Optimal Policies in an Insurance Stackelberg Game: Demand Response and Premium Setting

Abstract

This paper examines a stochastic Stackelberg differential game between an insurer and a pool of homogeneous policyholders. Policyholders dynamically optimize insurance coverage and risky asset allocations to minimize the probability of wealth shortfall, while the insurer, acting as the leader, sets the premium loading to maximize the expected exponential utility of terminal surplus. Employing dynamic programming techniques, we derive closed-form equilibrium strategies for both parties. The analysis reveals that a strong positive correlation between insurance claims and financial market returns incentivizes full coverage with modest premiums, whereas a strong negative correlation may induce market collapse as insurers exit underwriting to exploit natural hedging opportunities. Furthermore, larger policyholder pools generate diversification benefits that reduce equilibrium premiums and stimulate insurance demand.

1. Introduction

The study of insurance and investment decisions has long been an important research domain in economics and finance, with contributions from multiple influential scholars. The theoretical foundation was established in [1], the authors of which introduced the mean-variance framework for portfolio selection. This work was significantly advanced in [2,3], the authors of which developed continuous-time methods for analyzing intertemporal consumption and investment decisions. In parallel, Yaari [4,5] established how insurance purchases fit within lifetime financial planning under uncertain lifetimes, while Samuelson [6] provided mathematical tools for lifetime portfolio selection through dynamic stochastic programming. These foundational works created the basic framework for understanding individual financial decision-making under uncertainty.

Building on these foundations, researchers began exploring how insurance and financial decisions interact. Early work by Briys [7] examined insurance-consumption tradeoffs in continuous time. More recent studies have deepened this integration. Bayraktar et al. [8] investigated how term life insurance purchases interact with bequest motives and consumption decisions. Li et al. [9] explored optimal consumption and annuity purchases under mortality model uncertainty, recognizing that individuals face both financial market risks and longevity uncertainties. However, a common limitation of these models is their treatment of insurance pricing as exogenous, overlooking the strategic nature of premium setting by insurers.

The insurer’s perspective has been extensively studied through optimal control theory, with research progressing from models considering insurers in isolation to those incorporating strategic interactions. Early research focused on insurers optimizing their own decisions without considering strategic responses from policyholders. Martin-Löf [10] investigated dynamic premium control for dividend maximization, while applying linear control theory to insurance systems. More comprehensive approaches emerged later. Christenen et al. [11] examined the optimal control of investments, premiums, and deductibles for non-life insurers. Zhou et al. [12] studied optimal investment and premium control for insurers with nonlinear diffusion models, and Lin and Li [13] explored optimal reinsurance and investment for jump-diffusion risk processes. Similar research can also be found in [14,15,16,17,18,19] and so on. These studies typically assumed passive or exogenously specified policyholder behavior.

Recognizing the strategic nature of insurance markets, researchers began applying game theory, particularly Stackelberg differential games. This approach is natural for reinsurance markets where strategic interactions are prominent. Chen and Shen [20,21] pioneered this line with their work on stochastic Stackelberg differential games between insurers and reinsurers and under time-inconsistent mean-variance frameworks. Subsequent research has extended this methodology in various directions: Cao et al. [22] incorporated model ambiguity, Liang and Young [23] analyzed Stackelberg games in life insurance using mean-variance criteria, and Ghossoub and Zhu [24] specifically considered the case with multiple policyholders. Related work on reinsurance games includes Bai et al. [25] studying games with delay in defaultable markets, Yuan et al. [26] examining robust reinsurance contracts with asymmetric information, and Dong et al. [27] analyzing non-zero-sum reinsurance and investment games.

For other related research employing the Stackelberg framework, see, for instance, the studies by [28,29,30].

Many existing models typically assume that policyholder behavior is passive or exogenously given, failing to capture their demand response to pricing—a critical feature of real insurance markets. In actual markets, price changes directly affect consumers’ insurance purchase decisions, which in turn influences insurers’ business scale and risk profile. Models that ignore this dynamic connection struggle to accurately reflect how real markets operate. We develop a Stackelberg game framework that differs from related work in two key aspects. First, unlike the authors of [23], who studied life insurance with mean-variance objectives, we focus on non-life insurance and employ the probability of survival criterion, which directly addresses ruin avoidance. Second, in contrast to the authors of [24], who examined static contracting, our model features dynamic interactions where both premium setting and investment decisions are simultaneously endogenized for both parties, with explicit consideration of correlation between financial and insurance risks. Our analysis yields closed-form equilibrium solutions that reveal how risk correlation shapes market regimes—from full coverage to potential market collapse.

This paper develops a Stackelberg game framework to analyze strategic interactions between insurers and policyholders. The Stackelberg game framework is well-suited for our study because it captures the sequential interaction between the insurer (leader) and policyholders (followers). Unlike competitive models or single-agent models, it accounts for differences in market power and information. This allows us to analyze how premium setting affects demand and investment, which is key to studying demand response. We obtain explicit analytical solutions showing that, when policyholders maximize the probability of reaching a wealth target before ruin and insurers maximize CARA utility of terminal surplus, three distinct market regimes emerge depending on the correlation between financial and insurance risks. Specifically, high positive correlation leads to low premiums with full insurance coverage, and moderate correlation results in partial insurance markets, while negative correlation may cause market collapse as insurers exit underwriting to focus solely on financial investments. Our closed-form solutions reveal how both parties optimally balance investment returns against risk hedging, with insurance demand decreasing linearly with premium loading and financial investments providing natural hedging against uninsured risks.

The paper is organized as follows. Section 2 presents the model framework. Section 3 solves the policyholder’s probability maximization problem and derives the insurer’s utility maximization equilibrium strategies. Section 4 provides some numerical examples to illustrate the economic significance of the results. Finally, Section 5 concludes the paper.

2. Model Formulation

For reader convenience, we provide a summary of key parameters and variables used throughout the paper (see Abbreviations Section).

