2. Model and problem formulation

In this section, we introduce the models and some basic assumptions. We assume that trading in the reinsurance and financial markets is continuous, without taxes or transaction costs, and that all assets are infinitely divisible. Let $(\Omega,\mathcal{F},\mathbb{F}=\{\mathcal{F}_{t}\}_{t\geq0},\mathbb{P})$ be a filtered complete probability space satisfying the usual conditions of completeness and right continuity, where $\mathbb{P}$ is the real-world probability measure and $\mathcal{F}_{t}$ represents the information available until time t. All stochastic processes given in the following are assumed to be adapted on this space.

According to the classical Cramér–Lundberg model, the surplus process of the insurer $U=\{U_{t}\}_{t\geq0}$ adapted to the filtration $\mathbb{F}$ can be described as ${\mathrm{d}}U_{t}=c{\mathrm{d}}t-{\mathrm{d}}\sum_{i=1}^{N_{t}}Y_{i}$ , where $c \gt 0$ is the premium rate. $\{N_{t}\}_{t\geq0}$ is a homogeneous Poisson process with intensity $\lambda \gt 0$ and, for each $t\geq0$ , $N_{t}$ represents the total claims number at interval [0, t]. The claim sizes $\{Y_{i},i=1,2,\ldots\}$ are independent and identically distributed positive random variables following a common distribution $F_{Y}(y)$ with finite first- and second-order moments.

Consider now an insurer using reinsurance to manage its risk exposure. Without reinsurance, the insurer is fully responsible for all losses arising, $Y_{i}$ , $i=1,2,\ldots$ In the presence of reinsurance, a portion of each arising loss $Y_{i}$ will be ceded to the reinsurer. The precise coverage is dictated by the reinsurance policy $l_{t}(y)$ at time $t\geq0$ as a function of the (possible) claim $Y=y$ at that time. Thus, the reinsurer pays the insurer the amount $y-l_{t}(y)$ if there is a claim of size y at time $t\geq0$ . The reinsurance premium is computed according to the mean–variance premium principle, i.e.

(2.1) \begin{equation} (1+\theta)\lambda{\mathbb{E}}(Y-l_{t}(Y)) + \frac{\eta}{2}\lambda{\mathbb{E}}((Y-l_{t}(Y))^{2}),\end{equation}

in which $\theta$ and $\eta$ are the non-negative risk-loading parameters. If $\theta=0$ then (2.1) reduces to the variance premium principle; similarly, if $\eta=0$ then (2.1) reduces to the expected-value premium principle. Thus, the insurer’s surplus process $U_{t}^{l}$ in the presence of reinsurance becomes

\begin{equation*} {\mathrm{d}}U_{t}^{l} = \bigg[c - (1+\theta)\lambda{\mathbb{E}}(Y-l_{t}(Y)) - \frac{\eta}{2}\lambda{\mathbb{E}}((Y-l_{t}(Y))^{2})\bigg]\,\mathrm{d}t-\mathrm{d}\sum_{i=1}^{N_{t}}l_{t}(Y_{i}).\end{equation*}

By [Reference Grandell13], $U_{t}^{l}$ can be approximated by the following diffusion processes $\hat{U}_{t}^{l}$ :

\begin{equation*} {\mathrm{d}}\hat{U}_{t}^{l} = \bigg[\theta\lambda{\mathbb{E}}(l_{t}(Y)) + \eta\lambda\mathbb{E}(Yl_{t}(Y)) - \frac{\eta}{2}\lambda\mathbb{E}\big(l_{t}^{2}(Y)\big) - \delta\bigg]\,\mathrm{d}t + \sqrt{\lambda\mathbb{E}\big(l_{t}^{2}(Y)\big)}\,\mathrm{d}W_{t},\end{equation*}

in which $\delta=(1+\theta)\lambda{\mathbb{E}}(Y)+\frac12\eta\lambda{\mathbb{E}}\big(Y^{2}\big)-c$ and $\{W_{t}\}_{t\geq0}$ is a standard Brownian motion.

Assumption 2.1. The insurer’s premium income rate is greater than the expected value of the claims but less than the premium for full reinsurance, i.e.

\begin{align*}\lambda{\mathbb{E}}(Y) \lt c \lt (1+\theta)\lambda{\mathbb{E}}(Y)+\frac{\eta}{2}\lambda{\mathbb{E}}\big(Y^{2}\big).\end{align*}

Apart from reinsurance, we impose investment and assume that the price process of the risky asset satisfies a standard geometric Brownian motion, ${\mathrm{d}}P_{t}^{1}=P_{t}^{1}[\mu\,{\mathrm{d}}t+\sigma\,{\mathrm{d}}B_{t}]$ , where $\mu({ \gt }0)$ is the appreciation rate, $\sigma({ \gt }0)$ is the volatility rate, and $\{B_{t}\}_{t\geq0}$ is a standard Brownian motion, independent of $\{W_{t}\}_{t\geq0}$ .

In this paper, we suppose that the insurer can invest a non-negative amount in a risk-free asset that earns interest at the constant rate r. If the insurer borrows the money then it pays interest at a higher rate $\beta \gt r$ . Then, a risk-free asset has the dynamics ${\mathrm{d}}P_{t}^{0}=f(P_{t}^{0})\,{\mathrm{d}}t$ , in which the Lipschitz continuous function f is expressed as

\begin{equation*} f(P_{t}^{0}) = \left\{ \begin{array}{l@{\quad}l} rP_{t}^{0} & \mathrm{if}\ P_{t}^{0} \geq 0, \\[3pt] \beta P_{t}^{0} & \mathrm{if}\ P_{t}^{0} \lt 0. \end{array} \right.\end{equation*}

Here, we assume that $\mu \gt \beta \gt r \gt 0$ . The case of $\beta \gt \mu \gt r \gt 0$ can be reduced to the special case with no-borrowing.

Denote by $\pi_{t}$ the fraction of the wealth invested in the risky asset by the insurer. We can see that $0\leq\pi_{t}\leq1$ means investment without short-selling and borrowing; if $\pi_{t} \gt 1$ then it means that the insurer has to borrow money from the market and invests all the wealth in the risky asset. We define the control strategy of the insurer as $u_{t}=(\pi_{t},l_{t})$ . Thus, the wealth process $X^{u}_{t}$ of the insurer is described as

(2.2) \begin{equation}\left\{\begin{array}{l}\displaystyle{\mathrm{d}}X_{t}^{u} = \big\{f\big((1-\pi_{t})X_{t}^{u}\big) - \delta + \pi_{t}X_{t}^{u}\mu + \theta\lambda{\mathbb{E}}(l_{t}(Y)) + \eta\lambda{\mathbb{E}}(Yl_{t}(Y)) - \frac12\eta\lambda{\mathbb{E}}\big(l_{t}^{2}(Y)\big)\big\}\,\mathrm{d}t \\[3pt]\quad \qquad + \sigma X_{t}^{u}\pi_{t}{\mathrm{d}}B_{t} + \sqrt{\lambda{\mathbb{E}}\big(l_{t}^{2}(Y)\big)}\,{\mathrm{d}}W_{t}, \\[3pt]X^{u}_{0}=x,\end{array}\right.\end{equation}

in which

\begin{equation*} f\big((1-\pi_{t})X_{t}^{u}\big) = \left\{ \begin{array}{l@{\quad}l} r(1-\pi_{t})X_{t}^{u} & \mathrm{if}\ \pi_{t}\geq1, \\[4pt] \beta(1-\pi_{t})X_{t}^{u} & \mathrm{if}\ \pi_{t} \lt 1, \end{array} \right.\end{equation*}

and $x\ ({ \gt }0)$ is the initial wealth of the insurer.

