Delayed Reinsurance & Investment Optimization

• Delayed reinsurance and investment optimization is a framework integrating risk transfer and multi-asset allocation under decision delays and heterogeneous risk preferences.
• It employs stochastic delay differential equations to capture insurer wealth dynamics and delayed feedback, facilitating equilibrium strategy analyses via pseudo-HJB systems.
• Sensitivity analyses reveal that delays and memory parameters significantly influence reinsurance ratios and investment proportions, prompting more cautious or aggressive strategies.

Delayed reinsurance and investment optimization encompasses stochastic control problems in which insurance companies jointly determine the timing and magnitude of risk transfer via reinsurance, together with multi-asset investment allocations. Critical in these problems are time-lags (delays) in the effect or knowledge of decisions taken—for example, when reinsurance impacts capital positions through delayed cash flows, or when investment/reinsurance decisions are made in response to delayed or averaged historical surplus information. The paper of delayed reinsurance and investment optimization is motivated both by realistic financial-market frictions and by regulatory and organizational structures. Recent advances extend foundational stochastic control models to allow for delays, random risk aversion, bounded memory, and regime-switching, while also rigorously treating time inconsistency, stochastic volatility, and equilibrium in competitive environments.

1. Mathematical Framework with Delay and Random Risk Aversion

One major class of delayed reinsurance-investment problems is formulated in terms of a stochastic delay differential equation (SDDE), where both the insurer’s current wealth $X(t)$ and a weighted integral of past wealth $M_1(t)$, e.g., $$M_1^\mathbf{u}(t) = \int_{-h}^0 e^{\alpha s} X^\mathbf{u}(t+s)\,ds$$, serve as state variables. The model explicitly accounts for capital flows triggered by performance over a delay horizon, with dynamics (for a proportional reinsurance policy $q(t)$ and risky investment proportion $\pi(t)$ in a Black-Scholes market):

$$\begin{align*} dX_t &= \Big\lbrace X_t [r+\pi_t(\mu - r)] + a\eta + a\eta_2 q_t + BM_1(t) + CM_2(t) \Big\rbrace dt \ &\quad + b q_t dW_1(t) + \pi_t X_t \sigma dW_2(t), \end{align*}$$

where $M_2(t) = X(t-h)$, $a, b$ summarize the insurance market’s size and claim volatility, $(r, \mu, \sigma)$ are standard Black-Scholes parameters, and $W_1, W_2$ are independent Brownian motions. Delayed feedback enters via $BM_1(t) + CM_2(t)$, with $B, C$ reprising the role of memory parameters.

The insurer’s objective is formulated in a certainty equivalent form reflecting random risk aversion $\gamma$ (with distribution $\Gamma$) and extended utility on both $X^\mathbf{u}(T)$ and $M_1^\mathbf{u}(T)$:

$$J^\mathbf{u}(t,x,m_1,m_2) = \int \varphi_\gamma^{-1} \left( \mathbb{E}_t[\varphi_\gamma(X^\mathbf{u}(T) + \beta M_1^\mathbf{u}(T))] \right)\,d\Gamma(\gamma),$$

where $\varphi_\gamma$ is a parameterized utility function (e.g., exponential or power). The time-inconsistency induced by such objectives (nonlinear transformations of expected utility) precludes the direct application of classical dynamic programming and instead mandates a game-theoretic equilibrium approach (Kang et al., 19 Sep 2025).

2. Equilibrium Approach and Verification Theorem

To recover time-consistent strategies, a game-theoretic concept of equilibrium is adopted. The optimization problem is recast as an intra-personal non-cooperative game in which different points along time (“future selves” of the insurer) play against each other. Equilibrium strategies are characterized via a pseudo–Hamilton–Jacobi–BeLLMan (HJB) equation system: if $U$ and $Y_\gamma$ (the value functions) are sufficiently smooth and satisfy:

$$\sup_{\mathbf{u}} \left\{\mathcal{A}^{u} U - \mathcal{A}^{u} H + \int \iota_\gamma(Y_\gamma) \mathcal{A}^{u} Y_\gamma d\Gamma(\gamma)\right\} = 0,$$

where $H$ is a transformation of $Y_\gamma$ via $\varphi_\gamma$, and $\mathcal{A}^u$ is the infinitesimal generator associated with the SDDE, then the maximizing control $\hat{\mathbf{u}}$ attains the equilibrium [(Kang et al., 19 Sep 2025), Thm 3.1]. Explicit feedback forms for equilibrium controls can be given when the utility is exponential or power, and under further model restrictions (e.g., coefficients constant, delay functional linear).

3. Analytical and Semi-Analytical Equilibrium Strategies

Under exponential utility $\varphi_\gamma(x)=-\frac{1}{\gamma}e^{-\gamma x}$, ansatz solutions lead to linear SDE or SDDE systems whose solutions yield the closed-form equilibrium strategies, e.g.,

$$\hat{q}(t) = \frac{a\eta_2}{b^2 \mathbb{E}[\gamma]} e^{-(A+\beta)(T-t)}, \qquad \hat{\pi}(t) = \frac{\mu - r}{\sigma^2 x \mathbb{E}[\gamma]} e^{-(A+\beta)(T-t)},$$

where $A, \beta$ are functions of the insurance and delay parameters.