2.1. Financial and Insurance Markets

Let $(\Omega ,\mathcal{F},\mathbb{P})$ be a complete probability space supporting three standard Brownian motions $W_0\left(t\right)$, $W_1\left(t\right)$, and $W_2\left(t\right)$. The filtration ${\left\{{\mathcal{F}}_t\right\}}_{t\ge 0}$ is generated by these processes. In the financial market, there is a risky asset (stock) with price dynamics:

$${\displaystyle \frac{dP\left(t\right)}{P\left(t\right)}}=\mu dt+\sigma d{W}_0\left(t\right),$$

(1)

where $\mu >0$ and $\sigma >0$ are constants.

In the insurance market, there are n homogeneous policyholders. Each faces insurable losses modeled by a compound Poisson process ${\sum }_{i=1}^{N_1\left(t\right)}{X}_i$. The number of claims up to time t is $N_1\left(t\right)$, a Poisson process with intensity $\lambda >0$. Claim sizes $\left\{X_i\right\}$ are $i.i.d.$ positive random variables, independent of $N_1\left(t\right)$, with mean $\alpha :=\mathbb{E}\left[X_i\right]$ and second moment ${\beta }^2:=\mathbb{E}\left[X_i^2\right]$. Each policyholder can purchase proportional insurance coverage. At time t, they choose coverage proportion $a_t\in [0,1]$ (insurance demand). The insurer charges a premium at rate $(1+{\theta }_t)a_t\lambda \alpha$, where ${\theta }_t$ is the safety loading (premium markup) set by the insurer. When a claim of size X occurs, the insurer pays $a_{t}X$, and the policyholder bears $(1-a_t)X$. Hence, the wealth process of an insured with per-claim proportional coverage can be expressed as

$$\mathrm{d}{S}^a\left(t\right)=(I_t-c_t-(1+{\theta }_t)a_t\lambda \alpha )\mathrm{d}t-(1-a_t)\mathrm{d}\sum _{i=1}^{N_1\left(t\right)}{X}_i.$$

(2)

The model primarily describes a voluntary (non-compulsory) insurance setting, where policyholders optimally choose whether and how much to insure. The insurer’s aggregate claims come from all n policyholders. The total number of claims up to time t is $N_n\left(t\right)$, a Poisson process with intensity $n\lambda$, with claim sizes having the same distribution as $X_i$. Consequently, the dynamics of the insurer’s surplus can be modeled as

$$\mathrm{d}Y\left(t\right)=(1+{\theta }_t)a_{t}n\lambda \alpha \mathrm{d}t-a_t\mathrm{d}\sum _{i=1}^{N_n\left(t\right)}{X}_i.$$

(3)

2.2. Diffusion Approximations and Wealth Dynamics

Using diffusion approximations for the compound Poisson processes [14], we obtain continuous dynamics. For a policyholder with consumption rate $c_t>0$ (exogenously given), the wealth process without investment is approximately:

$$d{S}^a\left(t\right)=(I_t-c_t-{\theta }_t{a}_t\lambda \alpha -\lambda \alpha )dt+(1-a_t)\sqrt{\lambda }\beta d{W}_1\left(t\right),$$

(4)

where $I_t>0$ (exogenously given) is an income rate. The term $-\lambda \alpha$ represents the expected loss per unit time. Since financial markets and insurance claims are often influenced by common economic shocks, their associated risk processes are correlated. We capture this by introducing a correlation between the Brownian motions $W_0\left(t\right)$ (financial market) and $W_1\left(t\right)$ (insurance risk). The correlation structure is defined by

$$Cov\left(W_0\left(t\right),W_1\left(t\right)\right)=\rho t,\rho \in (-1,1),$$

(5)

where $\rho$ measures the strength and direction of the linkage. For analytical convenience, we represent $W_0\left(t\right)$ in terms of $W_1\left(t\right)$ and an independent Brownian motion $W_2\left(t\right)$ as follows:

$$W_0\left(t\right)=\rho W_1\left(t\right)+\sqrt{1-{\rho }^2}{W}_2\left(t\right).$$

(6)

This decomposition explicitly separates the common risk factor ($W_1$) from the idiosyncratic financial risk ($W_2$). The square root ensures that the variance of the resulting process remains equal to t, preserving its properties as a standard Brownian motion.

With investment in the financial market, let $f_t$ be the dollar amount invested in the risky asset. The policyholder’s wealth dynamics become

$$\begin{array}{cc}\hfill d{S}^{a,f}\left(t\right)& =\left[\mu f_t+I_t-c_t-{\theta }_t{a}_t\lambda \alpha -\lambda \alpha \right]dt\hfill \\ \hfill & +(1-a_t)\sqrt{\lambda }\beta d{W}_1\left(t\right)+\rho \sigma f_{t}d{W}_1\left(t\right)+\sigma f_t\sqrt{1-{\rho }^2}d{W}_2\left(t\right).\hfill \end{array}$$

(7)

For the insurer, the surplus process without investment is approximately

$$dY\left(t\right)={\theta }_t{a}_{t}n\lambda \alpha dt+a_t\sqrt{n\lambda }\beta d{W}_1\left(t\right).$$

(8)

With investment amount ${\pi }_t$ in the risky asset, the insurer’s surplus dynamics are

$$\begin{array}{c}\hfill d{Y}^{\pi ,\theta }\left(t\right)=\left[\mu {\pi }_t+{\theta }_t{a}_{t}n\lambda \alpha \right]dt+\left(a_t\sqrt{n\lambda }\beta +\rho \sigma {\pi }_t\right)d{W}_1\left(t\right)+\sigma {\pi }_t\sqrt{1-{\rho }^2}d{W}_2\left(t\right).\end{array}$$

(9)

In order to formulate the stochastic optimization problems, we define admissible strategies as follows:

Definition 1

(Admissible Strategies). A strategy tuple $(a,\theta ,f,\pi )$ is admissible if the following holds:

(i) 

$a_t$, ${\theta }_t$, $f_t$, ${\pi }_t$ are ${\mathcal{F}}_t$-predictable processes.