Let $\tau_{M}^{u}\,:\!=\,\inf\{t\mid t\geq0,X_{t}^{u}\leq M\}$ and $\tau_{N}^{u}\,:\!=\,\inf\{t\mid t\geq0,X_{t}^{u}\geq N\}$ be the first times the insurer’s wealth respectively hits a specified ruin level M and a high goal N under the control policy u. The insurer aims at minimizing the probability that ruin happens before the high goal N, i.e. $\tau_{M}^{u} \lt \tau_{N}^{u}$ , in a robust sense. More precisely, they suspect that the drifts of the risky asset and the surplus may be misspecified. Define $\tau^{u}=\tau_{M}^{u}\wedge\tau_{N}^{u}$ as the first exit time of the interval (M, N), and note that $\tau^{u}=\infty$ if $\tau_{M}^{u}=\tau_{N}^{u}=\infty$ . So instead of optimizing under the reference measure $\mathbb{P}$ , they consider a set $\mathscr{Q}$ of candidate measures that are locally equivalent to $\mathbb{P}$ , i.e. $\mathscr{Q}\,:\!=\,\{\mathbb{Q}\mid\mathbb{Q}\thicksim\mathbb{P}\}$ , and penalizes their deviation from $\mathbb{P}$ . To give the precise definition of the set $\mathscr{Q}$ of candidate measures, we first define $\phi_{t}=(\phi_{1t},\phi_{2t})$ , $t\geq0$ , as a so-called probability distortion process and let $\Phi$ denote the collection of $\phi$ satisfying the Novikov condition ${\mathbb{E}}^{\mathbb{P}}\big[\exp\big\{\int_0^{t}\frac{1}{2}\big(\phi_{1s}^{2}+\phi_{2s}^{2}\big)\,{\mathrm{d}}s\big\}\big] \lt \infty$ . So, for each $\phi_{t}\in\Phi$ , a probability measure $\mathbb{Q}\in\mathscr{Q}$ if

\begin{align*}\frac{{\mathrm{d}}\mathbb{Q}}{{\mathrm{d}}\mathbb{P}}\bigg|_{\mathcal{F}_{t}} =\exp\bigg\{\int_{0}^{t}\phi_{1s}\,{\mathrm{d}}B_{s} + \int_{0}^{t}\phi_{2s}\,{\mathrm{d}}W_{s} -\frac{1}{2}\int_{0}^{t}\big(\phi_{1s}^{2}+\phi_{2s}^{2}\big)\,{\mathrm{d}}s\bigg\}.\end{align*}

By Girsanov’s theorem, the Brownian motions under $\mathbb{Q}\in\mathscr{Q}$ are presented as

\begin{equation*} \left\{ \begin{array}{l} {\mathrm{d}}B_{t}^{\mathbb{Q}} = {\mathrm{d}}B_{t} - \phi_{1t}\,{\mathrm{d}}t, \\[3pt] {\mathrm{d}}W_{t}^{\mathbb{Q}} = {\mathrm{d}}W_{t} - \phi_{2t}\,{\mathrm{d}}t, \end{array} \right.\end{equation*}

where $\{B_{t}^{\mathbb{Q}}\}_{t\geq0}$ and $\{W_{t}^{\mathbb{Q}}\}_{t\geq0}$ are two $\mathbb{Q}$ -Brownian motions and are mutually independent. Therefore, we rewrite the wealth process $X_{t}^{u}$ under the probability measure $\mathbb{Q}$ as follows:

(2.3) \begin{equation}\begin{cases}{\mathrm{d}}X_{t}^{u} = \big\{f\big((1-\pi_{t})X_{t}^{u}\big) - \delta + \pi_{t}X_{t}^{u}\mu + \theta\lambda{\mathbb{E}}(l_{t}(Y)) + \eta\lambda{\mathbb{E}}(Yl_{t}(Y)) - \frac12\eta\lambda{\mathbb{E}}\big(l_{t}^{2}(Y)\big) \\[3pt]\quad \qquad + \sigma X_{t}^{u}\pi_{t}\phi_{1t} + \sqrt{\lambda{\mathbb{E}}\big(l_{t}^{2}(Y)\big)}\phi_{2t}\big\}\,\mathrm{d}t + \sigma X_{t}^{u}\pi_{t}{\mathrm{d}}B_{t}^{\mathbb{Q}} + \sqrt{\lambda{\mathbb{E}}\big(l_{t}^{2}(Y)\big)}\,{\mathrm{d}}W_{t}^{\mathbb{Q}}, \\[3pt]X^{u}_{0}=x.\end{cases}\end{equation}

The relative entropy between $\mathbb{Q}$ and $\mathbb{P}$ up to time t is given by

\begin{align*}{\mathbb{E}}^{\mathbb{Q}}\bigg(\ln\frac{{\mathrm{d}}\mathbb{Q}_{t}}{{\mathrm{d}}\mathbb{P}_{t}}\bigg) ={\mathbb{E}}^{\mathbb{Q}}\bigg(\frac{1}{2}\int_{0}^{t}\big(\phi_{1s}^{2}+\phi_{2s}^{2}\big)\,{\mathrm{d}}s\bigg), \quad t \lt \infty,\end{align*}

where $\mathbb{Q}_{t}$ is the probability measure $\mathbb{Q}$ restricted to $\mathcal{F}_{t}$ and $\mathbb{P}_{t}$ is the probability measure $\mathbb{P}$ restricted to $\mathcal{F}_{t}$ . We define a performance function for any $u\in\mathcal{U}$ and $\phi\in\Phi$ as

\begin{equation*} J^{u,\phi}(x) \,:\!=\, \mathbb{Q}^{x}\big(\tau_{M}^{u} \lt \tau_{N}^{u},\tau^{u} \lt \infty\big) - \frac{1}{\epsilon}\mathbb{E}^{\mathbb{Q}}\bigg(\frac{1}{2}\int_{0}^{\tau^{u}}\big(\phi_{1s}^{2}+\phi_{2s}^{2}\big) \,{\mathrm{d}}s\bigg),\end{equation*}

where $\mathbb{Q}^{x}(\cdot)=\mathbb{Q}(\cdot\mid X_{0}^{u}=x)$ and $\epsilon$ is the ambiguity-aversion coefficient for the insurer. The robust value function is then defined as

(2.4) \begin{equation} V(x)\,:\!=\,\inf_{u\in\mathcal{U}}\sup_{\phi\in\Phi}J^{u,\phi}(x).\end{equation}

Note that if the value of the wealth is greater than or equal to $x^{\ast}={\delta}/{{\beta}}$ , then the insurer can buy full reinsurance via income from the risk-free asset, and therefore the wealth will never drop below its current value. For this reason, we call $x^{\ast}$ the safe level. We generalize from this case in the following remark.

Remark 2.1. As the wealth increases towards ${\delta}/{{\beta}}$ , the optimal investment–reinsurance strategy approaches $u_{0}=(0,0)$ . It makes sense because when the value of the wealth increases, the insurer invests only in the risk-free asset and transfers all the risk to the reinsurer; from (2.3), the wealth process becomes an ordinary differential equation $\mathrm{d}X^{u_{0}}_{t}=\big(rX^{u_{0}}_{t}-\delta\big)\,\mathrm{d}t$ , and thus the wealth will never decrease, so ruin cannot happen. Indeed, on one hand, we have

\begin{align*} V(x) = \inf_{u\in\mathcal{U}}\sup_{\phi\in\Phi}J^{u,\phi}(x) \leq \sup_{\phi\in\Phi}J^{u_{0},\phi}(x) = \sup_{\phi\in\Phi}\bigg\{0-\frac{1}{\epsilon}{\mathbb{E}}^{\mathbb{Q}} \bigg(\frac{1}{2}\int_{0}^{\tau^{u_{0}}}\big(\phi_{1s}^{2}+\phi_{2s}^{2}\big)\,{\mathrm{d}}s\bigg)\bigg\} \leq 0. \end{align*}

On the other hand, note that $\mathbb{P}\in\mathscr{Q}$ with $\phi_{0}\equiv(0,0)$ , so

\begin{align*} V(x) = \inf_{u\in\mathcal{U}}\sup_{\phi\in\Phi}J^{u,\phi}(x) \geq \inf_{u\in\mathcal{U}}J^{u,\phi_{0}}(x) = \inf_{u\in\mathcal{U}}\mathbb{P}^{x}\big(\tau_{M}^{u} \lt \tau_{N}^{u},\tau^{u} \lt \infty\big)\geq0. \end{align*}

Thus, we have $V(x)\equiv0$ for any $x\geq{\delta}/{{\beta}}$ . As a consequence, we make the following assumption.

Let $C^{2}[M, N]$ be the space of any function F(x) such that F and its derivatives $F_{x}$ , $F_{xx}$ are continuous on [M, N]. To solve the above robust problem, we use the dynamic programming approach described in [Reference Fleming and Soner12]. From standard arguments, we see that if $V\in C^{2}$ then V satisfies the following HJB equation for $x\in[M, N]$ :

(2.5) \begin{equation} \inf_{u\in\mathcal{U}}\sup_{\phi\in\Phi}\{\mathcal{A}^{u,\phi}V(x)\}=0,\end{equation}

in which

\begin{align*} \mathcal{A}^{u,\phi}V(x) & = V_{x}(x)\bigg[f((1-\pi)x) - \delta + \pi x\mu + \theta\lambda{\mathbb{E}}(l(Y)) + \eta\lambda{\mathbb{E}}(Yl(Y)) - \frac{\eta}{2}\lambda{\mathbb{E}}\big(l^{2}(Y)\big) \\[5pt] & \quad + \sigma x\pi\phi_{1} + \sqrt{\lambda{\mathbb{E}}\big(l^{2}(Y)\big)}\phi_{2}\bigg] + \frac{1}{2}V_{xx}(x)\big[\sigma^{2}x^{2}\pi^{2} + \lambda{\mathbb{E}}\big(l^{2}(Y)\big)\big] - \frac{1}{2\epsilon}\big(\phi_{1}^{2}+\phi_{2}^{2}\big),\end{align*}

with the boundary conditions

(2.6) \begin{eqnarray} V(M)=1, \qquad V(N)=0.\end{eqnarray}

Obviously, the optimal control strategy u obtained through the dynamic programming approach is time-independent.

