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<p>This paper develops a novel entropy-regularized policy iteration algorithm (PIA) for solving the optimal dividend problem under the classical Compound-Poisson risk model. Building on Howard’s classical PIA framework, we resolve longstanding barriers to policy iteration in dividend optimization: entropy regularization guarantees smooth PIA iterates, eliminating historical nonsmoothness obstacles; first-claim truncation transforms the governing integro-differential equation into an exactly solvable ODE system, overcoming spatial nonlocality; and boundedness arguments establish unique closed-form solutions without ad hoc boundary specifications. Furthermore, we prove uniform convergence of both value function sequences and associated policies—ensuring algorithmic stability under general compound Poisson dynamics. Finally, asymptotic analysis demonstrates consistency with classical theory: as <InlineEquation ID="IEq1"> <EquationSource Format="TEX">\(\lambda \rightarrow 0^+\)</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>λ</mi> <mo stretchy="false">→</mo> <msup> <mn>0</mn> <mo>+</mo> </msup> </mrow> </math></EquationSource> </InlineEquation>, our regularized solutions converge to the discontinuous bang–bang strategy and its value function. Collectively, this work establishes the first provably convergent implementation of Howard’s policy iteration algorithm (PIA) for compound-Poisson dividend models, resolving the tripartite challenges of nonsmoothness, nonlocality, and nonlinearity while preserving compatibility with classical control theory through vanishing entropy regularization.</p>

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The Howard’s Policy Iteration and Convergence for Optimal Dividend Under Compound-Poisson Model

  • 期刊论文

  • 发表日期：

    2026年7月9日

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  • Lihua Bai,
  • Linyu Miao

摘要

This paper develops a novel entropy-regularized policy iteration algorithm (PIA) for solving the optimal dividend problem under the classical Compound-Poisson risk model. Building on Howard’s classical PIA framework, we resolve longstanding barriers to policy iteration in dividend optimization: entropy regularization guarantees smooth PIA iterates, eliminating historical nonsmoothness obstacles; first-claim truncation transforms the governing integro-differential equation into an exactly solvable ODE system, overcoming spatial nonlocality; and boundedness arguments establish unique closed-form solutions without ad hoc boundary specifications. Furthermore, we prove uniform convergence of both value function sequences and associated policies—ensuring algorithmic stability under general compound Poisson dynamics. Finally, asymptotic analysis demonstrates consistency with classical theory: as \(\lambda \rightarrow 0^+\) λ → 0 + , our regularized solutions converge to the discontinuous bang–bang strategy and its value function. Collectively, this work establishes the first provably convergent implementation of Howard’s policy iteration algorithm (PIA) for compound-Poisson dividend models, resolving the tripartite challenges of nonsmoothness, nonlocality, and nonlinearity while preserving compatibility with classical control theory through vanishing entropy regularization.