(ii) 

$a_t\in [0,1]$, ${\theta }_t\in [\underline{\theta },\overline{\theta }]$ for fixed bounds $\underline{\theta }<\overline{\theta }$.

(iii) 

$E{\int }_0^T(f_t^2+{\pi }_t^2)dt<\infty$.

(iv) 

The SDEs (7) and (9) have unique strong solutions.

Denote the set of admissible strategies by $\mathcal{A}$.

2.3. Stackelberg Game

In this Stackelberg game, the insurer (leader) determines the premium loading ${\theta }_t$ and the investment amount ${\pi }_t$ in the risky asset, aiming to maximize its expected utility of terminal surplus. The policyholders (followers), taking the insurer’s pricing strategy ${\theta }_t$ as given, choose their insurance coverage level $a_t$ and personal investment $f_t$ in the financial market. The objective of each policyholder is to maximize the probability of reaching a target wealth level $d>0$ before falling to ruin (wealth level 0) by the terminal time T. This sequential structure distinguishes our Stackelberg game from simultaneous-move games where all players’ objectives depend symmetrically on all decision variables.

Follower’s Problem (Policyholder). For given $\theta$, let ${\tau }_b^{a,f}$ and ${\tau }_d^{a,f}$ denote the first hitting times to levels b and d under strategy $(a,f)$, i.e.,

$${\tau }_b^{a,f}=inf\{t>0:S^{a,f}\left(t\right)=b\},{\tau }_d^{a,f}=inf\{t>0:S^{a,f}\left(t\right)=d\},$$

with $b=0$ (ruin level) and $d>0$ (a survival target level). Define ${\tau }^{a,f}=min({\tau }_b^{a,f},{\tau }_d^{a,f})$. The policyholder maximizes the probability of reaching d before hitting 0:

$$\begin{array}{c}\hfill V(x;\theta )=\underset{(a,f)\in \mathcal{A}}{sup}{\mathbb{P}}_{t,x}\left({\tau }^{a,f}={\tau }_d^{a,f}\right),\end{array}$$

(10)

where ${\mathbb{P}}_{t,x}$ denotes the conditional probability given $S^{a,f}\left(t\right)=x$, and we assume $0<x<d$ initially.

Leader’s Problem (Insurer). Anticipating the optimal response $a^{*}\left(\theta \right)$, the insurer maximizes expected utility of terminal surplus with CARA utility:

$$\begin{array}{c}\hfill Z(t,y)=\underset{(\theta ,\pi )\in \mathcal{A}}{sup}{\mathbb{E}}_{t,y}\left[-{\displaystyle \frac{1}{\gamma }}{e}^{-\gamma Y^{\pi ,\theta }\left(T\right)}\right],\end{array}$$

(11)

where $\gamma >0$ is the insurer’s absolute risk aversion coefficient.

3. Equilibrium Analysis

3.1. Policyholder’s Optimal Response

The policyholder’s value function $V(x;\theta )$ satisfies the HJB equation:

$$\begin{array}{c}\hfill \underset{(a,f)\in [0,1]\times \mathbb{R}}{sup}\left\{{\mathcal{L}}^{a,f}V\left(x\right)\right\}=0,0<x<d,\end{array}$$

(12)

where

$$\begin{array}{cc}\hfill {\mathcal{L}}^{a,f}V\left(x\right)=& \left[\mu f+I-c-\theta a\lambda \alpha -\lambda \alpha \right]V^{\prime }\left(x\right)\hfill \\ & +{\displaystyle \frac{1}{2}}\left[\lambda {(1-a)}^2{\beta }^2+{\sigma }^2{f}^2+2\rho \sigma \sqrt{\lambda }\beta (1-a)f\right]V^{\prime \prime }\left(x\right),\hfill \end{array}$$

with the following boundary conditions: $V\left(0\right)=0$, $V\left(d\right)=1$.

For maximizing the probability of reaching d before 0, the value function should be increasing and concave, i.e., $V^{\prime }\left(x\right)>0$, $V^{\prime \prime }\left(x\right)<0$. A standard ansatz (see [15]) is

$$V\left(x\right)={\displaystyle \frac{1-e^{-\eta x}}{1-e^{-\eta d}}},$$

(13)

where $\eta >0$ is a parameter to be determined. This satisfies the following:

$$\begin{array}{cc}\hfill V\left(0\right)& =0,V\left(d\right)=1,\hfill \\ \hfill V^{\prime }\left(x\right)& =\eta {\displaystyle \frac{e^{-\eta x}}{1-e^{-\eta d}}}>0,\hfill \\ \hfill V^{\prime \prime }\left(x\right)& =-{\eta }^2{\displaystyle \frac{e^{-\eta x}}{1-e^{-\eta d}}}<0.\hfill \end{array}$$

Let $C\left(x\right):={\displaystyle \frac{e^{-\eta x}}{1-e^{-\eta d}}}>0$. Then

$$V^{\prime }\left(x\right)=\eta C\left(x\right),V^{\prime \prime }\left(x\right)=-{\eta }^{2}C\left(x\right).$$

(14)

Substituting into (12) and dividing by $C\left(x\right)>0$, we have

$$\underset{(a,f)\in [0,1]\times \mathbb{R}}{sup}\left\{\eta \mu (a,f)-{\displaystyle \frac{{\eta }^2}{2}}{\sigma }^2(a,f)\right\}=0,$$

(15)

where

$$\begin{array}{cc}\hfill \mu (a,f)& =\mu f+I-c-\theta a\lambda \alpha -\lambda \alpha ,\hfill \\ \hfill {\sigma }^2(a,f)& =\lambda {(1-a)}^2{\beta }^2+{\sigma }^2{f}^2+2\rho \sigma \sqrt{\lambda }\beta (1-a)f.\hfill \end{array}$$

Let

$$\Phi (a,f):=\eta \mu (a,f)-{\displaystyle \frac{{\eta }^2}{2}}{\sigma }^2(a,f).$$

(16)

