In this paper, we consider the problem of investment and reinsurance with time delay under the compound Poisson model of two-dimensional dependent claims. Suppose an insurance company controls the claim risk of two kinds of dependent insurance businesses by purchasing proportional reinsurance and invests its wealth in a financial market composed of a risk-free asset and a risk asset. The risk asset price process obeys the geometric Brownian motion. By introducing the capital flow related to the historical performance of the insurer, the wealth process described by stochastic delay differential equation (SDDE) is obtained. The extended HJB equation is obtained by using the stochastic control theory under the framework of game theory. Under the reinsurance expected premium principle, optimal time-consistent investment and reinsurance strategy and the corresponding value function are obtained. Finally, the influence of model parameters on the optimal strategy is explained by numerical analysis.

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<p style='text-indent:20px;'>This paper studies the optimal reinsurance-investment problem under Heston's stochastic volatility (SV) in the framework of Stackelberg stochastic differential game, in which an insurer and a reinsurer are the two players. The aim of the insurer as the follower in the game is to find the optimal reinsurance strategy and investment strategy such that its mean-variance cost functional is maximized, while the aim of the reinsurer as the leader is to maximize its mean-variance cost functional through finding its optimal premium pricing strategy and investment strategy. To overcome the time-inconsistency problem in the game, we derive the optimization problems of the two players as embedded games and obtain the corresponding extended Hamilton–Jacobi–Bellman (HJB) equations. Then we present the verification theorem and get the equilibrium reinsurance-investment strategies and the corresponding value functions of both the insurer and the reinsurer. Finally, we provide some numerical examples to draw the findings.</p>

<p style='text-indent:20px;'>In this paper, we consider an optimal mean-variance investment and reinsurance problem with delay and Common Shock Dependence. An insurer can control the claim risk by purchasing proportional reinsurance. He/she invests his/her wealth on a risk-free asset and a risky asset, which follows the jump-diffusion process. By introducing a capital flow related to the historical performance of the insurer, the wealth process described by a stochastic differential equation with delay is obtained. By stochastic linear-quadratic control theory and stochastic control theory with delay, we achieve the explicit expression of the optimal strategy and value function in the framework of the viscosity solution. Furthermore, an efficient strategy and its efficient frontier are derived by Lagrange dual method. Finally, we analyze the influence of the parameters of our model on the efficient frontier by a numerical example.</p>

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