This paper investigates an optimal reinsurance-investment problem in relation to thinning dependent risks. The insurer's wealth process is described by a risk model with two dependent classes of insurance business. The insurer is allowed to purchase reinsurance and invest in one risk-free asset and one risky asset whose price follows CEV model. Our aim is to maximize the expected exponential utility of terminal wealth. Applying Legendre transform-dual technique along with stochastic control theory, we obtain the closed-form expression of optimal strategy. In addition, our wealth process will reduce to the classical Cramér-Lundberg (C-L) model when $ p = 0 $, in this case, we achieve the explicit expression of the optimal strategy for Hyperbolic Absolute Risk Aversion (HARA) utility by using Legendre transform. Finally, some numerical examples are presented to illustrate the impact of our model parameters (e.g., interest and volatility) on the optimal reinsurance-investment strategy.

The present paper investigates an optimal reinsurance-investment problem with Hyperbolic Absolute Risk Aversion (HARA) utility. The paper is distinguished from other literature by taking into account the interests of both an insurer and a reinsurer. The insurer is allowed to purchase reinsurance from the reinsurer. Both the insurer and the reinsurer are assumed to invest in one risk-free asset and one risky asset whose price follows Heston's SV model. Our aim is to seek optimal investment-reinsurance strategies to maximize the expected HARA utility of the insurer's and the reinsurer's terminal wealth. In the utility theory, HARA utility consists of power utility, exponential utility and logarithmic utility as special cases. In addition, HARA utility is seldom studied in the optimal investment and reinsurance problem due to its sophisticated expression. In this paper, we choose HARA utility as the risky preference of the insurer. Due to the complexity of the structure of the solution to the original Hamilton-Jacobi-Bellman (HJB) equation, we use Legendre transform to change the original non-linear HJB equation into its linear dual one, whose solution is easy to conjecture in the case of HARA utility. By calculations and deductions, we obtain the closed-form solutions of optimal investment-reinsurance strategies. Moreover, some special cases are also discussed in detail. Finally, some numerical examples are presented to illustrate the impacts of our model parameters (e.g., interest and volatility) on the optimal reinsurance-investment strategies.

The fracturing process of sandstone is inherently complex due to its loose internal structure and deformation adaptability. Liquid nitrogen pre-injection has emerged as a promising approach to damage reservoir rocks, effectively reducing fracture pressure and establishing intricate fracture networks, thus offering a potential solution for reservoir reconstruction. To unravel the fundamental mechanisms governing sandstone fracturing behaviors following liquid nitrogen pre-injection, sandstone fracturing experiments were conducted under varying durations of liquid nitrogen injection, rock temperature, and in-situ stress conditions. The experiments showcased the evolution of injection pressure and fracture characteristics under different testing conditions, complemented by electron microscope analysis to elucidate the factors driving the complex fracture characteristics of sandstone. The findings revealed a significant decrease in fracture pressure after liquid nitrogen pre-injection, accompanied by a notable increase in the complexity of the fracture network and the roughness of the fracture surface. Moreover, prolonging the duration of liquid nitrogen injection and elevating reservoir temperature further contributed to reducing fracture pressure, consequently enhancing fracture complexity and surface roughness. Conversely, the application of confining pressure amplified fracture pressure while intensifying the degree of fracturing. Notably, the investigation highlighted the increased presence of microcracks in sandstone resulting from liquid nitrogen pre-injection, facilitating fluid diffusion during fracturing and yielding lower fracture pressures, thereby enhancing the effectiveness of sandstone reservoir reformation. The research results can provide theoretical guidance for geothermal reservoir reconstruction.