Define the market price of risk $m:={\displaystyle \frac{\mu }{\sigma }}$. By first-order condition, we have

$${\displaystyle \frac{\partial \Phi }{\partial f}}=\eta \mu -{\eta }^2\left[{\sigma }^{2}f+\rho \sigma \sqrt{\lambda }\beta (1-a)\right]=0.$$

We have

$$f^{*}={\displaystyle \frac{\mu }{\eta {\sigma }^2}}-{\displaystyle \frac{\rho \sqrt{\lambda }\beta }{\sigma }}(1-a)={\displaystyle \frac{m}{\eta \sigma }}-{\displaystyle \frac{\rho \sqrt{\lambda }\beta }{\sigma }}(1-a).$$

(17)

Substitute $f^{*}$ into $\Phi (a,f)$, we have

$$\begin{array}{cc}\hfill \Phi (a,f^{*})& =\eta \left[{\displaystyle \frac{m^2}{\eta }}-m\rho \sqrt{\lambda }\beta (1-a)+I-c-\theta a\lambda \alpha -\lambda \alpha \right]\hfill \\ \hfill & -{\displaystyle \frac{{\eta }^2}{2}}\left[\lambda (1-{\rho }^2){(1-a)}^2{\beta }^2+{\displaystyle \frac{m^2}{{\eta }^2}}\right]\hfill \\ \hfill & ={\displaystyle \frac{m^2}{2}}-\eta m\rho \sqrt{\lambda }\beta (1-a)+\eta (I-c-\lambda \alpha )-\eta \theta a\lambda \alpha \hfill \\ \hfill & -{\displaystyle \frac{1}{2}}{\eta }^2\lambda (1-{\rho }^2){(1-a)}^2{\beta }^2.\hfill \end{array}$$

The first-order condition for maximizing the objective function with respect to a is

$${\displaystyle \frac{\partial \Phi }{\partial a}}=\eta m\rho \sqrt{\lambda }\beta -\eta \theta \lambda \alpha +{\eta }^2\lambda (1-{\rho }^2){\beta }^2(1-a)=0.$$

Therefore, the candidate insurance equilibrium demand policy satisfies

$$1-a^{\star }={\displaystyle \frac{\theta \alpha -m\rho \frac{\beta }{\sqrt{\lambda }}}{\eta (1-{\rho }^2){\beta }^2}}.$$

Then

$$a^{\star }=1-{\displaystyle \frac{\theta \alpha -\frac{m\rho \beta }{\sqrt{\lambda }}}{\eta (1-{\rho }^2){\beta }^2}}.$$

(18)

Denote $\underline{\theta }:={\displaystyle \frac{m\rho \beta }{\sqrt{\lambda }\alpha }}$ and $\overline{\theta }:=\underline{\theta }+{\displaystyle \frac{\eta (1-{\rho }^2){\beta }^2}{\alpha }}$. Then,

$$a^{\star }={\displaystyle \frac{\overline{\theta }-\theta }{\overline{\theta }-\underline{\theta }}}.$$

(19)

Since $0\le a\le 1$, the equilibrium insurance demand policy can be given by

$$a^{*}\left(\theta \right)=\left\{\begin{array}{cc}1,\hfill & \theta \le \underline{\theta },\hfill \\ {\displaystyle \frac{\overline{\theta }-\theta }{\overline{\theta }-\underline{\theta }},}\hfill & \underline{\theta }<\theta <\overline{\theta },\hfill \\ 0,\hfill & \theta \ge \overline{\theta }.\hfill \end{array}\right.$$

(20)

Denote

$$D=\left\{\begin{array}{cc}0,\hfill & \theta \le \underline{\theta },\hfill \\ {\displaystyle \frac{\alpha (\theta -\underline{\theta })}{(1-{\rho }^2){\beta }^2},}\hfill & \underline{\theta }<\theta <\overline{\theta },\hfill \\ \eta ,\hfill & \theta \ge \overline{\theta },\hfill \end{array}\right.$$

(21)

then $1-a^{*}=D/\eta$. Thus,

$$\begin{array}{cc}\hfill \Phi (a^{*},f^{*})& ={\displaystyle \frac{m^2}{2}}-\eta m\rho \sqrt{\lambda }\beta ·{\displaystyle \frac{D}{\eta }}+\eta (I-c-\lambda \alpha )-\eta \theta \lambda \alpha \left(1-{\displaystyle \frac{D}{\eta }}\right)-{\displaystyle \frac{1}{2}}{\eta }^2\lambda (1-{\rho }^2){\beta }^2·{\displaystyle \frac{D^2}{{\eta }^2}}\hfill \end{array}$$

Let $E:=I-c-\lambda \alpha >0$. In simple terms, $E>0$ means the net benefit is positive after accounting for all costs. Then

$$\eta (E-\theta \lambda \alpha )+\left[{\displaystyle \frac{m^2}{2}}-m\rho \sqrt{\lambda }\beta D+\theta \lambda \alpha D-{\displaystyle \frac{1}{2}}\lambda (1-{\rho }^2){\beta }^2{D}^2\right]=0.$$

Thus,

$$\eta ={\displaystyle \frac{-\frac{m^2}{2}+m\rho \sqrt{\lambda }\beta D-\theta \lambda \alpha D+\frac{1}{2}\lambda (1-{\rho }^2){\beta }^2{D}^2}{E-\theta \lambda \alpha }}.$$

(22)

For $\eta >0$, we need numerator and denominator to have the same sign. Since $E-\theta \lambda \alpha >0$ for survival, the numerator must be positive.

From (17) and $1-a^{*}=D/\eta$,

$$f^{*}\left(\theta \right)={\displaystyle \frac{m}{\eta \sigma }}-{\displaystyle \frac{\rho \sqrt{\lambda }\beta }{\sigma }}·{\displaystyle \frac{D}{\eta }}={\displaystyle \frac{1}{\eta \sigma }}\left[m-\rho \sqrt{\lambda }\beta D\right].$$

(23)

Theorem 1

(Policyholder’s Best Response). Assume $r=0$ for time-homogeneity. For given premium loading θ, the policyholder’s optimal insurance demand is given by (20), and optimal investment is given by (23), where $\underline{\theta }={\displaystyle \frac{m\rho \beta }{\sqrt{\lambda }\alpha }}$, $\overline{\theta }=\underline{\theta }+{\displaystyle \frac{\eta (1-{\rho }^2){\beta }^2}{\alpha }}$, and $\eta >0$ is determined by (22).