3. Explicit solution for the problem

In this section, we aim to derive the robust optimal reinsurance–investment strategy for the problem in (2.4). The HJB equation in (2.5) becomes

(3.1) \begin{align} \inf_{u\in\mathcal{U}}\sup_{\phi\in\Phi}\bigg\{ & \bigg[f((1-\pi)x) - \delta + \pi x\mu + \theta\lambda{\mathbb{E}}(l(Y)) + \eta\lambda{\mathbb{E}}(Yl(Y)) - \frac{\eta}{2}\lambda{\mathbb{E}}\big(l^{2}(Y)\big) + \sigma x\pi\phi_{1} \nonumber \\ & \ + \sqrt{\lambda{\mathbb{E}}\big(l^{2}(Y)\big)}\phi_{2}\bigg]V_{x}(x) + \frac{1}{2}\big[\sigma^{2}x^{2}\pi^{2} + \lambda{\mathbb{E}}\big(l^{2}(Y)\big)\big]V_{xx}(x) - \frac{1}{2\epsilon}\big(\phi_{1}^{2}+\phi_{2}^{2}\big)\bigg\} = 0.\end{align}

The first-order conditions of $\phi_{1}$ and $\phi_{2}$ yield

(3.2) \begin{equation} \left\{ \begin{array}{l} \hat{\phi}_{1}(x,\pi)=\epsilon\sigma x\pi V_{x}(x), \\[3pt] \hat{\phi}_{2}(x,l)=\epsilon\sqrt{\lambda{\mathbb{E}}\big(l^{2}(Y)\big)}V_{x}(x). \end{array} \right.\end{equation}

Plugging $\hat{\phi}_{1}$ and $\hat{\phi}_{2}$ into (3.1), we have

(3.3) \begin{align} \inf_{u\in\mathcal{U}}\bigg\{ & [f((1-\pi)x) - \delta + \pi x\mu + \theta\lambda{\mathbb{E}}(l(Y)) + \eta\lambda{\mathbb{E}}(Yl(Y))]V_{x}(x) \nonumber \\ & + \frac{1}{2}\sigma^{2}x^{2}\pi^{2}\big[V_{xx}(x) + \epsilon V_{x}^{2}(x)\big] + \frac{1}{2}\lambda{\mathbb{E}}\big(l^{2}(Y)\big)\big[V_{xx}(x) + \epsilon V_{x}^{2}(x) - \eta V_{x}(x)\big]\bigg\} = 0.\end{align}

By using the cumulative distribution function of Y, we define $g(x,l,\pi)$ as

(3.4) \begin{align} g(x,l,\pi) & = [f((1-\pi)x)-\delta+\pi x\mu]V_{x}(x) + \frac{1}{2}\sigma^{2}x^{2}\pi^{2}\big[V_{xx}(x) + \epsilon V_{x}^{2}(x)\big] \nonumber \\ & \quad + \int_{0}^{\infty}\big\{(\theta\lambda l(y) + \eta\lambda y l(y))V_{x}(x) + \frac{1}{2}\lambda l^{2}(y)\big[V_{xx}(x) + \epsilon V_{x}^{2}(x) - \eta V_{x}(x)\big]\big\}\,\mathrm{d}F_{Y}(y).\end{align}

According to the first-order optimality conditions, the minimizers of the function $g(x,l,\pi)$ are obtained at

(3.5) \begin{equation} \left\{ \begin{array}{l} \hat{l}(x,y)=\dfrac{\theta+\eta y}{\xi(x)}\wedge y, \\[12pt] \hat{\pi}^{r}(x)=-\dfrac{\mu-r}{\sigma^{2}x}\dfrac{1}{\eta-\xi(x)}, \\[14pt] \hat{\pi}^{\beta}(x)=-\dfrac{\mu-\beta}{\sigma^{2}x}\dfrac{1}{\eta-\xi(x)}, \end{array} \right.\end{equation}

where $\xi(x)=\eta-\epsilon V_{x}(x)-({V_{xx}(x)}/{V_{x}(x)})$ . Equation (3.5) holds if $\xi(x)\neq\eta$ , and $\xi(x)$ is well-defined if $V_{x}(x)\neq 0$ . These will be proved in the following lemma.

From Lemma 3.1, it immediately follows that $\hat{\pi}^{r}(x) \gt \hat{\pi}^{\beta}(x) \gt 0$ . Due to the constraint on the investment strategy, we define the following regions:

\begin{equation*} \left\{ \begin{array}{l} \Gamma_{1}\,:\!=\,\{x\in[M, N]\mid\hat{\pi}^{r}(x) \lt 1\}, \\[2pt] \Gamma_{2}\,:\!=\,\{x\in[M, N]\mid\hat{\pi}^{\beta}(x)\geq1+m\}, \\[2pt] \Gamma_{3}\,:\!=\,\{x\in[M, N]\mid1 \lt \hat{\pi}^{\beta}(x) \lt 1+m\}, \\[2pt] \Gamma_{4}\,:\!=\,\{x\in[M, N]\mid\hat{\pi}^{\beta}(x)\leq1\leq\hat{\pi}^{r}(x)\}. \end{array} \right.\end{equation*}

Based on the above analysis, we have the following cases to deal with. To simplify our analysis, we define the functions

(3.6) \begin{align} f_{1}(\xi) = {\mathbb{E}}(\hat{l}(Y)) & = \int_{0}^{{\theta}/({\xi-\eta})}\bar{F}_{Y}(y)\,\mathrm{d}y + \frac{\eta}{\xi}\int_{{\theta}/({\xi-\eta})}^{\infty}\bar{F}_{Y}(y)\,\mathrm{d}y,\qquad\qquad\end{align}

(3.7) \begin{align} f_{2}(\xi) = {\mathbb{E}}(Y\hat{l}(Y)) & = 2\int_{0}^{{\theta}/({\xi-\eta})}y\bar{F}_{Y}(y)\,\mathrm{d}y + \frac{1}{\xi}\int_{{\theta}/({\xi-\eta})}^{\infty}(\theta+2\eta y)\bar{F}_{Y}(y)\,\mathrm{d}y,\end{align}

(3.8) \begin{align} f_{3}(\xi) = {\mathbb{E}}(\hat{l}^{2}(Y)) & = 2\int_{0}^{{\theta}/({\xi-\eta})}y\bar{F}_{Y}(y)\,\mathrm{d}y + \frac{2\eta}{\xi^{2}}\int_{{\theta}/({\xi-\eta})}^{\infty}(\theta+\eta y)\bar{F}_{Y}(y)\,\mathrm{d}y, \end{align}

where $\bar{F}_{Y}(y)=1-F_{Y}(y)$ .

In order to derive robust optimal reinsurance and investment strategies, we first prove the following four lemmas, which are in one-to-one correspondence with the above four regions.

Lemma 3.2. For region $\Gamma_{1}$ , let $\xi_{1}^{{\ast}} \gt \eta$ be the unique solution of

(3.9) \begin{align} \int_{0}^{{\theta}/({\xi-\eta})}[1+(\xi-\eta)y]\bar{F}_{Y}(y)\,\mathrm{d}y & + \int_{{\theta}/({\xi-\eta})}^{+\infty}\bigg[1+\frac{\xi-\eta}{\xi}(\theta+\eta y)\bigg] \bar{F}_{Y}(y)\,\mathrm{d}y \nonumber \\ & = \frac1\lambda\bigg\{rx+c-\frac{(\mu-r)^{2}}{2\sigma^{2}(\eta-\xi)}\bigg\}. \end{align}

Then the optimal investment strategy and retention level are

\begin{equation*} (\pi^{\ast}(x),l^{\ast}(x,y)) = \bigg({-}\frac{\mu-r}{\sigma^{2}x}\frac{1}{\eta-\xi_{1}^{\ast}(x)}, \frac{\theta+\eta y}{\xi_{1}^{\ast}(x)}\wedge y\bigg), \end{equation*}

and the optimal drift distortion is

\begin{equation*} \big(\phi_{1}^{\ast}(x),\phi_{2}^{\ast}(x)\big) = \Bigg({-}\frac{\mu-r}{\sigma}\frac{\epsilon V_{x}(x)}{\eta-\xi_{1}^{\ast}(x)}, \epsilon\sqrt{\lambda{\mathbb{E}}\bigg(\bigg( \frac{\theta+\eta Y}{\xi_{1}^{\ast}(x)}\wedge Y\bigg)^{2}\bigg)}V_{x}(x)\Bigg). \end{equation*}

