STOCHASTIC DIFFERENTIAL GAMES BETWEEN TWO INSURERS WITH GENERALIZED MEAN-VARIANCE PREMIUM PRINCIPLE

2018 ◽  
Vol 48 (1) ◽  
pp. 413-434 ◽  
Author(s):  
Shumin Chen ◽  
Hailiang Yang ◽  
Yan Zeng
Keyword(s):  
Differential Game ◽  
Game Problem ◽  
Stochastic Differential Game ◽  
Value Functions ◽  
Programming Technique ◽  
Bellman Equations ◽  
Equilibrium Strategies ◽  
Generalized Mean ◽  
Hamilton Jacobi Bellman ◽  
Mean Variance

AbstractWe study a stochastic differential game problem between two insurers, who invest in a financial market and adopt reinsurance to manage their claim risks. Supposing that their reinsurance premium rates are calculated according to the generalized mean-variance principle, we consider the competition between the two insurers as a non-zero sum stochastic differential game. Using dynamic programming technique, we derive a system of coupled Hamilton–Jacobi–Bellman equations and show the existence of equilibrium strategies. For an exponential utility maximizing game and a probability maximizing game, we obtain semi-explicit solutions for the equilibrium strategies and the equilibrium value functions, respectively. Finally, we provide some detailed comparative-static analyses on the equilibrium strategies and illustrate some economic insights.

scholarly journals The Dynamic Programming Method of Stochastic Differential Game for Functional Forward-Backward Stochastic System

2013 ◽  
Vol 2013 ◽  
pp. 1-14 ◽  
Author(s):  
Shaolin Ji ◽  
Chuanfeng Sun ◽  
Qingmeng Wei
Keyword(s):  
Dynamic Programming ◽  
Differential Game ◽  
Backward Stochastic Differential Equation ◽  
Transformation Method ◽  
Dynamic Programming Method ◽  
Stochastic Differential Game ◽  
Value Functions ◽  
Girsanov Transformation ◽  
Path Dependent ◽  
Hamilton Jacobi Bellman

This paper is devoted to a stochastic differential game (SDG) of decoupled functional forward-backward stochastic differential equation (FBSDE). For our SDG, the associated upper and lower value functions of the SDG are defined through the solution of controlled functional backward stochastic differential equations (BSDEs). Applying the Girsanov transformation method introduced by Buckdahn and Li (2008), the upper and the lower value functions are shown to be deterministic. We also generalize the Hamilton-Jacobi-Bellman-Isaacs (HJBI) equations to the path-dependent ones. By establishing the dynamic programming principal (DPP), we derive that the upper and the lower value functions are the viscosity solutions of the corresponding upper and the lower path-dependent HJBI equations, respectively.

scholarly journals Precommitted Investment Strategy versus Time-Consistent Investment Strategy for a General Risk Model with Diffusion

2014 ◽  
Vol 2014 ◽  
pp. 1-15
Author(s):  
Lidong Zhang ◽  
Ximin Rong ◽  
Ziping Du
Keyword(s):  
Risk Model ◽  
Theoretical Perspective ◽  
Optimal Strategies ◽  
Investment Strategy ◽  
Value Functions ◽  
Lagrange Method ◽  
Bellman Equations ◽  
Strategy Versus ◽  
Hamilton Jacobi Bellman ◽  
Mean Variance

We mainly study a general risk model and investigate the precommitted strategy and the time-consistent strategy under mean-variance criterion, respectively. A lagrange method is proposed to derive the precommitted investment strategy. Meanwhile from the game theoretical perspective, we find the time-consistent investment strategy by solving the extended Hamilton-Jacobi-Bellman equations. By comparing the precommitted strategy with the time-consistent strategy, we find that the company under the time-consistent strategy has to give up the better current utility in order to keep a consistent satisfaction over the whole time horizon. Furthermore, we theoretically and numerically provide the effect of the parameters on these two optimal strategies and the corresponding value functions.

scholarly journals Optimal Execution considering Trading Signal and Execution Risk Simultaneously

2021 ◽  
Vol 2021 ◽  
pp. 1-12
Author(s):  
Yuan Cheng ◽  
Lan Wu
Keyword(s):  
Kernel Function ◽  
Price Impact ◽  
Value Functions ◽  
Feedback Form ◽  
Bellman Equations ◽  
Impact Function ◽  
Optimal Execution ◽  
Dirac Function ◽  
Hamilton Jacobi Bellman ◽  
Optimal Execution Problem

In this paper, we study the optimal execution problem by considering the trading signal and the transaction risk simultaneously. We propose an optimal execution problem by taking into account the trading signal and the execution risk with the associated decay kernel function and the transient price impact function being of generalized forms. In particular, we solve the stochastic optimal control problems under the assumptions that the decay kernel function is the Dirac function and the transient price function is a linear function. We give the optimal executing strategies in state-feedback form and the Hamilton‐Jacobi‐Bellman equations that the corresponding value functions satisfy in the cases of a constant execution risk and a linear execution risk. We also demonstrate that our results can recover previous results when the process of the trading signal degenerates.

scholarly journals Solution of Hamilton-Jacobi-Bellman Equation in Optimal Reinsurance Strategy under Dynamic VaR Constraint

