Optimal Time-Consistent Investment and Reinsurance Strategies with Default Risk and Delay under Heston’s SV Model

Sheng Li,Zhijian Qiu,Bekir Sahin

doi:10.1155/2021/8834842

Sheng Li, Zhijian Qiu + Show 1 more

Open Access

https://doi.org/10.1155/2021/8834842

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Abstract

Considering the influence of past information on the decision-making of insurers, the correlation between the insurance businesses owned by insurers, and the possible default faced by insurers, we investigate the mean-variance investment and reinsurance problem with the default risk, delay, and common shock dependence. We characterize the insurance market by two-dimensional dependent claims, the financial market by the Heston SV model, and default risk by reduced-form approach and then obtain the evolution equation of the insurer’s wealth. Based on the introduction of time delay, the insurer’s wealth dynamics characterized by a stochastic delay differential equation are obtained. Furthermore, applying stochastic control theory within the game-theoretic framework and stochastic control theory with delay, we derive optimal time-consistent investment and reinsurance strategies, as well as equilibrium value function and equilibrium efficient frontier. Finally, we use a numerical example to analyze the influence of parameters on the time-consistent equilibrium strategies and give an economic explanation.

Highlights

Ere are two deficiencies in the abovementioned literature, which are worthy of further discussion
Inspired by the above works, considering the influence of past information on the decision-making of insurers, the correlation between the insurance businesses owned by insurers, and the possible default faced by insurers, we investigate mean-variance investment and reinsurance problem with the default risk, delay, and common shock dependence
When our model degenerates into a model without considering the default risk, delay, and insurance risk common shock dependence, we can obtain the result of Li et al [14]. e remainder of this paper is organized as follows

Summary

General Formulation

We consider a filtered complete probability space (Ω, F, 􏼈Ft􏼉t∈[0,T], P) satisfying the usual conditions; that is, 􏼈Ft􏼉t∈[0,T] is right continuous and P-complete, where 􏼈Ft􏼉t∈[0,T] is the information of the market available up to time t and [0, T] is a fixed and finite time horizon. Define π(t) (p1(t), p2(t), q1(t), q2(t)) as the investment-reinsurance strategy at time t; the wealth process {Xπ(t)} of the insurer under a strategy π satisfies the following stochastic differential equation: dXπ(t). Suppose that f(t, X(t) − L(t), X(t) − M(t)) denotes the inflow/outflow of capital, the insurer’s wealth process satisfies the following stochastic delay differential equation (SDDE): dXπ(t) 􏼂rXπ(t) + δp1(t)v(t) + p2(t)(1 − Z(t))θ(1 − Δ) + 1 + η1􏼁a􏽰1q 1 ( t ) + 1 + η2􏼁a2q2(t). For the mean-variance problem (18), we assume that there exist two real-valued functions V(t, x, l, v, z) and g(t, x, l, v, z) ∈ C1,2,1,2([0, T] × R × R × R × {0, 1}) satisfying the following extended HJB equations:.

Solution to the Optimization

Sensitivity Analysis

Conclusion

Proof of Theorem 1

Proof of Theorem 2