The robust value function is equivalent to

(3.10) \begin{equation} V(x) = \frac{1}{\epsilon}\ln\Bigg[{\mathrm{e}}^{\epsilon} + (1-{\mathrm{e}}^{\epsilon})\frac{\int_{M}^{x}\exp\big\{{-}\int_{M}^{y}(\xi_{1}^{\ast}(z)-\eta)\,\mathrm{d}z\big\}\,\mathrm{d}y} {\int_{M}^{N}\exp\big\{{-}\int_{M}^{y}(\xi_{1}^{\ast}(z)-\eta)\,\mathrm{d}z\big\}\,\mathrm{d}y}\Bigg]. \end{equation}

Moreover, we deduce that $\Gamma_{1}=\big(x_{1}^{r},+\infty\big)\cap[M, N]$ , where $x_{1}^{r}$ satisfies

(3.11) \begin{align} \xi_{1}^{\ast}(x)=\eta+\frac{\mu-r}{x\sigma^{2}}. \end{align}

Proof. Note that in this case

\begin{align*}\xi(x)=\eta-\epsilon V_{x}(x)-\frac{V_{xx}(x)}{V_{x}(x)} \gt \eta.\end{align*}

For $x\in\Gamma_{1}$ , the minimum point of the left-hand side of (3.3) is attained at $(\pi^{\ast}(x),\,l^{\ast}(x,y))=(\hat{\pi}^{r}(x),\hat{l}(x,y))$ , in which $\hat{\pi}^{r}(x)$ and $\hat{l}(x,y)$ are defined in (3.5). Plugging these into (3.2) and (3.4), we have

\begin{equation*} \big(\phi_{1}^{\ast}(x),\phi_{2}^{\ast}(x)\big) = \Bigg({-}\frac{\mu-r}{\sigma}\frac{\epsilon V_{x}(x)}{\eta-\xi(x)}, \epsilon\sqrt{\lambda{\mathbb{E}}\bigg(\bigg( \frac{\theta+\eta Y}{\xi(x)}\wedge Y\bigg)^{2}\bigg)}V_{x}(x)\Bigg), \end{equation*}

and $g(x,\xi)=V_{x}(x)h_{1}(x,\xi)$ , where

\begin{equation*} h_{1}(x,\xi) = rx - \delta + \theta\lambda f_{1}(\xi) + \eta\lambda f_{2}(\xi) - \frac{\xi}{2}\lambda f_{3}(\xi) - \frac{(\mu-r)^{2}}{2\sigma^{2}}\frac{1}{\eta-\xi}, \end{equation*}

and we slightly abuse the notation of g by replacing its arguments $(\pi,l)$ with $\xi$ . Obviously, $g(x,\xi)=0$ is equal to $h_{1}(x,\xi)=0$ . Next, we wish to show that $h_{1}(x,\xi)=0$ has a unique solution at $\xi \gt \eta$ . To this end, we can obtain the following result from (3.6)–(3.8):

\begin{align*} \lim_{\xi\rightarrow\eta^{+}}rx - \delta + \theta\lambda f_{1}(\xi) + \eta\lambda f_{2}(\xi) - \frac{\xi}{2}\lambda f_{3}(\xi) = rx + c - \lambda E(Y). \end{align*}

Note that

\begin{align*}\lim_{\xi\rightarrow\eta^{+}}-\frac{(\mu-r)^{2}}{2\sigma^{2}}\frac{1}{\eta-\xi}=+\infty.\end{align*}

Thus, based on the above two conditions, we obtain $\lim_{\xi\rightarrow\eta^{+}}h_{1}(x,\xi)=+\infty$ . Also, we have $\lim_{\xi\rightarrow\infty}h_{1}(x,\xi)=rx-\delta \lt 0$ . By differentiating $h_{1}(x,\xi)$ with respect to $\xi$ , we obtain

\begin{align*} \frac{\partial h_{1}(x,\xi)}{\partial\xi} = -\frac{1}{2}\lambda f_{3}(\xi) - \frac{(\mu-r)^{2}}{2\sigma^{2}}\frac{1}{(\xi-\eta)^{2}} \lt 0, \end{align*}

and then $h_{1}(x,\xi)$ is a strictly decreasing function in $\xi$ . Thus, according to the analysis above, it follows that $h_{1}(x,\xi)$ has a unique zero at $\xi_{1}^{{\ast}} \gt \eta$ , i.e. $h_{1}(x,\xi_{1}^{{\ast}})=0$ . Since $\xi_{1}^{\ast}$ is a function of x, we take derivatives with respect to x and, simplifying the expression, we obtain

\begin{align*} \frac{r}{(\xi_{1}^{\ast}(x))^{\prime}} = \frac{1}{2}\lambda f_{3}(\xi_{1}^{\ast}(x)) + \frac{(\mu-r)^{2}}{2\sigma^{2}}\frac{1}{(\xi_{1}^{\ast}(x)-\eta)^{2}} \gt 0;\end{align*}

therefore, $(\xi_{1}^{\ast}(x))^{\prime} \gt 0$ . There is no doubt that $\xi_{1}^{\ast}(x)$ is strictly increasing with respect to x. $\hat{\pi}^{r}(x) \lt 1$ implies $x \gt x_{1}^{r}$ , where $x_{1}^{r}$ satisfies $\xi_{1}^{\ast}(x)=\eta+(({\mu-r})/{x\sigma^{2}})$ . Therefore, we have $\Gamma_{1}=[x_{1}^{r},+\infty)\cap[M, N]$ .

Under the optimal strategy $(\pi^{\ast}(x),l^{\ast}(x,y),\phi_{1}^{\ast}(x),\phi_{2}^{\ast}(x))$ , the corresponding HJB equation becomes

(3.12) \begin{equation} (\xi_{1}^{\ast}(x)-\eta)V_{x}(x)+\epsilon V_{x}^{2}(x)+V_{xx}(x)=0. \end{equation}

To find the solution of the above equation, we make the transformation $G(x)={\mathrm{e}}^{\epsilon V(x)}$ . Thus, we have

\begin{equation*} V(x) = \frac{\ln G(x)}{\epsilon}, \qquad V_{x}(x) = \frac{G_{x}(x)}{\epsilon G(x)}, \qquad V_{xx}(x) = \frac{G_{xx}(x)G(x)-(G_{x}(x))^{2}}{\epsilon G^{2}(x)}. \end{equation*}

Substituting the above equations into (3.12) and applying the boundary conditions in (2.6), we have

(3.13) \begin{equation} \frac{G_{xx}(x)}{G_{x}(x)}=-(\xi_{1}^{\ast}(x)-\eta), \end{equation}

(3.14) \begin{equation} G(M)={\mathrm{e}}^{\epsilon}, \qquad G(N)=1. \end{equation}

Notice that $G_{x}(x)=\epsilon V_{x}(x)G(x) \lt 0$ , and (3.13) implies $G_{xx}(x) \gt 0$ . Solving (3.13) under the boundary conditions (3.14), we obtain

\begin{equation*} G(x) = {\mathrm{e}}^{\epsilon} + (1-{\mathrm{e}}^{\epsilon}) \frac{\int_{M}^{x}\exp\big\{{-}\int_{M}^{y}(\xi_{1}^{\ast}(z)-\eta)\,\mathrm{d}z\big\}\,\mathrm{d}y} {\int_{M}^{N}\exp\big\{{-}\int_{M}^{y}(\xi_{1}^{\ast}(z)-\eta)\,\mathrm{d}z\big\}\,\mathrm{d}y}. \end{equation*}

Consequently, we can obtain the expression for V(x) in (3.10).

Lemma 3.3. For $x\in\Gamma_{2}$ , which is characterized in Remark 3.1, $\xi_{2}^{\ast}(x)$ uniquely solves

(3.15) \begin{align} &\int_{0}^{{\theta}/({\xi-\eta})}[1+(\xi-\eta)y]\bar{F}_{Y}(y)\,\mathrm{d}y + \int_{{\theta}/({\xi-\eta})}^{+\infty}\bigg[1+\frac{\xi-\eta}{\xi}(\theta+\eta y)\bigg] \bar{F}_{Y}(y)\,\mathrm{d}y \nonumber \\ & = \frac1\lambda\bigg\{c+\beta x+(\mu-\beta)x(1+m)+\frac{\sigma^{2}x^{2}(1+m)^{2}(\eta-\xi)}{2}\bigg\}. \end{align}

The corresponding optimal investment strategy and retention level are given by

\begin{equation*} (\pi^{\ast}(x),l^{\ast}(x,y))=\bigg(1+m,\frac{\theta+\eta y}{\xi_{2}^{\ast}(x)}\wedge y\bigg), \end{equation*}

and the optimal drift distortion is

\begin{equation*} \big(\phi_{1}^{\ast}(x),\phi_{2}^{\ast}(x)\big) = \Bigg(\epsilon\sigma x(1+m)V_{x}(x), \epsilon\sqrt{\lambda{\mathbb{E}}\bigg(\bigg( \frac{\theta+\eta Y}{\xi_{2}^{\ast}(x)}\wedge Y\bigg)^{2}\bigg)}V_{x}(x)\Bigg). \end{equation*}

Moreover, the robust value function V(x) is given in (3.10) with $\xi_{1}^{\ast}(x)$ replaced by $\xi_{2}^{\ast}(x)$ .

