pi\eta)^{\delta-2}\eta^2\nu_{e_i}(dz)\end{align*} $$

with

$$ \begin{align*}\left\{\begin{array}{@{}ll} \varphi(0)=\sigma^2+\displaystyle\int_{\mathbb{R}_0}\eta^2\nu_{e_i}(dz),\\ \varphi(+\infty)=\lim\limits_{\pi\rightarrow +\infty}\varphi(\pi)=\sigma^2,\\ \varphi'(\pi)=(\delta-2)\displaystyle\int_{\mathbb{R}_0}(1+\pi\eta)^{\delta-3}\eta^3\nu_{e_i}(dz)>0. \end{array}\right.\end{align*} $$

Thus, there exists a breaking point $\tilde {\pi }\in (0,+\infty )$ satisfying

$$ \begin{align*}\varphi(\tilde{\pi}-)=\lim\limits_{\pi\rightarrow \tilde{\pi}-}\varphi(\pi)=+\infty\end{align*} $$

and

$$ \begin{align*}\varphi(\tilde{\pi}+)=\lim\limits_{\pi\rightarrow \tilde{\pi}+}\varphi(\pi)=-\infty,\end{align*} $$

where $\varphi (\tilde {\pi }-)$ and $\varphi (\tilde {\pi }+)$ express the left limit and the right limit for the function $\varphi (\cdot )$ at $\tilde {\pi }$ , respectively. Then there is a $\tilde {\pi }_1>\tilde {\pi }$ such that $\varphi (\tilde {\pi }_1)=0.$ Define

$$ \begin{align*}\left\{\begin{array}{@{}ll} \tilde{\pi}_0=\inf\{\tilde{\pi}>0\mid\varphi(\tilde{\pi}-)=+\infty,~ \varphi(\tilde{\pi}+)=-\infty\},\\ \pi_1=\inf\{\tilde{\pi}_1>\tilde{\pi}_0\mid\varphi(\tilde{\pi}_1)=0\},\\ \pi_{\infty}=\inf\{\pi>\pi_1\mid\varphi(\pi-)=+\infty\} \end{array}\right.\end{align*} $$

and put $\pi _{\infty }=+\infty $ if there does not exist $\pi>\pi _1$ such that $\varphi (\pi -)=+\infty .$ Obviously, we have $\tilde {\pi }_0\geq \pi _0.$ Moreover, it is easy to see that $\phi (0)=0, \phi '(0)>0$ and

$$ \begin{align*}\left\{\begin{array}{@{}ll} \phi'(+\infty)= \lim\limits_{\pi\rightarrow +\infty}\phi'(\pi)=-\infty,\\ \phi''(\pi)= (\delta-1)\varphi(\pi),\\[3mm] \phi'''(\pi)=(\delta-1)\varphi'(\pi)<0. \end{array}\right.\end{align*} $$

Then, for $\pi \in [0,\tilde {\pi }_0), \phi ''(\pi )$ decreases and $\phi ''(\pi )<0,$ which in turn imply that $\phi '(\pi )$ is decreasing. On the other hand, for $\pi \in (\tilde {\pi }_0,\pi _\infty ), \phi (\pi )$ is convex in $(\tilde {\pi }_0,\pi _1)$ and concave in $(\pi _1,\pi _\infty ).$ Define $\Theta =\{\pi \geq 0:\varphi (\pi )>0\}.$ Then we have $[0,\tilde {\pi }_0)\cup (\pi _1,\pi _\infty )\subseteq \Theta $ . Note that the condition (4.4) holds for $\pi \in \Theta $ and $\phi (\pi )$ is concave in $\Theta .$

If $\phi '(\pi _1)\leq 0,$ then $\phi '(\tilde {\pi }_0+)=\lim \nolimits _{\pi \rightarrow \tilde {\pi }_0+}\phi '(\pi )<0$ and $\phi (\pi )$ decreases in $(\tilde {\pi }_0,\pi _\infty ).$ So, there exists a unique $\pi _1^*\in (0,\tilde {\pi }_0)$ such that $\phi '(\pi _1^*)=0$ and $\pi _1^*$ is the maximizer of $\phi (\pi ).$ Hence, the optimal portfolio strategy in this case is $\pi ^*=\min \{\pi _1^*,\pi _0\}$ and the trajectory of the function $\phi (\pi )$ is shown in Figure 1.