Remark 1.

(i) The investment strategy $f^{*}$ has a Merton-type component ${\displaystyle \frac{m}{\eta \sigma }}$ for return-seeking, and a hedging component $-{\displaystyle \frac{\rho \sqrt{\lambda }\beta }{\sigma }}(1-a^{*})$ for managing uninsured risk exposure. (ii) Insurance demand $a^{*}$ decreases linearly with premium loading θ, reflecting standard price sensitivity. (iii) The parameter η represents the effective risk aversion of the policyholder and is endogenously determined by market conditions and survival requirements. (iv) Notably, $\underline{\theta }$ can be negative when $\rho <0$, meaning even with subsidized premiums ($\theta <0$), full insurance may not be optimal. This occurs because negative correlation allows financial investments to hedge insurance risk, creating a natural risk management alternative.

3.2. Insurer’s Equilibrium Strategy

Anticipating the policyholder’s best response $a^{*}\left(\theta \right)$, the insurer (leader) solves the optimal control problem:

$$Z(t,y)=\underset{(\theta ,\pi )\in \mathcal{A}}{sup}{\mathbb{E}}_{t,y}\left[-{\displaystyle \frac{1}{\gamma }}{e}^{-\gamma Y^{\pi ,\theta }\left(T\right)}\right].$$

(24)

The corresponding HJB equation is

$$\begin{array}{c}\hfill \underset{(\theta ,\pi )}{sup}\left\{Z_t+\left[\mu \pi +\theta a^{*}\left(\theta \right)n\lambda \alpha \right]Z_y+{\displaystyle \frac{1}{2}}\left[\lambda n{\left(a^{*}\left(\theta \right)\right)}^2{\beta }^2+{\sigma }^2{\pi }^2+2\rho \sigma \sqrt{\lambda n}\beta a^{*}\left(\theta \right)\pi \right]Z_{yy}\right\}=0,\end{array}$$

(25)

with terminal condition $Z(T,y)=-{\displaystyle \frac{1}{\gamma }}{e}^{-\gamma y}$.

We conjecture that $Z(t,y)$ satisfies the following form.

$$Z(t,y)=-{\displaystyle \frac{1}{\gamma }}exp\left[-\gamma K\left(t\right)y+H\left(t\right)\right],$$

(26)

where $K\left(t\right)>0$ and $H\left(t\right)$ are to be determined, $K\left(T\right)=1$, $H\left(T\right)=0$. Then, we have

$$\begin{array}{cc}\hfill Z_t& =\left[-\gamma K^{\prime }\left(t\right)y+H^{\prime }\left(t\right)\right]Z,\hfill \\ \hfill Z_y& =-\gamma K\left(t\right)Z,\hfill \\ \hfill Z_{yy}& ={\gamma }^{2}K{\left(t\right)}^{2}Z.\hfill \end{array}$$

(27)

Substituting (27) into (25) and dividing by Z, we can obtain that

$$\begin{array}{cc}\hfill -\gamma K^{\prime }\left(t\right)y+H^{\prime }\left(t\right)& -\gamma K\left(t\right)\left[\mu \pi +\theta a^{*}\left(\theta \right)n\lambda \alpha \right]\hfill \\ \hfill & +{\displaystyle \frac{1}{2}}{\gamma }^{2}K{\left(t\right)}^2\left[\lambda n{\left(a^{*}\left(\theta \right)\right)}^2{\beta }^2+{\sigma }^2{\pi }^2+2\rho \sigma \sqrt{\lambda n}\beta a^{*}\left(\theta \right)\pi \right]=0.\hfill \end{array}$$

Since the equation must hold for all y,

$$-\gamma K^{\prime }\left(t\right)=0\Rightarrow K^{\prime }\left(t\right)=0.$$

Thus $K\left(t\right)$ is constant. From the terminal condition $Z(T,y)=-{\displaystyle \frac{1}{\gamma }}{e}^{-\gamma y}$, we have $K\left(T\right)=1$, so

$$K\left(t\right)\equiv 1.$$

Therefore, the HJB Equation (25) reduces to

$$H^{\prime }\left(t\right)+\underset{(\theta ,\pi )}{sup}\Psi (\theta ,\pi )=0,$$

(28)

where

$$\Psi (\theta ,\pi )=-\gamma \left[\mu \pi +\theta a^{*}\left(\theta \right)n\lambda \alpha \right]+{\displaystyle \frac{1}{2}}{\gamma }^2\left[\lambda n{\left(a^{*}\left(\theta \right)\right)}^2{\beta }^2+{\sigma }^2{\pi }^2+2\rho \sigma \sqrt{\lambda n}\beta a^{*}\left(\theta \right)\pi \right].$$

(29)

The first-order condition for $\pi$ is

$${\displaystyle \frac{\partial \Psi }{\partial \pi }}=-\gamma \mu +{\gamma }^2\left[{\sigma }^2\pi +\rho \sigma \sqrt{\lambda n}\beta a^{*}\left(\theta \right)\right]=0.$$

Solving for $\pi$,

$${\pi }^{*}\left(\theta \right)={\displaystyle \frac{\mu }{\gamma {\sigma }^2}}-{\displaystyle \frac{\rho \sqrt{\lambda n}\beta }{\sigma }}{a}^{*}\left(\theta \right)={\displaystyle \frac{m}{\gamma \sigma }}-{\displaystyle \frac{\rho \sqrt{\lambda n}\beta }{\sigma }}{a}^{*}\left(\theta \right).$$

(30)