Proof. For $x\in\Gamma_{2}$ , we have $(\pi^{\ast}(x),l^{\ast}(x,y))=(1+m,\,\hat{l}(x,y))$ and

\begin{align*} \big(\phi_{1}^{\ast}(x),\phi_{2}^{\ast}(x)\big) = \Bigg(\epsilon\sigma x(1+m)V_{x}(x), \epsilon\sqrt{\lambda{\mathbb{E}}\bigg(\bigg(\frac{\theta+\eta Y}{\xi(x)}\wedge Y\bigg)^{2}\bigg)}V_{x}(x)\Bigg). \end{align*}

Following the same steps as in the proof of Lemma 3.2, we can obtain the results.

Lemma 3.4. For region $\Gamma_{3}$ , $\xi_{3}^{{\ast}} \gt \eta$ is the unique solution of

(3.16) \begin{align} \int_{0}^{{\theta}/({\xi-\eta})}[1+(\xi-\eta)y]\bar{F}_{Y}(y)\,\mathrm{d}y & + \int_{{\theta}/({\xi-\eta})}^{+\infty}\bigg[1+\frac{\xi-\eta}{\xi}(\theta+\eta y)\bigg] \bar{F}_{Y}(y)\,\mathrm{d}y \nonumber \\ & = \frac1\lambda\bigg\{\beta x+c-\frac{(\mu-\beta)^{2}}{2\sigma^{2}(\eta-\xi)}\bigg\}. \end{align}

Then, the optimal investment strategy and retention level are

\begin{equation*} (\pi^{\ast}(x),l^{\ast}(x,y)) = \bigg({-}\frac{\mu-\beta}{\sigma^{2}x}\frac{1}{\eta-\xi_{3}^{\ast}(x)}, \frac{\theta+\eta y}{\xi_{3}^{\ast}(x)}\wedge y\bigg), \end{equation*}

the optimal drift distortion is

\begin{equation*} \big(\phi_{1}^{\ast}(x),\phi_{2}^{\ast}(x)\big) = \Bigg({-}\frac{\mu-\beta}{\sigma} \frac{\epsilon V_{x}(x)}{\eta-\xi_{3}^{\ast}(x)}, \epsilon\sqrt{\lambda{\mathbb{E}}\bigg(\bigg(\frac{\theta+\eta Y}{\xi_{3}^{\ast}(x)}\wedge Y\bigg)^{2}\bigg)} V_{x}(x)\Bigg), \end{equation*}

and the robust value function V(x) is given in (3.10) with $\xi_{3}^{\ast}(x)$ replaced by $\xi_{1}^{\ast}(x)$ . Moreover, we deduce that $\Gamma_{3}=\big(x_{2}^{\beta},x_{3}^{\beta}\big)\cap[M, N]$ , where $x_{2}^{\beta}$ and $x_{3}^{\beta}$ satisfy the following equations respectively:

(3.17) \begin{equation} \left\{ \begin{array}{l} \xi_{3}^{\ast}(x)=\eta+\dfrac{\mu-\beta}{x\sigma^{2}(1+m)}, \\[14pt] \xi_{3}^{\ast}(x)=\eta+\dfrac{\mu-\beta}{x\sigma^{2}}. \end{array} \right. \end{equation}

Proof. For $x\in\Gamma_{3}$ , we have $(\pi^{\ast}(x),l^{\ast}(x,y))=(\hat{\pi}^{\beta}(x),\hat{l}(x,y))$ and

\begin{align*} \big(\phi_{1}^{\ast}(x),\phi_{2}^{\ast}(x)\big) = \Bigg({-}\frac{\mu-\beta}{\sigma} \frac{\epsilon V_{x}(x)}{\eta-\xi(x)}, \epsilon\sqrt{\lambda{\mathbb{E}}\bigg(\bigg(\frac{\theta+\eta Y}{\xi(x)}\wedge Y\bigg)^{2}\bigg)}V_{x}(x)\Bigg). \end{align*}

Again, the results can be derived in the same way as in the proof of Lemma 3.2.

Lemma 3.5. For $x\in\Gamma_{4}$ , which is characterized in Remark 3.1, $\xi_{4}^{\ast}(x) \gt \eta$ uniquely solves

(3.18) \begin{align} \int_{0}^{{\theta}/({\xi-\eta})}[1+(\xi-\eta)y]\bar{F}_{Y}(y)\,\mathrm{d}y & + \int_{{\theta}/({\xi-\eta})}^{+\infty}\bigg[1+\frac{\xi-\eta}{\xi}(\theta+\eta y)\bigg] \bar{F}_{Y}(y)\,\mathrm{d}y \nonumber \\ & = \frac1\lambda\bigg\{\mu x+c+\frac{\sigma^{2}x^{2}(\eta-\xi)}{2}\bigg\}. \end{align}

The corresponding optimal investment strategy and retention level are given by

\begin{equation*} (\pi^{\ast}(x),l^{\ast}(x,y))=\bigg(1,\frac{\theta+\eta y}{\xi_{4}^{\ast}(x)}\wedge y\bigg), \end{equation*}

and the optimal drift distortion is

\begin{equation*} \big(\phi_{1}^{\ast}(x),\phi_{2}^{\ast}(x)\big) = \Bigg(\epsilon\sigma xV_{x}(x), \epsilon\sqrt{\lambda{\mathbb{E}}\bigg(\bigg(\frac{\theta+\eta Y}{\xi_{4}^{\ast}(x)}\wedge Y\bigg)^{2}\bigg)} V_{x}(x)\Bigg). \end{equation*}

Moreover, the robust value function V(x) is given in (3.10) with $\xi_{1}^{\ast}(x)$ replaced by $\xi_{4}^{\ast}(x)$ .

Proof. For $x\in\Gamma_{4}$ , we have $l^{\ast}(x,y)=\hat{l}(x,y)$ and $\phi^{\ast}(x)=\hat{\phi}(x)=(\hat{\phi}_{1}(x),\hat{\phi}_{2}(x))$ . Substituting these into the HJB equation (2.5), we have $\inf_{\pi\in(0, 1+m]}\mathcal{A}^{\pi,\hat{l},\hat{\phi}}V(x)=0$ . Referring to the idea in [Reference Luo18, Lemma 3.5], we can prove that $\pi^{\ast}(x)=1$ by contradiction. Suppose that the minimum on the left-hand side of the HJB equation is attained at the minimizer $\pi^{\ast}(x) \gt 1$ . Thus, by differentiation we have

\begin{align*}\frac{\partial\mathcal{A}^{\pi,\hat{l},\hat{\phi}}V(x)}{\partial\pi}\bigg|_{\pi=\pi^{\ast}}=0,\end{align*}

and $\pi^{\ast}(x)=\hat{\pi}^{\beta}(x)$ , so $\hat{\pi}^{\beta}(x) \gt 1$ , which contradicts the definition of $\Gamma_{4}$ . We can prove that $\pi^{\ast}(x) \lt 1$ is impossible by using the same method. Hence, the optimal investment strategy is obtained at the point of 1. Then, the results can be derived by the same analysis as in Lemma 3.2.

Based on the results in Lemmas 3.2–3.5, we summarize the optimal strategies in Theorem 3.1.