Figure 1

$\phi '(\pi _1)\leq 0$ .

If $\phi '(\pi _1)>0,$ then we have $\phi '(\tilde {\pi }_0+)\geq 0$ or $\phi '(\tilde {\pi }_0+)<0.$ For $\phi '(\tilde {\pi }_0+)\geq 0,$ there exists $\pi _1^*>\pi _1$ such that $\phi (\pi )$ is increasing in $[0,\pi _1^*)$ and decreasing in $(\pi _1^*,\pi _\infty ).$ Hence, $\pi _1^*$ is the unique point such that $\phi '(\pi _1^*)=0,$ which maximizes $\phi (\pi ).$ However, $\pi _1^*\notin [0,\pi _0].$ Thus, the optimal portfolio strategy is $\pi ^*=\pi _0$ in this case and the trajectory of the function $\phi (\pi )$ is presented in Figure 2(a). For $\phi '(\tilde {\pi }_0+)<0, \phi (\pi )$ increases first and then decreases in $\pi \in (0,\tilde {\pi }_0-)$ . Moreover, there are two points $\pi _1^*, \pi _2^*\in (0,\pi _\infty )$ such that $\phi '(\pi _1^*)=0$ and $\phi '(\pi _2^*)=0.$ Note that $\pi _2^*\notin [0,\pi _0].$ Thus, the optimal portfolio strategy is $\pi ^*=\min \{\pi _1^*,\pi _0\}$ and the trajectory of the function $\phi (\pi )$ in this case is presented in Figure 2(b).

Figure 2

$\phi '(\pi _1)>0$ .

With (4.6) and Proposition 4.2, by computing $c^*$ and $\pi ^*$ and putting them back into (4.3),

(4.9) $$ \begin{align} \begin{cases} k_t(t,e_i)+\dfrac{k(t,e_i)}{1-\delta}\bigg\{-\gamma+\delta(r+\pi^*(b-r)) +\dfrac{1}{2}\sigma^2 \delta(\delta-1){\pi^*}^2\\ ~~+\displaystyle\int_{\mathbb{R}_0}[(1+\pi^*\eta)^\delta-1]\nu_{e_i}(dz)\bigg\} + \dfrac{1}{1-\delta}\displaystyle\sum\limits_{j=1}^{l}q_{ij}k^{1-\delta}(t,e_j)k^\delta(t,e_i)+1=0,\\ k(T,e_i)=1. \end{cases} \end{align} $$

Proof. Note that the equation (4.9) can be rewritten as

(4.10) $$ \begin{align} \begin{cases} \dfrac{dk(t,e_i)}{dt}+M_{ii}k(t,e_i)+\displaystyle\sum_{j\neq i}M_{ij}k^{1-\delta}(t,e_j)k^{\delta}(t,e_i)+C=0,\\ k(T,e_i)=1, \end{cases} \end{align} $$

where

$$ \begin{align*}\left\{\begin{array}{@{}ll} \begin{aligned} M_{ii}&=\frac{1}{1-\delta}\bigg\{-\gamma+\delta(r+\pi^*(b-r))+\frac{1}{2}\delta(\delta-1)\sigma^2{\pi^*}^2 \\[2pt] &\quad +\displaystyle\int_{\mathbb{R}_0} [(1+{\pi^*}\eta)^\delta-1]\nu_{e_i}(dz)+q_{ii}\bigg\},\\[2pt] M_{ij}&=\frac{1}{1-\delta}q_{ij}\quad (j\neq i),\\ C&=1. \end{aligned} \end{array}\right.\end{align*} $$

