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Optimal Claim-Dependent Proportional Reinsurance Under a Self-Exciting Claim Model

JOURNAL OF OPTIMIZATION THEORY AND APPLICATIONS 2024年第3期201卷 1229-1255页

作者： Wu, Fan Shen, Yang Zhang, Xin Ding, Kai Southeast Univ Sch Math Nanjing 211189 Jiangsu Peoples R China Univ New South Wales Sch Risk & Actuarial Studies Sydney NSW 2052 Australia

This paper investigates an optimal reinsurance problem for an insurance company with self-exciting claims, where the insurer's historical claims affect the claim intensity itself. We focus on a claim-dependent proportional reinsurance contact, where the term "claim-dependent" signifies that the insurer's risk retention ratio is allowed to depend on claim size. The insurer aims to maximize the expected utility of terminal wealth. By utilizing the dynamic programming principle and verification theorem, we obtain the optimal reinsurance strategy and corresponding value function in closed-form from the Hamilton-Jacobi-Bellman equation under an exponential utility function. We show that the claim-dependent proportional reinsurance is optimal among all types of reinsurance under the exponential utility maximization criterion. In addition, we present several analytical properties and numerical examples of the derived optimal strategy and provide economic insights through analytical and numerical analyses. In particular, we show the optimal claim-dependent proportional reinsurance can be considered as a continuous approximation of the step-wise risk sharing rule between the insurer and the reinsurer.

关键词： Optimal reinsurance problem Self-exciting claim Claim-dependent proportional reinsurance dynamic programming principle Verification theorem

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A stochastic target problem for branching diffusion processes

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STOCHASTIC PROCESSES AND THEIR APPLICATIONS 2024年 170卷

作者： Kharroubi, Idris Ocello, Antonio Sorbonne Univ LPSM UMR CNRS 8001 Paris France Univ Paris Cite Paris France

We consider an optimal stochastic target problem for branching diffusion processes. This problem consists in finding the minimal condition for which a control allows the underlying branching process to reach a target set at a finite terminal time for each of its branches. This problem is motivated by an example from fintech where we look for the super-replication price of options on blockchain-based cryptocurrencies. We first state a dynamic programming principle for the value function of the stochastic target problem. Next, we show that the value function can be simplified into a novel function with the use of a finite-dimensional argument through a concept known as the branching property. Under wide conditions, this last function is shown to be the unique viscosity solution to an HJB variational inequality.

关键词： Stochastic target control Fintech Cryptocurrencies options Branching diffusion process dynamic programming principle Hamilton-Jacobi-Bellman equation Viscosity solution

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Uniqueness of lower semicontinuous viscosity solutions for the minimum time problem

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SIAM JOURNAL ON CONTROL AND OPTIMIZATION 2000年第2期38卷 470-481页

作者： Alvarez, O Koike, S Nakayama, I Univ Rouen UPRESA 60 85 F-76821 Mt St Aignan France Saitama Univ Dept Math Urawa Saitama 3388570 Japan

We obtain the uniqueness of lower semicontinuous (LSC) viscosity solutions of the transformed minimum time problem assuming that they converge to zero on a "reachable" part of the target in appropriate directions. We present a counter-example which shows that the uniqueness does not hold without this convergence assumption. It was shown by Soravia that the uniqueness of LSC viscosity solutions having a "subsolution property" on the target holds. In order to verify this subsolution property, we show that the dynamic programming principle (DPP) holds inside for any LSC viscosity solutions. In order to obtain the DPP, we prepare appropriate approximate PDEs derived through Barles' inf-convolution and its variant.