Theorem 3.1. Let $\xi_{1}^{\ast}(x)$ , $x_{1}^{r}$ , $\xi_{2}^{\ast}(x)$ , $\xi_{3}^{\ast}(x)$ , $x_{2}^{\beta}$ , $x_{3}^{\beta}$ , and $\xi_{4}^{\ast}(x)$ be as defined in (3.9), (3.11), (3.15), (3.16), (3.17), and (3.18). The robust value function solution V(x) is

\begin{equation*} V(x) = \frac{1}{\epsilon}\ln\Bigg[{\mathrm{e}}^{\epsilon} + (1-{\mathrm{e}}^{\epsilon})\frac{\int_{M}^{x}\exp\big\{{-}\int_{M}^{y}(\xi^{\ast}(z)-\eta)\,\mathrm{d}z\big\}\,\mathrm{d}y} {\int_{M}^{N}\exp\big\{{-}\int_{M}^{y}(\xi^{\ast}(z)-\eta)\,\mathrm{d}z\big\}\mathrm{d}y}\Bigg], \end{equation*}

where $\xi^{\ast}(x)$ is defined in following form:

(3.19) \begin{equation} \xi^{\ast}(x) = \left\{ \begin{array}{l@{\quad}l}\xi_{2}^{\ast}(x), \quad & x\in\big(0,x_{2}^{\beta}\big]\cap[M, N], \\[5pt] \xi_{3}^{\ast}(x), & x\in\big(x_{2}^{\beta},x_{3}^{\beta}\big)\cap[M, N], \\[5pt] \xi_{4}^{\ast}(x), & x\in\big[x_{3}^{\beta},x_{1}^{r}\big]\cap[M, N], \\[5pt] \xi_{1}^{\ast}(x), & x\in\big(x_{1}^{r},+\infty\big)\cap[M, N]. \end{array} \right. \end{equation}

The corresponding optimal investment strategy and retention levels are given by

(3.20) \begin{equation} (\pi^{\ast}(x),l^{\ast}(x,y)) = \left\{ \begin{array}{l@{\quad}l} \bigg(1+m,\dfrac{\theta+\eta y}{\xi_{2}^{\ast}(x)}\wedge y\bigg), & x\in\big(0,x_{2}^{\beta}\big]\cap[M, N] \\[4pt] \bigg({-}\dfrac{\mu-\beta}{\sigma^{2}x}\dfrac{1}{\eta-\xi_{3}^{\ast}(x)}, \dfrac{\theta+\eta y}{\xi_{3}^{\ast}(x)}\wedge y\bigg), \quad & x\in\big(x_{2}^{\beta},x_{3}^{\beta}\big)\cap[M, N], \\[4pt] \bigg(1,\dfrac{\theta+\eta y}{\xi_{4}^{\ast}(x)}\wedge y\bigg), & x\in\big[x_{3}^{\beta},x_{1}^{r}\big]\cap[M, N], \\[4pt] \bigg({-}\dfrac{\mu-r}{\sigma^{2}x}\dfrac{1}{\eta-\xi_{1}^{\ast}(x)}, \dfrac{\theta+\eta y}{\xi_{1}^{\ast}(x)}\wedge y\bigg), & x\in\big(x_{1}^{r},+\infty\big)\cap[M, N]. \end{array} \right. \end{equation}

The optimal drift distortions are as follows:

\begin{equation*} \big(\phi_{1}^{\ast}(x),\phi_{2}^{\ast}(x)\big) = \left\{ \begin{array}{l@{\quad}l} \Bigg(\epsilon\sigma x(1+m)V_{x}(x), \epsilon\sqrt{\lambda{\mathbb{E}} \bigg(\bigg(\dfrac{\theta+\eta Y}{\xi_{2}^{\ast}(x)}\wedge Y\bigg)^{2}\bigg)}V_{x}(x)\Bigg), \quad & x\in\big(0,x_{2}^{\beta}\big]\cap[M, N], \\[15pt] \Bigg({-}\dfrac{\mu-\beta}{\sigma}\dfrac{\epsilon V_{x}(x)}{\eta-\xi_{3}^{\ast}(x)}, \epsilon\sqrt{ \lambda{\mathbb{E}}\bigg(\bigg(\dfrac{\theta+\eta Y}{\xi_{3}^{\ast}(x)}\wedge Y\bigg)^{2}\bigg)} V_{x}(x)\Bigg), & x\in\big(x_{2}^{\beta},x_{3}^{\beta}\big)\cap[M, N], \\[15pt] \Bigg(\epsilon\sigma xV_{x}(x), \epsilon\sqrt{\lambda{\mathbb{E}} \bigg(\bigg(\dfrac{\theta+\eta Y}{\xi_{4}^{\ast}(x)}\wedge Y\bigg)^{2}\bigg)}V_{x}(x)\Bigg), & x\in\big[x_{3}^{\beta},x_{1}^{r}\big]\cap[M, N], \\[15pt] \Bigg({-}\dfrac{\mu-r}{\sigma}\dfrac{\epsilon V_{x}(x)}{\eta-\xi_{1}^{\ast}(x)}, \epsilon\sqrt{ \lambda{\mathbb{E}}\bigg(\bigg(\dfrac{\theta+\eta Y}{\xi_{1}^{\ast}(x)}\wedge Y\bigg)^{2}\bigg)} V_{x}(x)\Bigg), & x\in\big(x_{1}^{r},+\infty\big)\cap[M, N]. \end{array} \right. \end{equation*}

Remark 3.5. Note that $\phi_{1}^{\ast}(x)$ and $\phi_{2}^{\ast}(x)$ are finite. Taking $x_{1}^{r}\leq M$ in Theorem 3.1 as an example, indeed, the integrand in the expression for $\int_{M}^{x}\exp\big\{{-}\int_{M}^{y}(\xi_{1}^{\ast}(z)-\eta)\,\mathrm{d}z\big\}\,\mathrm{d}y$ is bounded above by 1 since $\xi_{1}^{\ast}(x) \gt \eta$ ; thus,

\begin{align*}\int_{M}^{N}\exp\bigg\{{-}\int_{M}^{y}(\xi_{1}^{\ast}(z)-\eta)\,\mathrm{d}z\bigg\}\,\mathrm{d}y\leq N-M \lt \infty.\end{align*}

Since $\xi_{1}^{\ast}(x)$ is an increasing function with respect to x, it is not difficult to obtain that $\xi_{1}^{\ast}(M)\leq\xi_{1}^{\ast}(x)\leq\xi_{1}^{\ast}(N)$ for $x\in[M, N]$ . According to the analysis above, it follows that

\begin{equation*} \phi_{1}^{\ast}(x) \leq \frac{\mu-r}{\sigma(\xi_{1}^{\ast}(M)-\eta)} \frac{1-{\mathrm{e}}^{\epsilon}}{(1+{\mathrm{e}}^{\epsilon})(N-M)}, \qquad \phi_{2}^{\ast}(x) \leq \sqrt{\lambda{\mathbb{E}}\big(Y^{2}\big)}\frac{1-{\mathrm{e}}^{\epsilon}}{(1+{\mathrm{e}}^{\epsilon})(N-M)} \end{equation*}

for $0 \lt \epsilon \lt \infty$ . So, we set

\begin{align*} C = \frac{1-{\mathrm{e}}^{\epsilon}}{(1+{\mathrm{e}}^{\epsilon})(N-M)} \max\bigg\{\frac{\mu-r}{\sigma(\xi_{1}^{\ast}(M)-\eta)},\sqrt{\lambda{\mathbb{E}}\big(Y^{2}\big)}\bigg\}, \end{align*}

and then $\phi_{1}^{\ast}(x)\leq C$ and $\phi_{2}^{\ast}(x)\leq C$ . The other cases are obtained similarly

Corollary 3.2. If $\theta=0$ , the robust optimal reinsurance strategy of (3.20) reduces to proportional reinsurance and falls into the interval [0, 1], and then the robust optimal control strategies take the following forms:

\begin{equation*} (\pi^{\ast}(x),l^{\ast}(x,y)) = \left\{ \begin{array}{l@{\quad}l} \bigg(1+m,\dfrac{\eta}{\hat{\xi}_{2}^{\ast}(x)}y\bigg), & x\in\big(0,\hat{x}_{2}^{\beta}\big]\cap[M, N], \\[9pt] \bigg({-}\dfrac{\mu-\beta}{\sigma^{2}x}\dfrac{1}{\eta-\hat{\xi}_{3}^{\ast}(x)}, \dfrac{\eta}{\hat{\xi}_{3}^{\ast}(x)}y\bigg), \quad & x\in\big(\hat{x}_{2}^{\beta},\hat{x}_{3}^{\beta}\big)\cap[M, N], \\[9pt] \bigg(1,\dfrac{\eta}{\hat{\xi}_{4}^{\ast}(x)}y\bigg), & x\in\big[\hat{x}_{3}^{\beta},\hat{x}_{1}^{r}\big]\cap[M, N], \\[9pt] \bigg({-}\dfrac{\mu-r}{\sigma^{2}x}\dfrac{1}{\eta-\hat{\xi}_{1}^{\ast}(x)}, \dfrac{\eta}{\hat{\xi}_{1}^{\ast}(x)}y\bigg), & x\in\big(\hat{x}_{1}^{r},+\infty\big)\cap[M, N], \end{array} \right. \end{equation*}

where $\hat{\xi}_{1}^{\ast}$ , $\hat{\xi}_{2}^{\ast}$ , $\hat{\xi}_{3}^{\ast}$ , and $\hat{\xi}_{4}^{\ast}$ are given by