We first prove the existence and uniqueness of $k(t,e_i)$ to the equation (4.10). Denote $\bar {k}(t,e_i)=k^{1-\delta }(t,e_i)$ and

$$ \begin{align*}\left\{\!\!\!\!\!\begin{array}{@{}ll} &\bar{M}_{ii}=(1-\delta)M_{ii},\\ &\bar{M}_{ij}=(1-\delta)M_{ij} \quad (j\neq i), \\ &\bar{C}=(1-\delta)C. \end{array}\right.\end{align*} $$

Then (4.10) can be rewritten as

(4.11) $$ \begin{align} \frac{d\bar{k}(t,e_i)}{dt}+\bar{M}_{ii}\bar{k}(t,e_i)+\sum_{j\neq i}\bar{M}_{ij}\bar{k}(t,e_j)+\bar{C}\bar{k}^{-\delta/(1-\delta)}(t,e_i)=0. \end{align} $$

This equation satisfies the Lipschitz condition, since all of the parameters are continuous and bounded and $\pi ^*\in (0,\pi _0]$ . Applying Lemma A.1 in the Appendix of Cajueiro and Yoneyama [Reference Cajueiro and Yoneyama3], one can conclude that the equation (4.11) has a unique solution and that there exists a unique solution to the equation (4.10).

We next prove that the solution $k(t,e_i)$ to (4.10) is positive. Setting $\tau =T-t$ in (4.10) yields $dt=-d\tau .$ Let $\tilde {k}(\tau ,e_i)=k(T-\tau ,e_i)=k(t,e_i).$ Then equation (4.10) can be expressed as

(4.12) $$ \begin{align} \begin{cases} \dfrac{d\tilde{k}(\tau,e_i)}{d\tau}=M_{ii}\tilde{k}(\tau,e_i)+\displaystyle\sum_{j\neq i}M_{ij}\tilde{k}^{1-\delta}(\tau,e_j)\tilde{k}^{\delta}(\tau,e_i)+C,\\ \tilde{k}(0,e_i)=1, \end{cases} \end{align} $$

in which $M_{ij}\geq 0$ for $j\neq i$ and C is a positive constant. Note that the equation (4.12) is the same as (A.3) in Lemma A.2 of Cajueiro and Yoneyama [Reference Cajueiro and Yoneyama3]. Therefore, one can show that the equation (4.12) has a positive solution and hence the equation (4.10) also has a positive solution.

As a result, the uniqueness and positivity of (4.10) imply that the equation (4.9) has a unique positive solution.

Finally, Propositions 4.2 and 4.6 together with equation (4.6) give the following result for the optimal problem with constraints, the problem (2.5).

Theorem 4.7. For the portfolio–consumption problem (2.5), the optimal strategy is attained at $u^*=(\pi ^*, c^*),$ where $\pi ^*$ and $c^*$ are given by Proposition 4.2 and equation (4.6), respectively. Moreover, the value function has the form

$$ \begin{align*} V(t,x,e_i)=e^{-\gamma t}k^{1-\delta}(t,e_i)\frac{x^\delta}{\delta}, \end{align*} $$

where $k(t,e_i)$ is the unique positive solution to the equation (4.9).

5. Comparison of optimal results

In this section, we carry out some comparison between the optimal results for the jump-diffusion risk model and the pure diffusion model.

Assume that the financial market has only one state, that is, $l=1$ . Let $\pi ^*, c^*$ and $V(t,x)$ be the optimal results in the jump-diffusion market with $\nu \neq 0.$ Let $ \bar {\pi }^*, \bar {c}^*$ and $\bar {V}(t,x)$ be the corresponding optimal solutions when there are no jumps, that is, $\nu =0.$ Recall $\phi (\pi )$ defined in Proposition 4.2. Similarly, we define

$$ \begin{align*}\begin{array}{@{}ll} \bar{\phi}(\pi)= \pi (b(t)-r(t)) +\frac{1}{2} \sigma^2\pi^2 (\delta-1) \end{array}\end{align*} $$

in the case of no jumps. Then we have $\phi (0)=\bar {\phi }(0)=0$ . In addition, we can rewrite the equation (4.9) as

