Dynamic Optimal Reinsurance and Dividend Payout in Finite Time Horizon

Abstract

This paper studies a dynamic optimal reinsurance and dividend-payout problem for an insurance company in a finite time horizon. The goal of the company is to maximize the expected cumulative discounted dividend payouts until bankruptcy or maturity, whichever comes earlier. The company is allowed to buy reinsurance contracts dynamically over the whole time horizon to cede its risk exposure with other reinsurance companies. This is a mixed singular–classical stochastic control problem, and the corresponding Hamilton–Jacobi–Bellman equation is a variational inequality with a fully nonlinear operator and subject to a gradient constraint. We obtain the [Formula: see text] smoothness of the value function and a comparison principle for its gradient function by the penalty approximation method so that one can establish an efficient numerical scheme to compute the value function. We find that the surplus-time space can be divided into three nonoverlapping regions by a risk-magnitude and time-dependent reinsurance barrier and a time-dependent dividend-payout barrier. The insurance company should be exposed to a higher risk as its surplus increases, be exposed to the entire risk once its surplus upward crosses the reinsurance barrier, and pay out all its reserves exceeding the dividend-payout barrier. The estimated localities of these regions are also provided.

Optimal dividend payout problem under both diffusion risk and Poisson risk in finite horizon

“…there exists a sufficiently large constant x m such that d(t) ≤ x m for all t where L > x m . This result implies that the solution of the problem (16) and the problem (15) are consistent in Q L T . In order to get an upper bound estimate of d(t), we construct a super solution of the problem (16) which takes value 1 when x is sufficiently large.…”

“…Thanks to Lemma 6.6, we have u(x, t) = 1 if x ≥ x m when L > x m , so we can extend our solution u to the unbounded domain Q T by setting u(x, t) = 1 for x > L. Then after extension, u ∈ W 2,1 p,loc (Q T ) is the solution of the problem (15), and d(t) defined in ( 25) is its free boundary line. Moreover, it is easy to check that the estimates ( 23)-( 24) remain true in Q T .…”

“…Proof of Theorem 3.1: Based on the above preparation, now we can construct a solution to the problem (1). Suppose u is the solution to the problem (15), let…”

“…x, from which it can be obtained that the right side of the free boundary is the dividend payout region and the left side is the non-dividend payout region. This is also the key that one can transform the singular control problem on v into an equivalence obstacle problem on its gradient u = v x , see e.g., [2,17,15,14]. However, because the integral term in the equation in our problem is nonlocal, the equation on v xx or its approximation analog to those in [17,15,14] cannot be obtained, which prevents us from deriving the concavity of the solution w.r.t.…”

Moral-hazard-free insurance: mean-variance premium principle and rank-dependent utility theory

“…An important feature of our model (same as Xu [32]) is that we take the so-called incentive compatibility constraint into account. In the insurance literature, many works ignore this constraint intentionally or unintentionally; see, Deprez and Gerber [8], Gajek and Zagrodny [11], Kaluszka [20], Liang, Liang and Young [21], Guan, Xu and Zhou [12], to name a few. We believe it is mainly due to the mathematical challenges arising from the constraint to force the authors to ignore this constraint.…”