scholarly journals Markov approximations of nonzero-sum differential games

2020 ◽  
Vol 30 (1) ◽  
pp. 3-17
Author(s):  
Yu.V. Averboukh
Keyword(s):  
Differential Games ◽  
Differential Inclusion ◽  
Continuous Time ◽  
Value Function ◽  
Dynamic Game ◽  
Approximate Solutions ◽  
Time Dynamic ◽  
Markov Game ◽  
Approximate Nash Equilibrium ◽  
The Value Function

The paper is concerned with approximate solutions of nonzero-sum differential games. An approximate Nash equilibrium can be designed by a given solution of an auxiliary continuous-time dynamic game. We consider the case when dynamics is determined by a Markov chain. For this game the value function is determined by an ordinary differential inclusion. Thus, we obtain a construction of approximate equilibria with the players' outcome close to the solution of the differential inclusion. Additionally, we propose a way of designing a continuous-time Markov game approximating the original dynamics.

One-Player Differential Games

2017 ◽  
Author(s):  
João P. Hespanha
Keyword(s):  
Dynamical System ◽  
Differential Games ◽  
Continuous Time ◽  
Information Structure ◽  
Dynamic Games ◽  
State Feedback ◽  
Open Loop ◽  
Linear Quadratic ◽  
Time Dynamic ◽  
Termination Time

This chapter focuses on one-player continuous time dynamic games, that is, the optimal control of a continuous time dynamical system. It begins by considering a one-player continuous time differential game in which the (only) player wants to minimize either using an open-loop policy or a state-feedback policy. It then discusses continuous time cost-to-go, with the following conclusion: regardless of the information structure considered (open loop, state feedback, or other), it is not possible to obtain a cost lower than cost-to-go. It also explores continuous time dynamic programming, linear quadratic dynamic games, and differential games with variable termination time before concluding with a practice exercise and the corresponding solution.

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.

scholarly journals Basel III and the Net Stable Funding Ratio

2013 ◽  
Vol 2013 ◽  
pp. 1-20 ◽  
Author(s):  
F. Gideon ◽  
Mark A. Petersen ◽  
Janine Mukuddem-Petersen ◽  
LNP Hlatshwayo
Keyword(s):  
Optimal Control ◽  
Optimal Control Problem ◽  
Continuous Time ◽  
Value Function ◽  
Analytic Solution ◽  
Market Liquidity ◽  
Basel Iii ◽  
Quantitative Manner ◽  
The Value Function

We validate the new Basel liquidity standards as encapsulated by the net stable funding ratio in a quantitative manner. In this regard, we consider the dynamics of inverse net stable funding ratio as a measure to quantify the bank’s prospects for a stable funding over a period of a year. In essence, this justifies how Basel III liquidity standards can be effectively implemented in mitigating liquidity problems. We also discuss various classes of available stable funding and required stable funding. Furthermore, we discuss an optimal control problem for a continuous-time inverse net stable funding ratio. In particular, we make optimal choices for the inverse net stable funding targets in order to formulate its cost. This is normally done by obtaining analytic solution of the value function. Finally, we provide a numerical example for the dynamics of the inverse net stable funding ratio to identify trends in which banks behavior convey forward looking information on long-term market liquidity developments.

scholarly journals When Inaccuracies in Value Functions Do Not Propagate on Optima and Equilibria

Mathematics ◽  
2020 ◽  
Vol 8 (7) ◽  
pp. 1109 ◽  
Author(s):  
Agnieszka Wiszniewska-Matyszkiel ◽  
Rajani Singh
Keyword(s):  
Dynamic Optimization ◽  
Dynamic Games ◽  
Value Function ◽  
Optimization Problems ◽  
A Priori ◽  
Value Functions ◽  
Feedback Controls ◽  
Time Dynamic ◽  
Coupled Equations ◽  
The Value Function

We study general classes of discrete time dynamic optimization problems and dynamic games with feedback controls. In such problems, the solution is usually found by using the Bellman or Hamilton–Jacobi–Bellman equation for the value function in the case of dynamic optimization and a set of such coupled equations for dynamic games, which is not always possible accurately. We derive general rules stating what kind of errors in the calculation or computation of the value function do not result in errors in calculation or computation of an optimal control or a Nash equilibrium along the corresponding trajectory. This general result concerns not only errors resulting from using numerical methods but also errors resulting from some preliminary assumptions related to replacing the actual value functions by some a priori assumed constraints for them on certain subsets. We illustrate the results by a motivating example of the Fish Wars, with singularities in payoffs.

scholarly journals OPTIMAL LIQUIDATION UNDER STOCHASTIC PRICE IMPACT

2019 ◽  
Vol 22 (02) ◽  
pp. 1850059 ◽  
Author(s):  
WESTON BARGER ◽  
MATTHEW LORIG
Keyword(s):  
Continuous Time ◽  
Value Function ◽  
Price Impact ◽  
Trading Strategies ◽  
Impact Model ◽  
Optimal Liquidation ◽  
Special Cases ◽  
Time Price ◽  
Hamilton Jacobi Bellman ◽  
The Value Function

We assume a continuous-time price impact model similar to that of Almgren–Chriss but with the added assumption that the price impact parameters are stochastic processes modeled as correlated scalar Markov diffusions. In this setting, we develop trading strategies for a trader who desires to liquidate his inventory but faces price impact as a result of his trading. For a fixed trading horizon, we perform coefficient expansion on the Hamilton–Jacobi–Bellman (HJB) equation associated with the trader’s value function. The coefficient expansion yields a sequence of partial differential equations that we solve to give closed-form approximations to the value function and optimal liquidation strategy. We examine some special cases of the optimal liquidation problem and give financial interpretations of the approximate liquidation strategies in these cases. Finally, we provide numerical examples to demonstrate the effectiveness of the approximations.

Optimal stopping with random intervention times

2002 ◽  
Vol 34 (01) ◽  
pp. 141-157 ◽  
Author(s):  
Paul Dupuis ◽  
Hui Wang
Keyword(s):  
Optimal Stopping ◽  
Continuous Time ◽  
Value Function ◽  
Value Functions ◽  
Optimal Exercise Boundary ◽  
Exercise Boundary ◽  
Time Optimal ◽  
Optimal Stopping Problems ◽  
Necessary And Sufficient ◽  
The Value Function

We consider a class of optimal stopping problems where the ability to stop depends on an exogenous Poisson signal process - we can only stop at the Poisson jump times. Even though the time variable in these problems has a discrete aspect, a variational inequality can be obtained by considering an underlying continuous-time structure. Depending on whether stopping is allowed at t = 0, the value function exhibits different properties across the optimal exercise boundary. Indeed, the value function is only 𝒞 0 across the optimal boundary when stopping is allowed at t = 0 and 𝒞 2 otherwise, both contradicting the usual 𝒞 1 smoothness that is necessary and sufficient for the application of the principle of smooth fit. Also discussed is an equivalent stochastic control formulation for these stopping problems. Finally, we derive the asymptotic behaviour of the value functions and optimal exercise boundaries as the intensity of the Poisson process goes to infinity or, roughly speaking, as the problems converge to the classical continuous-time optimal stopping problems.

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