(5.1) $$ \begin{align} \begin{cases} k_t(t)+\dfrac{\Gamma(t)}{1-\delta} k(t)+1=0,\\[.2cm] k(T)=1, \end{cases} \end{align} $$

where $\Gamma (t)=-\gamma +\delta r(t)+\delta \phi (\pi ^*).$ Solving the equation (5.1) by using the variation of constants method yields

(5.2) $$ \begin{align} k(t)=\exp\bigg\{\int_t^T\frac{\Gamma(s)}{1-\delta}\,ds\bigg\}+\int_t^T\exp\bigg\{\int_t^s\frac{\Gamma(\tau)}{1-\delta}d\tau\bigg\}\,ds. \end{align} $$

It follows from (5.2) that the function $k(t)$ increases as $\Gamma (t)$ increases. In order to make comparison of the optimal results more explicitly, we need to discuss the following three cases.

Case I $\eta (t,\cdot )\geq 0$ .

In this case, $\phi (\pi )$ is concave and the inequality $\phi '(\pi )\geq \bar {\phi }'(\pi )$ holds for all $\pi \in (0,+\infty ),$ which implies that $\bar {\phi }'(\pi ^*)\leq \phi '(\pi ^*)=0.$ Since $\bar {\phi }(\pi )$ is concave and $\bar {\pi }^*$ is the maximum point with $\bar {\phi }'(\bar {\pi }^*)=0$ , we conclude that $\bar {\pi }^*\leq \pi ^*.$ Moreover, we have $\phi (\pi ^*)\geq \phi (\bar {\pi }^*)\geq \bar {\phi }(\bar {\pi }^*)$ and thus

$$ \begin{align*}\Gamma(t)\geq\bar{\Gamma}(t)=-\gamma+\delta r(t)+\delta\bar{\phi}(\bar{\pi}^*).\end{align*} $$

In other words, the value of $k(t)$ in the jump case is no less than that in the case without jumps. Therefore, we have $V(t,x)\geq \bar {V}(t,x)$ and $c^*\leq \bar {c}^*.$

Case III General $\eta (t,\cdot ),$ that is, $\eta (t,\cdot )> -1$ .

Put

$$ \begin{align*}\tilde{\Phi}(\pi)=E[(1+\pi\eta)^{\delta-1}\eta]=\displaystyle\int_{\mathbb{R}_0}(1+\pi\eta)^{\delta-1}\eta\nu(dz).\end{align*} $$

For any $\pi \in [0,\pi _0],$ we have $\tilde {\Phi }(\pi _0)\leq \tilde {\Phi }(\pi )\leq E[\eta ].$

1. (a) If $\phi '(\pi _0)<0$ holds, the optimal strategy $\pi ^*\in (0,\pi _0)$ satisfies $\phi '(\pi ^*)=0.$

   1. (i) If $E[\eta ]<0,$ that is, the average jump size of the risky asset is negative, the financial market is bad, so we have $\tilde {\Phi }(\pi )<0,$ from which we conclude that $\phi '(\pi )<\bar {\phi }'(\pi )$ . Therefore, $\bar {\pi }^*>\pi ^*$ and $\phi (\pi ^*)<\bar {\phi }(\bar {\pi }^*)$ , which lead to the two inequalities shown in Case II, that is, $V(t,x)<\bar {V}(t,x)$ and $c^*>\bar {c}^*.$

   2. (ii) If $\tilde {\Phi }(\pi _0)>0$ , which implies that $\tilde {\Phi }(\pi )>0,$ then $\phi '(\pi )>\bar {\phi }'(\pi ).$ Along the same lines, one can show that the two inequalities in this case are the same as those in Case I, that is, $V(t,x)\geq \bar {V}(t,x)$ and $c^*\leq \bar {c}^*.$

   3. (iii) If $E[\eta ]\geq 0$ and $\tilde {\Phi }(\pi _0)\leq 0,$ it is hard to decide the sign of $\tilde {\Phi }(\pi )$ and hence we cannot tell which one of $\bar {\pi }^*$ and $\pi ^*$ ( $\phi (\pi ^*)$ and $\bar {\phi }\,(\bar {\pi }^*)$ ) is larger. Hence, the relationship between $c^*$ and $\bar {c}^*$ and that of $V(t,x)$ and $\bar {V}(t,x)$ cannot be determined.