We substitute ${\pi }^{*}\left(\theta \right)$ into $\Psi (\theta ,\pi )$ to obtain $\Psi \left(\theta \right)$:

$$\Psi \left(\theta \right)=-{\displaystyle \frac{m^2}{2}}+\gamma m\rho \sigma \sqrt{\lambda n}\beta a^{*}\left(\theta \right)-\gamma \theta a^{*}\left(\theta \right)n\lambda \alpha +{\displaystyle \frac{1}{2}}{\gamma }^2\lambda n(1-{\rho }^2){\beta }^2{\left[a^{*}\left(\theta \right)\right]}^2.$$

(31)

Let

$$\Phi \left(\theta \right):=\gamma m\rho \sigma \sqrt{\lambda n}\beta a^{*}\left(\theta \right)-\gamma \theta a^{*}\left(\theta \right)n\lambda \alpha +{\displaystyle \frac{1}{2}}{\gamma }^2\lambda n(1-{\rho }^2){\beta }^2{\left[a^{*}\left(\theta \right)\right]}^2.$$

(32)

Recall the policyholder’s best response:

$$a^{*}\left(\theta \right)=\left\{\begin{array}{cc}1,\hfill & \theta \le \underline{\theta },\hfill \\ {\displaystyle \frac{\overline{\theta }-\theta }{\overline{\theta }-\underline{\theta }},}\hfill & \underline{\theta }<\theta <\overline{\theta },\hfill \\ 0,\hfill & \theta \ge \overline{\theta },\hfill \end{array}\right.$$

where

$$\underline{\theta }={\displaystyle \frac{m\rho \beta }{\sqrt{\lambda }\alpha }},\overline{\theta }=\underline{\theta }+{\displaystyle \frac{\eta (1-{\rho }^2){\beta }^2}{\alpha }}.$$

Case 1: $\theta \le \underline{\theta }$, $a^{*}=1$.

$${\Phi }_1\left(\theta \right)=\gamma m\rho \sigma \sqrt{\lambda n}\beta -\gamma \theta n\lambda \alpha +{\displaystyle \frac{1}{2}}{\gamma }^2\lambda n(1-{\rho }^2){\beta }^2.$$

Since ${\displaystyle \frac{d{\Phi }_1}{d\theta }}=-\gamma n\lambda \alpha <0$, ${\Phi }_1$ is decreasing in $\theta$. The maximum in this interval occurs at the left boundary $\theta =\underline{\theta }$.

Case 2: $\theta \ge \overline{\theta }$, $a^{*}=0$.

$${\Phi }_2\left(\theta \right)=0.$$

Case 3: $\underline{\theta }<\theta <\overline{\theta }$, $a^{*}={\displaystyle \frac{\overline{\theta }-\theta }{\Delta }}$ with $\Delta =\overline{\theta }-\underline{\theta }={\displaystyle \frac{\eta (1-{\rho }^2){\beta }^2}{\alpha }}$.

Substituting into $\Phi \left(\theta \right)$, we have

$$\begin{array}{cc}\hfill {\Phi }_3\left(\theta \right)& =\gamma m\rho \sigma \sqrt{\lambda n}\beta ·{\displaystyle \frac{\overline{\theta }-\theta }{\Delta }}-\gamma \theta n\lambda \alpha ·{\displaystyle \frac{\overline{\theta }-\theta }{\Delta }}+{\displaystyle \frac{1}{2}}{\gamma }^2\lambda n(1-{\rho }^2){\beta }^2·{\displaystyle \frac{{(\overline{\theta }-\theta )}^2}{{\Delta }^2}}.\hfill \end{array}$$

If we differentiate with respect to $\theta$, then

$$\begin{array}{cc}\hfill {\displaystyle \frac{d{\Phi }_3}{d\theta }}& =-{\displaystyle \frac{\gamma m\rho \sigma \sqrt{\lambda n}\beta }{\Delta }}+{\displaystyle \frac{\gamma n\lambda \alpha }{\Delta }}(\overline{\theta }-\theta )-\gamma n\lambda \alpha ·{\displaystyle \frac{\overline{\theta }-\theta }{\Delta }}+\gamma n\lambda \alpha ·{\displaystyle \frac{\theta }{\Delta }}\hfill \\ \hfill & -{\displaystyle \frac{{\gamma }^2\lambda n(1-{\rho }^2){\beta }^2}{{\Delta }^2}}(\overline{\theta }-\theta ).\hfill \end{array}$$

If we set ${\displaystyle \frac{d{\Phi }_3}{d\theta }}=0$, then the optimal premium loading is

$${\theta }^{*}={\displaystyle \frac{\frac{\underline{\theta }\sigma }{\sqrt{n}}+\frac{\gamma }{\eta }\overline{\theta }}{1+\frac{\gamma }{\eta }}}={\displaystyle \frac{\eta \underline{\theta }\sigma /\sqrt{n}+\gamma \overline{\theta }}{\eta +\gamma }}.$$

(33)

Next, we use $\rho$ to distinguish the critical conditions for ${\theta }^{*}$. Since $a^{*}\left({\theta }^{*}\right)=0$ corresponds to ${\theta }^{*}=\overline{\theta }$, the relevant condition is $\rho ={\rho }_1$ (to be determined in the following). In this situation, setting ${\theta }^{*}=\overline{\theta }$, we have

$$\begin{array}{c}\hfill \overline{\theta }=\underline{\theta }\sigma /\sqrt{n}.\end{array}$$

(34)

Substituting $\overline{\theta }=\underline{\theta }+{\displaystyle \frac{\eta (1-{\rho }^2){\beta }^2}{\alpha }}$ and $\underline{\theta }={\displaystyle \frac{m\rho \beta }{\sqrt{\lambda }\alpha }}$ into (34), we can obtain

$$\eta (1-{\rho }^2)\beta ={\displaystyle \frac{m\rho }{\sqrt{\lambda }}}\left({\displaystyle \frac{\sigma }{\sqrt{n}}}-1\right).$$