2019 ◽  
Vol 2019 ◽  
pp. 1-7 ◽  
Author(s):  
Yuzhen Wen ◽  
Chuancun Yin
Keyword(s):  
Bellman Equation ◽  
Closed Form Expression ◽  
Proportional Reinsurance ◽  
Form Expression ◽  
Optimal Reinsurance ◽  
Surplus Process ◽  
Hamilton Jacobi Bellman Equation ◽  
Generalized Mean ◽  
Hamilton Jacobi Bellman ◽  
Mean Variance

This paper analyzes the optimal reinsurance strategy for insurers with a generalized mean-variance premium principle. The surplus process of the insurer is described by the diffusion model which is an approximation of the classical Cramér-Lunderberg model. We assume the dynamic VaR constraints for proportional reinsurance. We obtain the closed form expression of the optimal reinsurance strategy and corresponding survival probability under proportional reinsurance.

scholarly journals Equilibrium Investment Strategy for DC Pension Plan with Inflation and Stochastic Income under Heston’s SV Model

2016 ◽  
Vol 2016 ◽  
pp. 1-18 ◽  
Author(s):  
Jingyun Sun ◽  
Zhongfei Li ◽  
Yongwu Li
Keyword(s):  
Pension Plan ◽  
Investment Strategy ◽  
Investment Strategies ◽  
Value Functions ◽  
Defined Contribution ◽  
Hjb Equations ◽  
Plan Member ◽  
The Mean ◽  
Hamilton Jacobi Bellman ◽  
Mean Variance

We consider a portfolio selection problem for a defined contribution (DC) pension plan under the mean-variance criteria. We take into account the inflation risk and assume that the salary income process of the pension plan member is stochastic. Furthermore, the financial market consists of a risk-free asset, an inflation-linked bond, and a risky asset with Heston’s stochastic volatility (SV). Under the framework of game theory, we derive two extended Hamilton-Jacobi-Bellman (HJB) equations systems and give the corresponding verification theorems in both the periods of accumulation and distribution of the DC pension plan. The explicit expressions of the equilibrium investment strategies, corresponding equilibrium value functions, and the efficient frontiers are also obtained. Finally, some numerical simulations and sensitivity analysis are presented to verify our theoretical results.

scholarly journals Partial Information Stochastic Differential Games for Backward Stochastic Systems Driven by Lévy Processes

2020 ◽  
Vol 2020 ◽  
pp. 1-7
Author(s):  
Fu Zhang ◽  
QingXin Meng ◽  
MaoNing Tang
Keyword(s):  
Differential Game ◽  
Stochastic Systems ◽  
Partial Information ◽  
Backward Stochastic Differential Equation ◽  
Game Problem ◽  
Stochastic Differential Game ◽  
Linear Quadratic ◽  
Independent Brownian Motion ◽  
Teugels Martingales ◽  
Zero Sum

In this paper, we consider a partial information two-person zero-sum stochastic differential game problem, where the system is governed by a backward stochastic differential equation driven by Teugels martingales and an independent Brownian motion. A sufficient condition and a necessary one for the existence of the saddle point for the game are proved. As an application, a linear quadratic stochastic differential game problem is discussed.

scholarly journals Utility Indifference Option Pricing Model with a Non-Constant Risk-Aversion under Transaction Costs and Its Numerical Approximation

2021 ◽  
Vol 14 (9) ◽  
pp. 399
Author(s):  
Pedro Pólvora ◽  
Daniel Ševčovič
Keyword(s):  
Risk Aversion ◽  
Nonlinear Partial Differential Equation ◽  
Option Price ◽  
Transformation Method ◽  
Value Functions ◽  
Hjb Equations ◽  
Bellman Equations ◽  
Option Pricing Model ◽  
Utility Indifference ◽  
Hamilton Jacobi Bellman

Our goal is to analyze the system of Hamilton-Jacobi-Bellman equations arising in derivative securities pricing models. The European style of an option price is constructed as a difference of the certainty equivalents to the value functions solving the system of HJB equations. We introduce the transformation method for solving the penalized nonlinear partial differential equation. The transformed equation involves possibly non-constant the risk aversion function containing the negative ratio between the second and first derivatives of the utility function. Using comparison principles we derive useful bounds on the option price. We also propose a finite difference numerical discretization scheme with some computational examples.

scholarly journals Nonzero-Sum Stochastic Differential Game between Controller and Stopper for Jump Diffusions

2013 ◽  
Vol 2013 ◽  
pp. 1-7 ◽  
Author(s):  
Yan Wang ◽  
Aimin Song ◽  
Cheng-De Zheng ◽  
Enmin Feng
Keyword(s):  
Dynamic Programming ◽  
Differential Game ◽  
Control Process ◽  
Programming Approach ◽  
Stochastic Differential Game ◽  
Jump Diffusions ◽  
Verification Theorem ◽  
Dynamic Programming Approach ◽  
Hamilton Jacobi Bellman ◽  
Markov Control

We consider a nonzero-sum stochastic differential game which involves two players, a controller and a stopper. The controller chooses a control process, and the stopper selects the stopping rule which halts the game. This game is studied in a jump diffusions setting within Markov control limit. By a dynamic programming approach, we give a verification theorem in terms of variational inequality-Hamilton-Jacobi-Bellman (VIHJB) equations for the solutions of the game. Furthermore, we apply the verification theorem to characterize Nash equilibrium of the game in a specific example.

Sign in / Sign up

Export Citation Format

Share Document