2. (b) If $\phi '(\pi _0)\geq 0,$ the optimal strategy $\pi ^*=\pi _0.$

   1. (i) If $E[\eta ]<0,$ similar to Case III(a)(i), one can obtain the result in Case II.

   2. (ii) If $\tilde {\Phi }(\pi _0)>0,$ one can obtain $\phi '(\pi )>\bar {\phi }'(\pi )$ and $\phi '(\bar {\pi }^*)>\bar {\phi }'(\bar {\pi }^*)=0$ . Unfortunately, we still do not know whether $\pi _0$ is larger than $\bar {\pi }^*$ or vice versa, since $\phi '(\pi _0)\geq 0,$ and $\phi (\pi _0)$ may be greater or less than $\bar {\phi }(\bar {\pi }^*)$ .

   3. (iii) If $E[\eta ]\geq 0$ and $\tilde {\Phi }(\pi _0)\leq 0,$

      $$ \begin{align*} \phi'(\pi_0)&= b-r+(\delta-1)\sigma^2\pi_0+\displaystyle\int_{\mathbb{R}_0}(1+\pi_0\eta)^{\delta-1}\eta\nu(dz)\geq0\\ &\Rightarrow~ b-r+(\delta-1)\sigma^2\pi_0\geq0=b-r+(\delta-1)\sigma^2\bar{\pi}^*\\ &\Rightarrow~ \pi_0\leq\bar{\pi}^*. \end{align*} $$
      However, we cannot judge which one of $\phi (\pi _0)$ and $\bar {\phi }(\bar {\pi }^*)$ is larger. Therefore, we cannot determine the relationship between $c^*$ and $\bar {c}^*$ and that of $V(t,x)$ and $\bar {V}(t,x)$ .

The following theorem summarizes the results of the comparison.

Remark 5.3 In this section, we present some comparison results to illustrate the effects of the jump on the optimal results under the single-regime case, that is, $l=1$ . If multiple regimes are considered, the corresponding equation (5.1) will be

$$ \begin{align*} \left\{\begin{array}{@{}ll} k_t(t,e_i)+\dfrac{\Gamma(t)}{1-\delta}k(t,e_i)+ \dfrac{1}{1-\delta}\displaystyle\sum\limits_{j=1}^{l}q_{ij}k^{1-\delta}(t,e_j)k^\delta(t,e_i)+1=0,\\ k(T,e_i)=1,~~i=1,2,\ldots,l. \end{array} \right. \end{align*} $$

For this case, although Proposition 4.6 guarantees its existence and uniqueness, it is very hard to obtain its explicit expression and not easy to analyse its properties as well, for example, the monotonicity with respect to $\Gamma (t)$ . As a result, it is difficult and even impossible to make comparisons for the optimal results. Thus, in order to obtain some comparative results, we only discuss the case with a single state.

6. Some special cases

In this section, we present some special cases of our model and compare the optimal results with those in the literature.

Example 6.1 (without jumps) Suppose that the risky asset’s price process is simply modulated by a geometric Brownian motion. Write $b(t,\alpha (t))=b(\alpha (t))$ , $r(t,\alpha (t))=r(\alpha (t))$ and $\sigma (t,\alpha (t))=\sigma (\alpha (t)).$ Then the optimal strategy in Theorem 4.7 has the form

$$ \begin{align*}(\pi^*,c^*)=\bigg(\dfrac{b(e_i)-r(e_i)}{(1-\delta)\sigma^2(e_i)},k^{-1}(t,e_i)x\bigg)\end{align*} $$

and the value function is given by

$$ \begin{align*}V(t,x,e_i)=e^{-\gamma t}k^{1-\delta}(t,e_i)\frac{x^\delta}{\delta},\end{align*} $$

where $k(t,e_i)$ satisfies the following system:

(6.1) $$ \begin{align} \begin{cases} k_t(t,e_i)+\dfrac{1}{1-\delta}\bigg[-\gamma+\delta r(e_i) + \dfrac{\delta(b(e_i)-r(e_i))^2}{2(1-\delta)\sigma^2(e_i)}\bigg]k(t,e_i) \\ \qquad \qquad +\dfrac{1}{1-\delta}\displaystyle\sum\limits_{j=1}^{l}q_{ij}k^{1-\delta}(t,e_j)k^\delta(t,e_i)+1=0,\\ k(T,e_i)=1. \end{cases} \end{align} $$

These optimal results are the same as those shown by Cajueiro and Yoneyama [Reference Cajueiro and Yoneyama3, Theorem 5.1].

Example 6.2 (without regime switching) Consider the case that Markov regime switching does not come into play, that is, the number of the market states is only one $(l = 1)$ , and that all the parameters $r(t), b(t), \sigma (t), \eta (t)$ are constants. Let $T\rightarrow +\infty $ and $k(t,e_i)=\tilde {k}.$ By putting $\tilde {k}$ into (4.9) and solving the corresponding equation, the optimal consumption strategy given in Theorem 4.7 becomes

$$ \begin{align*}c^*=(C\delta)^{1/(\delta-1)}x,\end{align*} $$

where

$$ \begin{align*}C=\frac{1}{\delta}\bigg(\frac{1-\delta}{\lambda}\bigg)^{1-\delta}\end{align*} $$

with

$$ \begin{align*}\lambda=\gamma-\delta [r+{\pi^*}(b-r)]+\frac{1}{2}{\pi^*}^2\sigma^2 \delta(1-\delta)-\int_{\mathbb{R}_0} [(1+{\pi^*}\eta)^\delta-1]\nu(dz).\end{align*} $$

Also, under the condition that $\phi '(\pi _0)<0$ , the optimal portfolio strategy $\pi ^*$ is the unique positive solution to the following equation:

$$ \begin{align*}b-r+{\pi}\sigma^2(\delta-1)+\displaystyle\int_{\mathbb{R}_0}(1+{\pi}\eta)^{\delta-1}\eta\nu(dz)=0.\end{align*} $$

Moreover, the value function is given by

$$ \begin{align*}V(t,x)=Cx^\delta.\end{align*} $$

With $\pi _0=1,$ these results are in line with those by Framstad et al. [Reference Framstad, Øksendal and Sulem9].

Example 6.3 (without jumps and regime switching) In this example, we assume that there are no jumps and no regime switching, and that the parameters $r(\alpha (t)), b(\alpha (t)), \sigma (\alpha (t))$ are constants. Then one can solve the equation (6.1) and obtain

$$ \begin{align*}k(t)=e^{A(T-t)}+\frac{1}{A}e^{A(T-t)}-\frac{1}{A}\end{align*} $$

with

$$ \begin{align*}A=\frac{1}{1-\delta}\bigg[-\gamma+\delta r+\frac{\delta(b-r)^2}{2(1-\delta)\sigma^2}\bigg].\end{align*} $$

If the following condition holds:

$$ \begin{align*}\gamma>\delta r+\frac{\delta(b-r)^2}{2(1-\delta)\sigma^2},\end{align*} $$

then $A<0.$ When $T\rightarrow +\infty ,$ we have $k(t)\rightarrow - A^{-1}$ . Therefore, we obtain the optimal strategy

$$ \begin{align*}\begin{array}{@{}ll} (\pi^*,c^*)= \bigg(\dfrac{b-r}{(1-\delta)\sigma^2},(\delta v)^{1/(\delta-1)}x\bigg) \end{array}\end{align*} $$

and the value function $V(x)=vx^\delta ,$ where

$$ \begin{align*}v=\frac{1}{\delta(1-\delta)^{\delta-1}}\bigg(\gamma-\delta r-\frac{\delta(b-r)^2}{2(1-\delta)\sigma^2}\bigg)^{\delta-1}.\end{align*} $$

These results are consistent with the optimal results by Merton [Reference Merton23].