arXiv subjects

Lihua Bai

Publications and source records attributed to Lihua Bai.

7 recordsLinked to original sources

arXiv2026

Reinforcement Learning for optimal dividend problem under diffusion model

In this paper, we study the optimal dividend problem under the continuous time diffusion model with the bounded dividend rate from the Reinforcement Learning (RL) perspective. Unlike the standard literature, our main focus will be on numerical algorithms that allow part or all of the system parameters to be unspecified so that the optimal control cannot be explicitly determined. Following the RL literature we introduce the entropy-regularized exploratory control problem, which randomizes the control actions and balances the levels of exploitation and exploration, and carry out a theoretical analysis of the associated Policy Improvement (PI) and Policy Evaluation (PE) devices and the corresponding sequence of the approximating optimal strategies. Specifically, our algorithm will be based on two independent neural networks that approximate the value function and its derivative simultaneously. Such an algorithm, to the best of our knowledge, is new in the context of the optimal dividend problems, and can be effective even for the situation when the premium and/or interest rate is state dependent, hence beyond reach of the standard statistical methods. Some numerical experiments are presented to empirically demonstrate the effectiveness of our RL algorithm.

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arXiv2020

Minimizing the Ruin Probability under the Sparre Andersen Model

In this paper, we consider the problem of minimizing the ruin probability of an insurance company in which the surplus process follows the Sparre Andersen model. Similar to Bai et al. \cite{bai2017optimal}, we recast this problem in a Markovian framework by adding another dimension representing the time elapsed since the last claim. After Markovization, We investigate the regularity properties of the value function and state the dynamic programming principle. Furthermore, we show that the value function is the unique constrained viscosity solution to the associated Hamilton-Jacobi-Bellman equation. It should be noted that there is no discount factor in our paper, which makes it tricky to prove the uniqueness. To overcome this difficulty, we construct the strict viscosity supersolution. Then instead of comparing the usual viscosity supersolution and subsolution, we compare the supersolution and the strict subsolution. Eventually we show that all viscosity subsolution is less than the supersolution.

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arXiv2019

On Optimal Dividend and Investment Strategy under Renewal Risk Models

In this paper we continue investigating the optimal dividend and investment problems under the Sparre Andersen model. More precisely, we assume that the claim frequency is a renewal process instead of a standard compound Poisson process, whence semi-Markovian. Building on our previous work \cite{BaiMa17}, where we established the dynamic programming principle via a {\it backward Markovization} procedure and proved that the value function is the unique {\it constrained} viscosity solution of the HJB equation, in this paper we focus on the construction of the optimal strategy. The main difficulties in this effort is two fold: the regularity of the viscosity solution to a non-local, nonlinear, and degenerate parabolic PDE on an unbounded domain, which seems to be new in its own right; and the well-posedness of the closed-loop stochastic system. By introducing an auxiliary PDE, we construct an $\e$-optimal strategy, and prove the well-posedness of the corresponding closed-loop system, via a "bootstrap" technique with the help of a Krylov estimate.

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arXiv2018

Optimal Singular Dividend Problem under the Sparre Anderson Model

Consider an insurance company for which the reserve process follows the Sparre Anderson model. In this paper, we study the optimal dividend problem for such a company as Bai, Ma and Xing [9] do. However, we remove the constant restriction on the dividend rates, i.e. the optimization problem is of singular type. In this case, the value function is no longer bounded and the associated HJB equation is a variational inequality involving a first order integro-differential operator and a gradient constraint. We use other techniques to prove the regularity properties for the value function and show that the value function is a constrained viscosity solution of the associated HJB equation. In addition, we show that the value function is the upper semi-continuous envelop of the supremum for a class of subsolutions.

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arXiv2016

Optimal Dividend and Investment Problems under Sparre Andersen Model

In this paper we study a class of optimal dividend and investment problems assuming that the underlying reserve process follows the Sparre Andersen model, that is, the claim frequency is a "renewal" process, rather than a standard compound Poisson process. The main feature of such problems is that the underlying reserve dynamics, even in its simplest form, is no longer Markovian. By using the backward Markovization technique we recast the problem in a Markovian framework with expanded dimension representing the time elapsed after the last claim, with which we investigate the regularity of the value function, and validate the dynamic programming principle. Furthermore, we show that the value function is the unique constrained viscosity solution} to the associated HJB equation on a cylindrical domain on which the problem is well-defined.

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arXiv2015

Stochastic differential equations driven by fractional Brownian motion and Poisson point process

In this paper, we study a class of stochastic differential equations with additive noise that contains a fractional Brownian motion (fBM) and a Poisson point process of class (QL). The differential equation of this kind is motivated by the reserve processes in a general insurance model, in which the long term dependence between the claim payment and the past history of liability becomes the main focus. We establish some new fractional calculus on the fractional Wiener-Poisson space, from which we define the weak solution of the SDE and prove its existence and uniqueness. Using an extended form of Krylov-type estimate for the combined noise of fBM and compound Poisson, we prove the existence of the strong solution, along the lines of Gyöngy and Pardoux (Probab. Theory Related Fields 94 (1993) 413-425). Our result in particular extends the one by Mishura and Nualart (Statist. Probab. Lett. 70 (2004) 253-261).

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arXiv2012

On non-trivial barrier solutions of the dividend problem for a diffusion under constant and proportional transaction costs

In Bai and Paulsen (SIAM J. Control optim. 48, 2010) the optimal dividend problem under transaction costs was analyzed for a rather general class of diffusion processes. It was divided into several subclasses, and for the majority of subclasses the optimal policy is a simple barrier policy; whenever the process hits an upper barrier $\bar{u}^*$, reduce it to $\bar{u}^*-ξ$ through a dividend payment. After transaction costs, the shareholder receives $kξ-K$. It was proved that a simple barrier strategy is not always optimal, and here these more difficult cases are solved. The optimal solutions are rather complicated, but interesting.

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