\begin{equation*} \begin{cases} \hat{\xi}_{1}^{\ast}(x) = \dfrac{(\hat{\delta}-rx)\eta + \frac12{\lambda\eta^{2}{\mathbb{E}}\big(Y^{2}\big)} + ({(\mu-r)^{2}}/{2\sigma^{2}}) + \sqrt{\bigtriangleup_{1}}}{2(\hat{\delta}-rx)}, \\[9pt] \hat{\xi}_{2}^{\ast}(x) = \dfrac{\beta x - \hat{\delta} + (\mu-\beta)x(1+m) + \frac12{\eta\sigma^{2}x^{2}(1+m)^{2}} + \sqrt{\bigtriangleup_{2}}}{\sigma^{2}x^{2}(1+m)^{2}}, \\[9pt] \hat{\xi}_{3}^{\ast}(x) = \dfrac{(\hat{\delta}-\beta x)\eta + \frac12{\lambda\eta^{2}{\mathbb{E}}\big(Y^{2}\big)} + ({(\mu-\beta)^{2}}/{2\sigma^{2}}) + \sqrt{\bigtriangleup_{3}}}{2(\hat{\delta}-\beta x)}, \\[9pt] \hat{\xi}_{4}^{\ast}(x) = \dfrac{-\hat{\delta} + x\mu + \frac12{\sigma^{2}x^{2}\eta} + \sqrt{\bigtriangleup_{4}}}{\sigma^{2}x^{2}}, \end{cases} \end{equation*}

and $\hat{x}_{1}^{r}$ , $\hat{x}_{2}^{\beta}$ , and $\hat{x}_{3}^{\beta}$ are the same forms as in (3.11) and (3.17), in which

\begin{equation*} \begin{cases} \bigtriangleup_{1} = \bigg[\dfrac{\lambda\eta^2\mathbb{E}(Y^2)}{2}+\dfrac{(\mu-r)^2}{2\sigma^2}-(\hat{\delta}-rx)\eta\bigg]^2 + 2(\hat{\delta}-rx)\eta\dfrac{(\mu-r)^{2}}{\sigma^{2}}, \\[13pt] \bigtriangleup_{2} = \big[\beta x-\hat{\delta}+(\mu-\beta)x(1+m)+\frac12{\eta\sigma^{2}x^{2}(1+m)^{2}}\big]^{2} + \sigma^{2}x^{2}(1+m)^{2}\lambda\eta^{2}{\mathbb{E}}\big(Y^{2}\big), \\[13pt] \bigtriangleup_{3} = \bigg[\dfrac{\lambda\eta^{2}{\mathbb{E}}\big(Y^{2}\big)}{2} + \dfrac{(\mu-\beta)^{2}}{2\sigma^{2}} - (\hat{\delta}-\beta x)\eta\bigg]^{2}+2(\hat{\delta}-\beta x)\eta\dfrac{(\mu-\beta)^{2}}{\sigma^{2}}, \\[13pt] \bigtriangleup_{4} = \big({-}\hat{\delta}+x\mu+\frac12{\sigma^{2}x^{2}\eta}\big)^{2} + \sigma^{2}x^{2}\lambda\eta^{2}{\mathbb{E}}\big(Y^{2}\big), \end{cases} \end{equation*}

and $\hat{\delta}=\lambda{\mathbb{E}}(Y)+\frac12{\eta}\lambda{\mathbb{E}}\big(Y^{2}\big)-c$ .

Corollary 3.3. If $\eta=0$ , the optimal reinsurance strategy of (3.20) reduces to excess-of-loss reinsurance. Hence, the robust optimal investment–reinsurance strategy is

\begin{equation*} (\pi^{\ast}(x),l^{\ast}(x,y)) = \left\{ \begin{array}{l@{\quad}l} \bigg(1+m,\dfrac{\theta}{\tilde{\xi}_{2}^{\ast}(x)}\wedge y\bigg), & x\in\big(0,\tilde{x}_{2}^{\beta}\big]\cap[M, N], \\[13pt] \bigg(\dfrac{\mu-\beta}{\sigma^{2}x}\dfrac{1}{\tilde{\xi}_{3}^{\ast}(x)}, \dfrac{\theta}{\tilde{\xi}_{3}^{\ast}(x)}\wedge y\bigg), \quad & x\in\big(\tilde{x}_{2}^{\beta},\tilde{x}_{3}^{\beta}\big)\cap[M, N], \\[13pt] \bigg(1,\dfrac{\theta}{\tilde{\xi}_{4}^{\ast}(x)}\wedge y\bigg), & x\in\big[\tilde{x}_{3}^{\beta},\tilde{x}_{1}^{r}\big]\cap[M, N] \\[13pt] \bigg(\dfrac{\mu-r}{\sigma^{2}x}\dfrac{1}{\tilde{\xi}_{1}^{\ast}(x)}, \dfrac{\theta}{\tilde{\xi}_{1}^{\ast}(x)}\wedge y\bigg), & x\in\big(\tilde{x}_{1}^{r},+\infty\big)\cap[M, N], \end{array} \right. \end{equation*}

where $\tilde{\xi}_{1}^{\ast}(x)$ , $\tilde{\xi}_{2}^{\ast}(x)$ , $\tilde{\xi}_{3}^{\ast}(x)$ , and $\tilde{\xi}_{4}^{\ast}(x)$ satisfy

\begin{equation*} \begin{cases} \theta\displaystyle\int_{0}^{{\theta}/{\xi}}\bigg(1-\dfrac{y}{{\theta}/{\xi}}\bigg)\bar{F}_{Y}(y)\,\mathrm{d}y = \dfrac1\lambda\bigg\{\tilde{\delta}-rx-\dfrac{(\mu-r)^{2}}{2\sigma^{2}\xi}\bigg\}, \\[10pt] \theta\displaystyle\int_{0}^{{\theta}/{\xi}}\bigg(1-\frac{y}{{\theta}/{\xi}}\bigg)\bar{F}_{Y}(y)\,\mathrm{d}y = \frac1\lambda\bigg\{\tilde{\delta}-\beta x-(\mu-\beta)x(1+m)+ \frac{\sigma^{2}x^{2}(1+m)^{2}\xi}{2}\bigg\},\\[10pt] \theta\displaystyle\int_{0}^{{\theta}/{\xi}}\bigg(1-\frac{y}{{\theta}/{\xi}}\bigg)\bar{F}_{Y}(y)\,\mathrm{d}y = \frac1\lambda\bigg\{\tilde{\delta}-\beta x-\frac{(\mu-\beta)^{2}}{2\sigma^{2}\xi}\bigg\}, \\[10pt] \theta\displaystyle\int_{0}^{{\theta}/{\xi}}\bigg(1-\frac{y}{{\theta}/{\xi}}\bigg)\bar{F}_{Y}(y)\,\mathrm{d}y = \frac1\lambda\bigg\{\tilde{\delta}-\mu x+\frac{\sigma^{2}x^{2}\xi}{2}\bigg\}, \end{cases} \end{equation*}

and $\tilde{x}_{1}^{r}$ , $\tilde{x}_{2}^{\beta}$ , and $\tilde{x}_{3}^{\beta}$ take the same forms as in (3.11) and (3.17), in which $\tilde{\delta}=\lambda(1+\theta){\mathbb{E}}(Y)-c$ .

4. Two extreme cases

In this section, we compute V(x) in the two extreme cases $\epsilon\rightarrow0$ and $\epsilon\rightarrow\infty$ . In the first case, we know that the corresponding results reduce to those in the benchmark case without model ambiguity. In the second case, the insurer has less faith in the reference model and a sequence of alternative measures can be selected.

Let $V_{0}(x)$ denote the non-robust ( $\epsilon\rightarrow0$ ) value function. In the benchmark case, the wealth process evolves by (2.2) and the value function is described by

(4.1) \begin{equation} V_{0}(x)\,:\!=\,\inf_{u\in\mathcal{U}}\mathbb{P}^{x}\big(\tau_{M}^{u} \lt \tau_{N}^{u},\tau^{u} \lt \infty\big).\end{equation}

Letting $\epsilon\rightarrow0$ in Theorem 3.1, we have the following theorem.