Let $C={\displaystyle \frac{m}{\sqrt{\lambda }\eta \beta }}\left({\displaystyle \frac{\sigma }{\sqrt{n}}}-1\right)$. Then

$$1-{\rho }^2=C\rho .$$

This gives

$$\rho ={\displaystyle \frac{-C\pm \sqrt{C^2+4}}{2}}.$$

Since $\rho \in (-1,1)$ and the product of roots is −1, one root is positive and one negative. For the lower threshold where the insurer exits the market, we expect ${\rho }_1<0$. Thus,

$$\begin{array}{c}\hfill {\rho }_1={\displaystyle \frac{-C-\sqrt{C^2+4}}{2}}(\mathrm{negative}\mathrm{root}).\end{array}$$

(35)

Since $a^{*}\left({\theta }^{*}\right)=1$ corresponds to ${\theta }^{*}=\underline{\theta }$, the relevant condition is $\rho ={\rho }_2$ (to be determined in the following). In this situation, setting ${\theta }^{*}=\underline{\theta }$, we have

$${\displaystyle \frac{m\rho (\sigma -\sqrt{n})}{\sqrt{\lambda n}\beta }}+\gamma (1-{\rho }^2)=0.$$

Let $k={\displaystyle \frac{m(\sigma -\sqrt{n})}{\sqrt{\lambda n}\beta }}(ingeneral\sigma <1,thenk<0)$. Then,

$$k\rho +\gamma (1-{\rho }^2)=0\Rightarrow -\gamma {\rho }^2+k\rho +\gamma =0.$$

Thus,

$$\rho ={\displaystyle \frac{k\pm \sqrt{k^2+4{\gamma }^2}}{2\gamma }}.$$

Since ${\rho }_2>0$ (the upper threshold for a positive correlation), we take the positive root

$$\begin{array}{c}\hfill {\rho }_2={\displaystyle \frac{k+\sqrt{k^2+4{\gamma }^2}}{2\gamma }}.\end{array}$$

(36)

Furthermore, $H\left(t\right)$ can be given by $H^{\prime }\left(t\right)=-\Psi ({\theta }^{*},{\pi }^{*})$ and $H\left(T\right)=0$. Consequently, we have the following theorem.

Theorem 2

(Insurer’s Equilibrium Strategy). The equilibrium premium loading and investment strategy for the insurer are

$$\begin{array}{cc}\hfill {\theta }^{*}& =\left\{\begin{array}{cc}\underline{\theta },\hfill & \rho \ge {\rho }_2,\hfill \\ {\displaystyle \frac{\eta \underline{\theta }\sigma /\sqrt{n}+\gamma \overline{\theta }}{\eta +\gamma },}\hfill & {\rho }_1<\rho <{\rho }_2,\hfill \\ \overline{\theta },\hfill & \rho \le {\rho }_1,\hfill \end{array}\right.\hfill \end{array}$$

(37)

$$\begin{array}{cc}\hfill {\pi }^{*}& ={\displaystyle \frac{m}{\gamma \sigma }}-{\displaystyle \frac{\rho \sqrt{\lambda n}\beta }{\sigma }}{a}^{*},\hfill \end{array}$$

(38)

where $a^{*}=a^{*}\left({\theta }^{*}\right)$ is the policyholder’s optimal response to the equilibrium premium, and the threshold correlations ${\rho }_1,{\rho }_2$ are determined by (35) and (36).

Remark 2.

(i) High Positive Correlation ($\rho \ge {\rho }_2$): The insurer sets the lowest premium $\underline{\theta }$ and policyholders purchase full coverage. When financial and insurance risks are strongly positively correlated, the insurer faces concentrated risk exposure. By pricing low to attract full insurance, the insurer transfers most underwriting risk away, and significantly reduces stock investments ${\pi }^{*}$, which has a large negative hedging component to avoid amplifying remaining risk. This suggests that during periods of high systemic risk, insurance markets may exhibit price competition as insurers prioritize risk reduction over premium income.

(ii) Moderate Correlation (${\rho }_1<\rho <{\rho }_2$): The insurer sets an intermediate premium and policyholders purchase partial coverage. This represents the typical market equilibrium where the insurer balances premium income against risk exposure. The interior solution ${\theta }^{*}$ in (33) can be rewritten as

$${\theta }^{*}=\underline{\theta }+{\displaystyle \frac{\gamma }{\eta +\gamma }}·{\displaystyle \frac{\eta (1-{\rho }^2){\beta }^2}{\alpha }}-{\displaystyle \frac{\eta }{\eta +\gamma }}·\underline{\theta }\left(1-{\displaystyle \frac{\sigma }{\sqrt{n}}}\right).$$

The second term represents the risk premium weighted by the relative risk aversion ratio $\gamma /(\eta +\gamma )$. The third term captures diversification benefits: when the insurer can effectively pool risks (n large) or risk asset is less volatile (σ small), it can charge lower premiums while maintaining profitability.

(iii) Negative/Low Positive Correlation ($\rho \le {\rho }_1$): The insurer sets the highest premium $\overline{\theta }$ and policyholders exit the market ($a^{*}=0$). When financial and insurance risks are negatively correlated, underwriting and investment activities naturally hedge each other. Surprisingly, the insurer’s optimal strategy is to withdraw from underwriting entirely and focus solely on financial investments. This reveals a potential market failure: when capital markets provide natural hedging for insurance risks, traditional insurance markets may collapse as capital moves to pure investment activities. This has important implications for insurance availability during economic conditions where such negative correlations emerge.

4. Numerical Analysis

This section presents a series of numerical examples from the perspectives of both the individual and the insurer. The parameter settings are as follows. The policyholder’s claim arrival rate is $\lambda =0.5$. The expected property loss is characterized by parameters $\alpha =5$ and $\beta =8$ for the loss distribution. The risky asset has an expected return $\mu =0.08$ and volatility $\sigma =0.25$. Risk aversion coefficient $\gamma =3$ and the number of simulation paths $n=1000$ (A similar setting is used in [21,30]).

