[Paper Review] Optimal Parisian-type dividends payments discounted by the number of claims for the perturbed classical risk process

Irmina Czarna, Yanhong Li+2 more|arXiv (Cornell University)|Mar 22, 2016

Probability and Risk Models15 references3 citations

TL;DR

This paper studies optimal dividend strategies in a perturbed classical risk process with Parisian ruin and claim-count-dependent discounting. It characterizes the value function under a barrier strategy, derives sufficient conditions for optimality, and provides explicit solutions for exponential claim sizes, showing that the optimal barrier shifts under Parisian delay.

ABSTRACT

In this paper we consider a classical risk process perturbed by a Brownian motion. We analyze the value function describing the mean of the cumulative discounted dividend payments paid up to Parisian ruin time and further discounted by the number of claims appeared up to this ruin time. We identify this value function for the barrier strategy and find the sufficient conditions for this strategy to be optimal. We also consider few particular examples.

Motivation & Objective

• To model dividend payments in a risk process perturbed by Brownian motion, incorporating both Parisian ruin and claim-count-dependent discounting.
• To analyze the value function representing cumulative discounted dividends up to Parisian ruin, with additional weighting by the number of claims.
• To identify the barrier strategy as optimal under sufficient conditions, extending classical results to a more realistic ruin and discounting framework.
• To provide explicit solutions for the value function in the case of exponentially distributed claim sizes.
• To validate the optimality of the barrier strategy through numerical examples and verification of the HJB equation.

Proposed method

• Formulates a surplus process combining a compound Poisson process for large claims and a Brownian motion for small fluctuations.
• Defines Parisian ruin as the first time the surplus stays negative for a continuous duration $ d \geq 0 $, generalizing classical ruin ($ d = 0 $).
• Introduces a value function that discounts cumulative dividends by $ r^{N_{\tau^{\pi,d}}} $, where $ N_{\tau^{\pi,d}} $ is the number of claims until ruin.
• Applies the Hamilton-Jacobi-Bellman (HJB) equation framework to derive the necessary conditions for optimality of the barrier strategy.
• Uses integral transforms and recursive convolution techniques to solve for the value function in the case of exponential claim sizes.
• Employs numerical verification by checking $ (\Gamma - q)v \leq 0 $ to confirm optimality of the derived barrier levels.

Experimental results

Research questions

• RQ1 How does introducing Parisian ruin affect the optimal dividend strategy in a perturbed classical risk model?
• RQ2 What is the impact of discounting dividends by the number of claims until ruin on the optimal strategy and value function?
• RQ3 Under what conditions is the barrier strategy optimal when both Parisian ruin and claim-count discounting are considered?
• RQ4 How do the optimal barrier levels change with the Parisian delay parameter $ d $?
• RQ5 Can explicit solutions be derived for the value function when claim sizes are exponentially distributed?

Key findings

• The value function under the barrier strategy is explicitly derived for both $ d = 0 $ (classical ruin) and $ d > 0 $ (Parisian ruin).
• For exponential claim sizes, the value function components $ \vartheta(x) $ and $ \varrho(x) $ are expressed via infinite series involving modified Bessel functions and exponential integrals.
• The optimal barrier level for $ d = 0 $ is numerically found to be $ a^* \approx 0.7693 $, and for $ d = 2 $, $ b^* \approx 0.52202 $, under the parameter set $ \lambda = 10, \mu = 1, c = 15, q = 0.1, r = 0.8 $.
• The condition $ (\Gamma - q)v \leq 0 $ is satisfied for both $ a^* $ and $ b^* $, confirming their optimality via the HJB verification theorem.
• The inclusion of claim-count discounting reduces the optimal barrier level compared to classical models, reflecting the increased penalty for high claim frequency.
• Numerical results confirm the feasibility of the algorithm for practical implementation in actuarial risk management.