关键词： semicontinuous viscosity solutions dynamic programming principle minimum time problem

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Differential games of inf-sup type and Isaacs equations

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APPLIED MATHEMATICS AND OPTIMIZATION 2005年第1期52卷 1-22页

作者： Kaise, H Sheu, SJ Nagoya Univ Grad Sch Informat Sci Chikusa Ku Nagoya Aichi 4648601 Japan Acad Sinica Inst Math Taipei 11529 Taiwan

Motivated by the work of Fleming [6], we provide a general framework to associate inf-sup type values with the Isaacs equations. We show that upper and lower bounds for the generators of inf-sup type are upper and lower Hamiltonians, respectively. In particular, the lower (resp. upper) bound corresponds to the progressive (resp. strictly progressive) strategy. By the dynamic programming principle and identification of the generator, we can prove that the inf-sup type game is characterized as the unique viscosity solution of the Isaacs equation. We also discuss the Isaacs equation with a Hamiltonian of a convex combination between the lower and upper Hamiltonians.

关键词： differential game dynamic programming principle viscosity solutions infinitesimal generators strategy classes

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Mathematical analysis and validation of an exactly solvable model for upstream migration of fish schools in one-dimensional rivers

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MATHEMATICAL BIOSCIENCES 2016年第0期281卷 139-148页

作者： Yoshioka, Hidekazu Shimane Univ Fac Life & Environm Sci Nishikawatsu Cho 1060 Matsue Shimane 6908504 Japan

Upstream migration of fish schools in 1-D rivers as an optimal control problem is formulated where their swimming velocity and the horizontal oblateness are taken as control variables. The objective function to be maximized through a migration process consists of the biological and ecological profit to be gained at the upstream-end of a river, energetic cost of swimming against the flow, and conceptual cost of forming a school. Under simplified conditions where the flow is uniform in both space and time and the profit to be gained at the goal of migration is sufficiently large, the optimal control variables are determined from a system of algebraic equations that can be solved in a cascading manner. Mathematical analysis of the system reveals that the optimal controls are uniquely found and the model is exactly solvable under certain conditions on the functions and parameters, which turn out to be realistic and actually satisfied in experimental fish migration. Identification results of the functional shapes of the functions and the parameters with experimentally observed data of swimming schools of Plecoglossus altivelis (Ayu) validate the present mathematical model from both qualitative and quantitative viewpoints. The present model thus turns out to be consistent with the reality, showing its potential applicability to assessing fish migration in applications. (C) 2016 Elsevier Inc. All rights reserved.

关键词： Fish school Upstream migration Optimal control problem dynamic programming principle Hamilton-jacobi-Bellman equation

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Relationship Between General MP and DPP for the Stochastic Recursive Optimal Control Problem with Jumps

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Journal of Systems Science & Complexity 2024年第6期37卷 2466-2486页

作者： WANG Bin SHI Jingtao School of Mathematics Shandong UniversityJinan 250100China

This paper is concerned with the relationship between general maximum principle and dynamic programming principle for the stochastic recursive optimal control problem with jumps,where the control domain is not necessarily *** among the adjoint processes,the generalized Hamiltonian function and the value function are proven,under the assumption of a smooth value function and within the framework of viscosity solutions,*** examples are given to illustrate the theoretical results.

关键词： Backward stochastic differential equation with jumps dynamic programming principle maximum principle recursive optimal control viscosity solution

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OPTIMAL STOCHASTIC CONTROL WITH RECURSIVE COST FUNCTIONALS OF STOCHASTIC DIFFERENTIAL SYSTEMS REFLECTED IN A DOMAIN

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ESAIM-CONTROL OPTIMISATION AND CALCULUS OF VARIATIONS 2015年第4期21卷 1150-1177页

作者： Li, Juan Tang, Shanjian Shandong Univ Sch Math & Stat Weihai 264200 Weihai Peoples R China Fudan Univ Sch Math Sci Inst Math Shanghai 200433 Peoples R China Fudan Univ Sch Math Sci Dept Finance & Control Sci Shanghai 200433 Peoples R China

The paper is concerned with optimal control of a stochastic differential system reflected in a domain. The cost functional is implicitly defined via a generalized backward stochastic differential equation developed by Pardoux and Zhang [Probab. Theory Relat. Fields 110 (1998) 535-558]. The value function is shown to be the unique viscosity solution to the associated Hamilton-Jacobi-Bellman equation, which is a fully nonlinear parabolic partial differential equation with a nonlinear Neumann boundary condition. The proof requires new estimates for the reflected stochastic differential system.

