scholarly journals Stochastic differential games and viscosity solutions of Isaacs equations

1988 ◽  
Vol 110 ◽  
pp. 163-184 ◽  
Author(s):  
Makiko Nisio
Keyword(s):  
Differential Games ◽  
Differential Game ◽  
Value Function ◽  
Nonlinear Partial Differential Equations ◽  
Nonlinear Function ◽  
Stochastic Differential Games ◽  
Order Partial Differential Equation ◽  
Partial Differential ◽  
Representation Formulas ◽  
The Value Function

Recently P. L. Lions has demonstrated the connection between the value function of stochastic optimal control and a viscosity solution of Hamilton-Jacobi-Bellman equation [cf. 10, 11, 12]. The purpose of this paper is to extend partially his results to stochastic differential games, where two players conflict each other. If the value function of stochatic differential game is smooth enough, then it satisfies a second order partial differential equation with max-min or min-max type nonlinearity, called Isaacs equation [cf. 5]. Since we can write a nonlinear function as min-max of appropriate affine functions, under some mild conditions, the stochastic differential game theory provides some convenient representation formulas for solutions of nonlinear partial differential equations [cf. 1, 2, 3].

Verification Theorems for Optimal Feedback Strategies in Differential Games

2003 ◽  
Vol 05 (02) ◽  
pp. 167-189 ◽  
Author(s):  
Ştefan Mirică
Keyword(s):  
Differential Games ◽  
Differential Game ◽  
Value Function ◽  
Generalized Derivatives ◽  
Differential Inequalities ◽  
Value Functions ◽  
Optimal Solutions ◽  
Optimal Feedback ◽  
Feedback Strategies ◽  
The Value Function

We give complete proofs to the verification theorems announced recently by the author for the "pairs of relatively optimal feedback strategies" of an autonomous differential game. These concepts are considered to describe the possibly optimal solutions of a differential game while the corresponding value functions are used as "instruments" for proving the relative optimality and also as "auxiliary characteristics" of the differential game. The 6 verification theorems in the paper are proved under different regularity assumptions accompanied by suitable differential inequalities verified by the generalized derivatives, mainly of contingent type, of the value function.

scholarly journals Viscosity Solution of Mean-Variance Portfolio Selection of a Jump Markov Process with No-Shorting Constraints

2016 ◽  
Vol 2016 ◽  
pp. 1-14 ◽  
Author(s):  
Moussa Kounta
Keyword(s):  
Differential Equation ◽  
Partial Differential Equation ◽  
Portfolio Selection ◽  
Viscosity Solution ◽  
Value Function ◽  
Auxiliary Problem ◽  
Partial Differential ◽  
Mean Variance Portfolio ◽  
Mean Variance ◽  
The Value Function

We consider the so-called mean-variance portfolio selection problem in continuous time under the constraint that the short-selling of stocks is prohibited where all the market coefficients are random processes. In this situation the Hamilton-Jacobi-Bellman (HJB) equation of the value function of the auxiliary problem becomes a coupled system of backward stochastic partial differential equation. In fact, the value functionVoften does not have the smoothness properties needed to interpret it as a solution to the dynamic programming partial differential equation in the usual (classical) sense; however, in such casesVcan be interpreted as a viscosity solution. Here we show the unicity of the viscosity solution and we see that the optimal and the value functions are piecewise linear functions based on some Riccati differential equations. In particular we solve the open problem posed by Li and Zhou and Zhou and Yin.

THE VALUE FUNCTION OF A DIFFERENTIAL GAME WITH SIMPLE MOTIONS AND AN INTEGRO-TERMINAL COST

2018 ◽  
pp. 877-890
Author(s):  
Lyubov Gennad’evna Shagalova
Keyword(s):  
Differential Equation ◽  
State Space ◽  
Differential Game ◽  
Value Function ◽  
Piecewise Linear ◽  
The State ◽  
The Value Function

An antagonistic positional differential game of two persons is considered. The dynamics of the system is described by a differential equation with simple motions, and the payoff functional is integro-terminal. For the case when the terminal function and the Hamiltonian are piecewise linear, and the dimension of the state space is two, a finite algorithm for the exact construction of the value function is proposed.

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