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De Finetti's Control for Refracted Skew Brownian Motion

Zhongqin Gao, Yan Lv, Xiaowen Zhou

TL;DR

The paper investigates optimal dividend strategies for a risk model driven by refracted skew Brownian motion, a process with endogenous regime switching through a two-valued drift relative to a threshold $a$ and a local-time term capturing regime transitions. By deriving explicit exit-time Laplace transforms and formulating Hamilton-Jacobi-Bellman inequalities, the authors identify sufficient conditions under which barrier and band strategies are optimal, showing how skewness $\beta$ and drift parameters shape strategy selection. The main contributions include closed-form expressions for barrier and band value functions, a detailed analysis of $W'$-based optimality criteria, and existence results for parameter regimes yielding optimal multi-barrier bands, complemented by numerical illustrations. These results enhance understanding of asymmetry-driven stochastic control and offer practical insights for dividend optimization under endogenous regime switching. The refracted skew Brownian framework provides a tractable yet rich setting to study how non-symmetric dynamics influence optimal control in risk processes.

Abstract

In this paper we propose a refracted skew Brownian motion as a risk model with endogenous regime switching, which generalizes the refracted diffusion risk process introduced by Gerber and Shiu. We consider an optimal dividend problem for the refracted skew Brownian risk model and identify sufficient conditions, respectively, for barrier strategy, band strategy and their variants to be optimal.

De Finetti's Control for Refracted Skew Brownian Motion

TL;DR

The paper investigates optimal dividend strategies for a risk model driven by refracted skew Brownian motion, a process with endogenous regime switching through a two-valued drift relative to a threshold and a local-time term capturing regime transitions. By deriving explicit exit-time Laplace transforms and formulating Hamilton-Jacobi-Bellman inequalities, the authors identify sufficient conditions under which barrier and band strategies are optimal, showing how skewness and drift parameters shape strategy selection. The main contributions include closed-form expressions for barrier and band value functions, a detailed analysis of -based optimality criteria, and existence results for parameter regimes yielding optimal multi-barrier bands, complemented by numerical illustrations. These results enhance understanding of asymmetry-driven stochastic control and offer practical insights for dividend optimization under endogenous regime switching. The refracted skew Brownian framework provides a tractable yet rich setting to study how non-symmetric dynamics influence optimal control in risk processes.

Abstract

In this paper we propose a refracted skew Brownian motion as a risk model with endogenous regime switching, which generalizes the refracted diffusion risk process introduced by Gerber and Shiu. We consider an optimal dividend problem for the refracted skew Brownian risk model and identify sufficient conditions, respectively, for barrier strategy, band strategy and their variants to be optimal.
Paper Structure (28 sections, 26 theorems, 158 equations, 2 tables)

This paper contains 28 sections, 26 theorems, 158 equations, 2 tables.

Table of Contents

  1. Introduction
  2. Solutions to the exit problems
  3. Conditions for optimal dividend strategies
  4. Optimal barrier strategies
  5. Expected discounted dividend function for barrier strategies
  6. Optimal barrier strategies
  7. Optimal band strategies
  8. Expected discounted dividend function for band strategies
  9. Optimal band strategies.
  10. The case $b_1\leq a_1<a$.
  11. The case $b_1< a \leq a_1$.
  12. Examples
  13. Proofs
  14. Proof of Lemma \ref{['c1c201']}
  15. Proof of Lemma \ref{['value function']}
  16. ...and 13 more sections

Key Result

Lemma 2.1

$c_i(q)<1$ for all $q>0$ and $i=1, 2$. Further, we have $c_i(q)<0$ if and only if $(1+\beta) \rho_{i}^+<(1-\beta)\rho_{i}^-$.

Theorems & Definitions (67)

  • Lemma 2.1
  • Theorem 2.1
  • proof
  • Theorem 2.2
  • Theorem 2.3
  • proof
  • Remark 2.1
  • Remark 2.2
  • Lemma 3.1
  • proof
  • ...and 57 more