关键词： Hamilton-Jacobi-Bellman equation nonlinear Neumann boundary value function backward stochastic differential equations dynamic programming principle viscosity solution

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A stochastic control model of investment, production, and consumption on a finite horizon

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MATHEMATICAL METHODS IN THE APPLIED SCIENCES 2015年第6期38卷 1070-1080页

作者： Han, Xiaoru Yi, Fahuai Foshan Univ Dept Math Foshan 528000 Guangdong Peoples R China S China Normal Univ Sch Math Sci Guangzhou 510631 Guangdong Peoples R China

In this paper, we consider a stochastic control problem on a finite time horizon. The unit price of capital obeys a logarithmic Brownian motion, and the income from production is also subject to the random Brownian fluctuations. The goal is to choose optimal investment and consumption policies to maximize the finite horizon expected discounted hyperbolic absolute risk aversion utility of consumption. A dynamic programming principle is used to derive a time-dependent Hamilton-Jacobi-Bellman equation. The Leray-Schauder fixed point theorem is used to obtain existence of solution of the HJB equation. At last, we derive the optimal investment and consumption policies by the verification theorem. The main contribution in this paper is the use of PDE technique to the finite time problem for obtaining optimal polices. Copyright (c) 2014 John Wiley & Sons, Ltd.

关键词： PDE technique dynamic programming principle optimal policies

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OPTIMAL CONTROL PROBLEMS OF FULLY COUPLED FBSDEs AND VISCOSITY SOLUTIONS OF HAMILTON-JACOBI-BELLMAN EQUATIONS

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SIAM JOURNAL ON CONTROL AND OPTIMIZATION 2014年第3期52卷 1622-1662页

作者： Li, Juan Wei, Qingmeng Shandong Univ Sch Math & Stat Weihai 264209 Peoples R China NE Normal Univ Sch Math & Stat Changchun 130024 Peoples R China Shandong Univ Sch Math Jinan 250100 Peoples R China

In this paper we study stochastic optimal control problems of fully coupled forward-backward stochastic differential equations (FBSDEs). The recursive cost functionals are defined by controlled fully coupled FBSDEs. We use a new method to prove that the value functions are deterministic, satisfy the dynamic programming principle, and are viscosity solutions to the associated generalized Hamilton-Jacobi-Bellman (HJB) equations. For this we generalize the notion of stochastic backward semigroup introduced by Peng Topics on Stochastic Analysis, Science Press, Beijing, 1997, pp. 85-138. We emphasize that when sigma depends on the second component of the solution (Y, Z) of the BSDE it makes the stochastic control much more complicated and has as a consequence that the associated HJB equation is combined with an algebraic equation. We prove that the algebraic equation has a unique solution, and moreover, we also give the representation for this solution. On the other hand, we prove a new local existence and uniqueness result for fully coupled FBSDEs when the Lipschitz constant of sigma with respect to z is sufficiently small. We also establish a generalized comparison theorem for such fully coupled FBSDEs.

关键词： fully coupled FBSDEs value functions stochastic backward semigroup dynamic programming principle viscosity solution

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Stochastic differential games with competing Brownian particles and related Isaacs' equations

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OPTIMAL CONTROL APPLICATIONS & METHODS 2018年第2期39卷 519-536页

作者： Feng, Xinwei Shandong Univ Sch Math Jinan 250100 Shandong Peoples R China

In this paper, we study zero-sum two-player stochastic differential games in which the state equations are competing Brownian particles and the cost functional is defined by generalized backward stochastic differential equations with more than one increasing process. After we study the regularity of competing Brownian particles, we establish the dynamic programming principle for the upper and lower value functions and show that these are the unique viscosity solution of the associated upper and lower Isaacs' equations, which are fully nonlinear parabolic partial differential equations with nonlinear Neumann boundary conditions.

关键词： competing Brownian particles dynamic programming principle generalized backward stochastic differential equations partial differential equations stochastic differential games

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