For power utility, an analogous approach yields [Proposition 4.2; (Kang et al., 19 Sep 2025)]:

$$\hat{q}(t) = \frac{a\eta_2 (x + \beta m_1)}{b^2 \omega(t,\Gamma)}, \qquad \hat{\pi}(t) = \frac{\mu - r}{\sigma^2 x \omega(t,\Gamma)} (x + \beta m_1),$$

where $\omega(t,\Gamma)$ represents a weighted average of risk aversion over the support of $\Gamma$.

The crucial point is that, because of the random risk aversion and path-dependence, the equilibrium reinsurance and investment strategies become mutually dependent—unlike classical memoryless, constant risk aversion cases.

4. Sensitivity to Delay, Memory, and Market Parameters

Numerical experiments and analytical sensitivities demonstrate the following:

• Both the optimal reinsurance ratio $\hat{q}$ and risky investment proportional $\hat{\pi}$ are decreasing functions of the delay-averaging parameter $\beta$: more weight on delayed (historical) wealth means more conservative current actions.
• The effect of the memory-weighting parameter $\alpha$ is non-monotone: initially, as $\alpha$ increases, risk-taking declines, then increases as agents become less sensitive to longer-past performance; the length of the delay window $h$ exerts a “smoothing” effect.
• Higher first moments of claim sizes or reinsurance safety loadings typically reduce equilibrium reinsurance demand and encourage more investment in the risky asset, but these effects are more complex in the presence of random risk aversion and delay.

This complex sensitivity illustrates the critical importance of accommodating both delay and heterogeneity of preferences in practical risk management.

5. Theoretical Innovations and Broader Implications

This regime significantly extends classical Markovian stochastic control models by incorporating SDDE dynamics, certainty equivalents with random risk aversion, and by rigorously solving for equilibrium rather than precommitment-optimal strategies. The derived verification theorems and feedback solutions provide a theoretical foundation for real-world reinsurance contracts where decision delay, past performance, and heterogeneity of stakeholder risk preferences are pronounced.

Notably, even when insurance and financial market shocks are independent, the equilibrium strategy couples investment and reinsurance controls due to random risk aversion and delayed information—a qualitative departure from classical models where these decisions decouple [(Kang et al., 19 Sep 2025), Cor. 4.3].

6. Connections to Contemporary Literature

Recent literature expands this delayed equilibrium framework:

• Mean-field games for robust, competitive reinsurance-investment among insurers under model uncertainty (Guan et al., 12 Dec 2024) introduce relative performance and ambiguity aversion.
• Hybrid reinsurance-investment games with delay and bounded memory (Bai et al., 2019) show that delay may encourage or discourage investment, with “herd effects” in competitive settings.
• Monotone and robust mean-variance optimization with delay (zhang et al., 2021, Shi et al., 28 May 2024, Shi et al., 15 Jun 2024) establish that optimal controls in delayed, jump-diffusion models can be equivalently characterized via backward stochastic differential equations with jumps, Riccati equations, or even Malliavin calculus for non-smooth optimization criteria.
• Regime-switching dynamic utility and forward preferences (Colaneri et al., 2021, Colaneri et al., 2022) allow for dynamic updating of risk aversion and environmental dependence, facilitating models where both preferences and information evolve conditionally on market regimes.
• Model uncertainty, estimation, and learning (robustification) all further interact with delayed optimization and risk aversion heterogeneity (see, e.g., (Bäuerle et al., 2020, Ceci et al., 14 Aug 2024)).

7. Numerical and Implementation Aspects

Numerical illustrations consistently confirm that delays and random risk aversion can significantly alter strategic allocations compared to memoryless models. For tractability, equilibrium strategies are implemented via:

• Closed- or semi-closed form feedback policies in the exponential/power utility case;
• Solution of systems of Riccati BSDEs (for variance-based objectives);
• Stochastic gradient or projection methods allowing for non-smooth or simulation-based objectives, particularly in Malliavin-calculus-based methods minimizing ruin probability (Otsuki et al., 8 Nov 2024).

Explicit code implementation in practice typically involves discretization of the SDDEs, solution of associated ODEs or BSDEs, Monte Carlo or finite-difference approximation of path-dependent/averaged wealth, and estimation of expected utility or certainty equivalent over random risk aversion.

Summary

Delayed reinsurance and investment optimization is defined by the joint, dynamically consistent management of risk transfer and multi-asset investment in the presence of delayed information flows and heterogeneous, random risk aversion. Such problems are naturally time-inconsistent and require equilibrium (game-theoretic) solutions—characterized via verification theorems involving pseudo-HJB systems or BSDEs with delay and jumps. Analytical and numerical methods yield explicit forms for equilibrium strategies under exponential and power utility, and sensitivity analysis highlights how delay and preferences interact to determine managerial behavior. These advances directly inform the design of optimal dynamic reinsurance-contracting and asset allocation for insurers operating in complex financial and regulatory environments.