Theorem 4.1. Let $\xi^{\ast}(x)$ be as given in (3.19). For $x\in[M, N]$ , the ambiguity-neutral value function is expressed as follows:

\begin{equation*} V_{0}(x) = 1 - \frac{\int_{M}^{x}\exp{\big\{{-}\int_{M}^{y}(\xi^{\ast}(z)-\eta)\,\mathrm{d}z\big\}}\,\mathrm{d}y} {\int_{M}^{N}\exp{\big\{{-}\int_{M}^{y}(\xi^{\ast}(z)-\eta)\,\mathrm{d}z\big\}}\,\mathrm{d}y}. \end{equation*}

The corresponding optimal investment strategy and retention level are given by

\begin{equation*} (\pi^{\ast}(x),l^{\ast}(x,y)) = \left\{ \begin{array}{l@{\quad}l} \bigg(1+m,\dfrac{\theta+\eta y}{\xi_{2}^{\ast}(x)}\wedge y\bigg), & x\in\big(0,x_{2}^{\beta}\big]\cap[M, N], \\[10pt] \bigg({-}\dfrac{\mu-\beta}{\sigma^{2}x}\dfrac{1}{\eta-\xi_{3}^{\ast}(x)}, \dfrac{\theta+\eta y}{\xi_{3}^{\ast}(x)}\wedge y\bigg), \quad & x\in\big(x_{2}^{\beta},x_{3}^{\beta}\big)\cap[M, N], \\[10pt] \bigg(1,\dfrac{\theta+\eta y}{\xi_{4}^{\ast}(x)}\wedge y\bigg), & x\in\big[x_{3}^{\beta},x_{1}^{r}\big]\cap[M, N], \\[10pt] \bigg({-}\dfrac{\mu-r}{\sigma^{2}x}\dfrac{1}{\eta-\xi_{1}^{\ast}(x)}, \dfrac{\theta+\eta y}{\xi_{1}^{\ast}(x)}\wedge y\bigg), & x\in\big(x_{1}^{r},+\infty\big)\cap[M, N]. \end{array} \right. \end{equation*}

Let $V_{\infty}(x)$ denote the value function in the case $\epsilon\rightarrow\infty$ . Under this case, the insurer becomes most ambiguity averse towards the model uncertainty. Then, the value function can be given as follows:

\begin{equation*} V_{\infty}(x)\,:\!=\,\inf_{u\in\mathcal{U}}\sup_{\phi\in\Phi}\mathbb{Q}^{x}\big(\tau_{M}^{u} \lt \tau_{N}^{u},\tau^{u} \lt \infty\big).\end{equation*}

It follows that the penalty term completely disappears. Letting $\epsilon\rightarrow\infty$ in (2.4), we have the following proposition.

Proof. The disappearance of the penalty term will result in the drift coefficients $\phi_{1}$ and $\phi_{2}$ in (2.3) derived from Girsanov’s theorem being positive or negative, and the previous constraints on $\phi_{1}$ and $\phi_{2}$ become meaningless. It makes sense that if $\phi_{1}$ is negative, the investment term coefficient is negative in (2.3), i.e. the investment does not reach the income target but makes the wealth value less. Similarly, when $\phi_{2}$ is negative, improper reinsurance strategies can also cause wealth losses. Therefore, the optimal investment strategy is not to invest at all, while the optimal reinsurance retention level is 0, which means transferring all risks, i.e. $u_{0}=(0,0)$ . Furthermore, the wealth process becomes $\mathrm{d}X^{u_{0}}(t)=(rX^{u_{0}}(t)-\delta)\,\mathrm{d}t$ . It is not difficult to derive through simple calculation that

\begin{align*} \tau_{M}^{u_{0}}=\frac{1}{r}\ln\frac{rM-\delta}{rx-\delta}, \qquad \tau_{N}^{u_{0}}=\frac{1}{r}\ln\frac{rN-\delta}{rx-\delta} \end{align*}

for $x\in[M, N]$ . Then, for all $\mathbb{Q}\in\mathscr{Q}$ , we have $\mathbb{Q}^{x}(\tau_{M}^{u_{0}} \lt \tau_{N}^{u_{0}},\tau^{u_{0}} \lt \infty)=1$ . Thus, for all $x\in[M, N]$ , $V_{\infty}(x)=1$ .

Remark 4.3. Note that $\mathbb{P}\in\mathscr{Q}$ with $\phi_{0}=(0,0)$ . From (2.4), we have

\begin{align*} V_{0}(x)=\inf_{u\in\mathcal{U}}J^{u,\phi_{0}}(x)\leq\inf_{u\in\mathcal{U}}\sup_{\phi\in\Phi}J^{u,\phi}(x)=V(x)\leq1=V_{\infty}(x). \end{align*}

This relation is naturally expected since V is non-decreasing with respect to $\epsilon$ . And the robust goal-reaching probability V(x) is always a conservative estimate for the non-robust value function $V_{0}(x)$ .

Remark 4.4. Let $\phi_{1}^{\ast}(x,\epsilon)=\phi_{1}^{\ast}(x)$ and $\phi_{1}^{\ast}(x,\epsilon)=\phi_{2}^{\ast}(x)$ . Taking $x_{1}^{r}\leq M$ in Theorem 3.1 as an example, when $\epsilon\rightarrow\infty$ we can obtain the following results:

\begin{align*} \lim_{\epsilon\rightarrow\infty}\phi_{1}^{\ast}(x,\epsilon) & = -\frac{\mu-r}{\sigma(\xi_{1}^{\ast}(x)-\eta)} \cdot \frac{\exp{\big\{{-}\int_{M}^{x}(\xi_{1}^{\ast}(z)-\eta)\,\mathrm{d}z\big\}}} {\int_{x}^{N}\exp{\big\{{-}\int_{M}^{y}(\xi_{1}^{\ast}(z)-\eta)\,\mathrm{d}z\big\}}\,\mathrm{d}y}, \\ \lim_{\epsilon\rightarrow\infty}\phi_{2}^{\ast}(x,\epsilon) & = -\sqrt{\lambda{\mathbb{E}}\bigg[\bigg(\frac{\theta+\eta Y}{\xi_{1}^{\ast}(x)}\wedge Y\bigg)^{2}\bigg]}\cdot \frac{\exp{\big\{{-}\int_{M}^{x}(\xi_{1}^{\ast}(z)-\eta)\,\mathrm{d}z\big\}}} {\int_{x}^{N}\exp{\big\{{-}\int_{M}^{y}(\xi_{1}^{\ast}(z)-\eta)\mathrm{d}z\big\}}\,\mathrm{d}y}. \end{align*}

It is easy to derive that the optimal drift distortions $\phi_{1}^{\ast}(x)$ and $\phi_{2}^{\ast}(x)$ are all negative and decrease to finite limits for any $x\in[M, N]$ as $\epsilon\rightarrow\infty$ .

5. Numerical analysis

In this section, we investigate the effect of higher borrowing rate and risk loading parameters on the optimal control strategies. In the following context, we assume that the claim size random variable Y is uniformly distributed in the interval [0, 2], and so we have $E(Y)=1$ and $E\big(Y^{2}\big)=\frac{4}{3}$ . And unless otherwise stated, we set the basic parameters as $\lambda=3$ and $\mu=0.5$ . The notations $\pi^{0}$ and $l^{0}$ respectively represent the optimal investment and reinsurance strategies without a higher borrowing rate.

Figure 1 shows how the higher borrowing rate affects the optimal investment strategy. It is evident from the figure that the optimal investment strategy with a higher borrowing rate is lower than the unconstrained strategy in both the borrowing and full-investment regions. This is to be expected: because of the higher borrowing rate, the insurer becomes more conservative and hesitates to borrow money. Moreover, the optimal investment strategy is a decreasing and continuous function with respect to x. Only if the wealth level falls below the borrowing level will borrowing occur. These results are natural consequences of Remark 3.2.

Figure 1. The influence of higher borrowing rate on the optimal investment strategies.

Figure 2. The influence of higher borrowing rate on the optimal reinsurance strategies.

Figure 2 investigates the impact of the higher borrowing rate on the optimal reinsurance strategy according to the variance premium principle. It can be seen that the increase in the wealth of the insurer leads directly to a decrease of the optimal reinsurance strategy. We can also observe that $l^{\ast} \gt l^{0}$ in the full-investment and borrowing regions, which means that with the higher borrowing rate the insurer is willing to keep more insurance business. Furthermore, it follows from the figure that the optimal reinsurance strategy is the proportional reinsurance and falls into the interval [0, 1], which is also a natural consequence of Corollary 3.2.

Figure 3. The influence of $\theta$ on the optimal control strategies.

Figure 3 illustrates that a higher value of $\theta$ yields greater values of the optimal investment and reinsurance strategies. An explanation for this phenomenon is that as $\theta$ increases, the reinsurance premium becomes more expensive, and hence the insurer would rather retain a larger share of each claim. However, if the reinsurance premium keeps increasing, to avoid ruin the insurer might optimally increase the investment in the risky asset to increase its profit.

Figure 4. The influence of $\eta$ on the optimal control strategies.

Figure 4 presents the effects of the parameter $\eta$ on the optimal control strategies. We can see that with the increase of the risk loading parameter $\eta$ , the corresponding investment proportion and retention level increase. The explanation for this result is similar to that in Figure 3. Since the reinsurance premium and the ruin probability of the insurer would increase with the increase of $\eta$ , so its investment proportion and retention level will improve naturally to increase its profit.