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.

Nested variational inequalities and related optimal starting–stopping problems

1992 ◽  
Vol 29 (01) ◽  
pp. 104-115 ◽  
Author(s):  
M. Sun
Keyword(s):  
Differential Equation ◽  
Variational Inequalities ◽  
Stochastic Differential Equation ◽  
Differential Game ◽  
Diffusion Model ◽  
Value Function ◽  
Optimal Strategies ◽  
Stochastic Differential Game ◽  
The Value Function

This paper introduces several versions of starting-stopping problem for the diffusion model defined in terms of a stochastic differential equation. The problem could be regarded as a stochastic differential game in which the player can only decide when to start the game and when to quit the game in order to maximize his fortune. Nested variational inequalities arise in studying such a problem, with which we are able to characterize the value function and to obtain optimal strategies.

ON THE SINGULARITIES OF AN IMPULSIVE DIFFERENTIAL GAME ARISING IN MATHEMATICAL FINANCE

2006 ◽  
Vol 08 (02) ◽  
pp. 219-229 ◽  
Author(s):  
PIERRE BERNHARD
Keyword(s):  
Option Pricing ◽  
State Space ◽  
Differential Game ◽  
Impulse Control ◽  
Value Function ◽  
Mathematical Finance ◽  
The State

We investigate an impulse control differential game arising in a problem of option pricing in mathematical finance. In a previous paper, it was shown that its Value function in ℝ3 could be described as a pair of functions affine in one of the variables, joined on a 2D manifold. Depending on the regions of the state space, this manifold is either a dispersal one, an equivocal one or a 2D focal manifold. A pair of PDE's were derived for the focal part. Here we show that irrespective of the nature of this manifold, it has to satisfy this same set of PDE's.

Nested variational inequalities and related optimal starting–stopping problems

1992 ◽  
Vol 29 (1) ◽  
pp. 104-115 ◽  
Author(s):  
M. Sun
Keyword(s):  
Differential Equation ◽  
Variational Inequalities ◽  
Stochastic Differential Equation ◽  
Differential Game ◽  
Diffusion Model ◽  
Value Function ◽  
Optimal Strategies ◽  
Stochastic Differential Game ◽  
The Value Function

This paper introduces several versions of starting-stopping problem for the diffusion model defined in terms of a stochastic differential equation. The problem could be regarded as a stochastic differential game in which the player can only decide when to start the game and when to quit the game in order to maximize his fortune. Nested variational inequalities arise in studying such a problem, with which we are able to characterize the value function and to obtain optimal strategies.

scholarly journals Dynkin game under g-expectation in continuous time

2020 ◽  
Vol 9 (2) ◽  
pp. 459-470
Author(s):  
Helin Wu ◽  
Yong Ren ◽  
Feng Hu
Keyword(s):  
Differential Equation ◽  
Stochastic Differential Equation ◽  
Continuous Time ◽  
Value Function ◽  
Backward Stochastic Differential Equation ◽  
Saddle Points ◽  
Value Functions ◽  
Dynkin Game ◽  
The Value Function

Abstract In this paper, we investigate some kind of Dynkin game under g-expectation induced by backward stochastic differential equation (short for BSDE). The lower and upper value functions $$\underline{V}_t=ess\sup \nolimits _{\tau \in {\mathcal {T}_t}} ess\inf \nolimits _{\sigma \in {\mathcal {T}_t}}\mathcal {E}^g_t[R(\tau ,\sigma )]$$ V ̲ t = e s s sup τ ∈ T t e s s inf σ ∈ T t E t g [ R ( τ , σ ) ] and $$\overline{V}_t=ess\inf \nolimits _{\sigma \in {\mathcal {T}_t}} ess\sup \nolimits _{\tau \in {\mathcal {T}_t}}\mathcal {E}^g_t[R(\tau ,\sigma )]$$ V ¯ t = e s s inf σ ∈ T t e s s sup τ ∈ T t E t g [ R ( τ , σ ) ] are defined, respectively. Under some suitable assumptions, a pair of saddle points is obtained and the value function of Dynkin game $$V(t)=\underline{V}_t=\overline{V}_t$$ V ( t ) = V ̲ t = V ¯ t follows. Furthermore, we also consider the constrained case of Dynkin game.

ON THE CREDIT RISK OF SECURED LOANS WITH MAXIMUM LOAN-TO-VALUE COVENANTS

2014 ◽  
Vol 17 (08) ◽  
pp. 1450055
Author(s):  
Fabian Astic ◽  
Agnès Tourin
Keyword(s):  
Differential Equation ◽  
Credit Risk ◽  
Value Function ◽  
Expected Loss ◽  
Trade Off ◽  
Closed Form Solutions ◽  
Control Techniques ◽  
Lending Policy ◽  
Lending Decisions ◽  
The Value Function

We propose a framework for analyzing the credit risk of secured loans with maximum loan-to-value covenants. Here, we do not assume that the collateral can be liquidated as soon as the maximum loan-to-value is breached. Closed-form solutions for the expected loss are obtained for nonrevolving loans. In the revolving case, we introduce a minimization problem with an objective function parameterized by a risk reluctance coefficient, capturing the trade-off between minimizing the expected loss incurred in the event of liquidation and maximizing the interest gain. Using stochastic control techniques, we derive the partial integro-differential equation satisfied by the value function, and solve it numerically with a finite difference scheme. The experimental results and their comparison with a standard loan-to-value-based lending policy suggest that stricter lending decisions would benefit the lender.

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.

Geometric Asymptotic Approximation of Value Functions

2009 ◽  
Vol 9 (1) ◽  
Author(s):  
Axel Anderson
Keyword(s):  
Value Function ◽  
Payoff Function ◽  
The State ◽  
Second Derivative ◽  
Value Functions ◽  
State Variable ◽  
Specific Formula ◽  
Geometric Term ◽  
Dynamic Stochastic ◽  
The Value Function

This paper characterizes the behavior of value functions in dynamic stochastic discounted programming models near fixed points of the state space. When the second derivative of the flow payoff function is bounded, the value function is proportional to a linear function plus geometric term. A specific formula for the exponent of this geometric term is provided. This exponent continuously falls in the rate of patience.If the state variable is a martingale, the second derivative of the value function is unbounded. If the state variable is instead a strict local submartingale, then the same holds for the first derivative of the value function. Thus, the proposed approximation is more accurate than Taylor series approximation.The approximation result is used to characterize locally optimal policies in several fundamental economic problems.

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.

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