Thiele's PIDE for unit-linked policies in the Heston-Hawkes stochastic volatility model

The main purpose of the paper is to derive Thiele's differential equation for unit-linked policies in the Heston-Hawkes stochastic volatility model presented in arXiv:2210.15343. This model is an extension of the well-known Heston model that incorporates the volatility clustering feature by adding a compound Hawkes process in the volatility. Since the model is arbitrage-free, pricing unit-linked policies via the equivalence principle under $\mathbb{Q}$ is possible. Some integrability conditions are checked and a suitable family of risk neutral probability measures is found to obtain Thiele's differential equation. The established and practical method to compute reserves in life insurance is by solving Thiele's equation, which is crucial to guarantee the solvency of the insurance company.