In Figure 1, we can see that the equilibrium premium loading ${\theta }^{*}$ increases with claim intensity $\lambda$, reflecting the insurer’s need for higher compensation when facing more frequent losses. The relationship between ${\theta }^{*}$ and correlation $\rho$ exhibits an inverted U-shape. ${\theta }^{*}$ is relatively high when $\rho$ is near zero, decreases as $\rho$ approaches 1, and decreases as $\rho$ approaches −1. This pattern arises because extreme correlations (either positive or negative) create opportunities for risk management that allow the insurer to charge lower premiums while maintaining profitability. Policyholder insurance demand $a^{*}$ decreases with $\lambda$ due to the price elasticity of demand. However, $a^{*}$ increases monotonically with $\rho$. When $\rho$ is negative, insurance and financial investments serve as substitutes in risk management, reducing the optimal level of insurance coverage. When $\rho$ is positive, they become complements, leading to higher insurance demand as protection against correlated risks.

Both the insurer’s investment ${\pi }^{*}$ and the policyholder’s investment $f^{*}$ show similar patterns with respect to $\rho$. As $\rho$ increases from negative to positive values, both parties reduce their positions in the risky asset, with the insurer potentially taking short positions when $\rho$ is strongly positive. This adjustment serves as a hedge against the correlation between insurance losses and financial market returns. As $\lambda$ increases, both ${\pi }^{*}$ and $f^{*}$ generally decrease across most values of $\rho$, indicating a general reduction in risk-taking when insurance claims become more frequent.

Figure 2 demonstrates that a higher risk aversion coefficient $\gamma$, indicating a more risk-averse insurer, leads to an increase in the equilibrium premium ${\theta }^{*}$. This reflects the insurer’s need for greater compensation when bearing underwriting risks. Consequently, the higher premium reduces the policyholder’s optimal insurance demand $a^{*}$, due to the price sensitivity of insurance products. As $\gamma$ increases, the insurer’s short positions in risky assets decrease, as more risk-averse insurers are less willing to take aggressive hedging positions. Similarly, the policyholder’s investment $f^{*}$ declines with higher $\gamma$, primarily because elevated premiums reduce disposable wealth for investment and because the policyholder’s own risk management becomes more conservative in response to the insurer’s risk-averse pricing behavior.

Figure 3 illustrates how the expected return of risky assets ($\mu$) interacts with risk correlation ($\rho$) to shape equilibrium strategies. When $\rho$ approaches $-1$, higher $\mu$ leads to a lower equilibrium premium ${\theta }^{*}$, as the insurer can rely on natural hedging between underwriting and investment, allowing it to reduce premiums while maintaining profitability. Conversely, when $\rho$ approaches 1, higher $\mu$ results in a higher ${\theta }^{*}$ because the insurer faces compounded correlated risks and demands greater compensation. The insurance demand $a^{*}$ follows a similar pattern: it decreases with $\mu$ when $\rho$ is negative (substitution effect) but increases with $\mu$ when $\rho$ is positive (complementarity effect). The insurer’s investment ${\pi }^{*}$ shows larger short positions as $\mu$ increases, reflecting a stronger hedging motive against positively correlated risks. Meanwhile, the policyholder’s investment $f^{*}$ rises with $\mu$, demonstrating a shift toward financial markets for higher returns, particularly when insurance provides limited hedging benefits under negative correlation.

5. Conclusions

This paper develops a Stackelberg stochastic differential game between an insurer and policyholders, providing an in-depth analysis of how risk correlation shapes insurance market equilibrium. The findings demonstrate that the correlation coefficient $\rho$ between financial market returns and insurance claims is not only a key determinant of insurance demand and pricing but also a critical threshold defining distinct market regimes. When $\rho$ is highly positive, the market reaches a competitive equilibrium characterized by “low premiums and full coverage.” In this regime, the insurer lowers premiums to transfer underwriting risk and significantly reduces investments in risky assets to avoid amplifying correlated exposures. When $\rho$ is moderately positive, a classic partial-insurance equilibrium emerges, where premiums balance risk premiums against diversification benefits. However, when $\rho$ turns negative, the market may collapse, as the insurer finds it optimal to exit underwriting entirely and focus solely on financial investments for natural hedging. This reveals a profound structural issue: when capital markets themselves provide effective hedging instruments for insurance risks, the traditional risk-transfer mechanism may be displaced, leading to market failure. This insight carries important warnings for regulators: during periods when systemic risk correlations shift, the potential weakening of traditional insurance market functions requires vigilant attention. Such market collapse could justify regulatory interventions, such as capital requirements that account for natural hedging benefits or reinsurance mandates, to preserve insurance availability and protect policyholder welfare. Our results also imply that solvency regulation should consider the hedging value of negatively correlated assets when setting minimum capital requirements, as ignoring this could lead to excessive capital charges that ironically discourage underwriting.

The model further highlights the constraint effect of household financial resilience on insurance decisions under the fixed-consumption assumption. When net income (after accounting for consumption and expected losses) is limited, insurance plays a strengthened role as a tool for maximizing survival probability, yet its demand becomes more sensitive to premium rates. Additionally, expanding the insurance pool generates significant diversification benefits, which not only lower equilibrium premiums but also stimulate insurance demand. This provides a theoretical basis for insurers to achieve sustainable operations by expanding their customer base. From an investment perspective, the insurer’s optimal portfolio pursues both return-seeking and risk-hedging objectives. The size and direction of its hedging positions are directly influenced by risk correlation and pool size, indicating that an insurer’s asset management must be deeply coordinated with the risk profile of its underwriting business.

Our model has several limitations that suggest directions for future research. The assumptions of homogeneous policyholders, a single risky asset, and constant correlation structure may not fully capture real-world complexity. Additionally, the diffusion approximation for claims may overlook extreme events. Future work could extend the model to include heterogeneous agents, competitive markets with multiple insurers, or more realistic asset models with stochastic volatility. An important empirical direction would be to calibrate the model using real data—such as S&P 500 returns for financial parameters, insurance claim data from regulatory filings, and historical correlations between market indices and insurance loss ratios—to test the predicted market regimes against actual market behavior.