scholarly journals Constrained BSDEs representation of the value function in optimal control of pure jump Markov processes

2017 ◽  
Vol 127 (5) ◽  
pp. 1441-1474 ◽  
Author(s):  
Elena Bandini ◽  
Marco Fuhrman
Keyword(s):  
Optimal Control ◽  
Markov Processes ◽  
Value Function ◽  
The Value Function ◽  
Jump Markov Processes

scholarly journals Optimal control of piecewise deterministic Markov processes: a BSDE representation of the value function

2018 ◽  
Vol 24 (1) ◽  
pp. 311-354 ◽  
Author(s):  
Elena Bandini
Keyword(s):  
Optimal Control ◽  
Control Problem ◽  
Markov Processes ◽  
Value Function ◽  
Backward Stochastic Differential Equation ◽  
Infinite Horizon ◽  
Random Measure ◽  
Uniqueness Results ◽  
Piecewise Deterministic Markov Processes ◽  
The Value Function

We consider an infinite-horizon discounted optimal control problem for piecewise deterministic Markov processes, where a piecewise open-loop control acts continuously on the jump dynamics and on the deterministic flow. For this class of control problems, the value function can in general be characterized as the unique viscosity solution to the corresponding Hamilton−Jacobi−Bellman equation. We prove that the value function can be represented by means of a backward stochastic differential equation (BSDE) on infinite horizon, driven by a random measure and with a sign constraint on its martingale part, for which we give existence and uniqueness results. This probabilistic representation is known as nonlinear Feynman−Kac formula. Finally we show that the constrained BSDE is related to an auxiliary dominated control problem, whose value function coincides with the value function of the original non-dominated control problem.

scholarly journals Hamilton-Jacobi equations for optimal control on networks with entry or exit costs

2019 ◽  
Vol 25 ◽  
pp. 15 ◽  
Author(s):  
Manh Khang Dao
Keyword(s):  
Optimal Control ◽  
Partial Differential Equations ◽  
Recent Work ◽  
Comparison Principle ◽  
Value Function ◽  
Exit Costs ◽  
Theory Of Optimal Control ◽  
New Feature ◽  
Jacobi Equations ◽  
The Value Function

We consider an optimal control on networks in the spirit of the works of Achdou et al. [NoDEA Nonlinear Differ. Equ. Appl. 20 (2013) 413–445] and Imbert et al. [ESAIM: COCV 19 (2013) 129–166]. The main new feature is that there are entry (or exit) costs at the edges of the network leading to a possible discontinuous value function. We characterize the value function as the unique viscosity solution of a new Hamilton-Jacobi system. The uniqueness is a consequence of a comparison principle for which we give two different proofs, one with arguments from the theory of optimal control inspired by Achdou et al. [ESAIM: COCV 21 (2015) 876–899] and one based on partial differential equations techniques inspired by a recent work of Lions and Souganidis [Atti Accad. Naz. Lincei Rend. Lincei Mat. Appl. 27 (2016) 535–545].

scholarly journals Averaging of the Hamilton-Jacobi equation in infinite dimensions and an application

2000 ◽  
Vol 41 (3) ◽  
pp. 372-385
Author(s):  
Shihong Wang ◽  
Zuoyi Zhou
Keyword(s):  
Optimal Control ◽  
Optimal Control Problem ◽  
Control Problem ◽  
Viscosity Solution ◽  
Jacobi Equation ◽  
Value Function ◽  
Limit Problem ◽  
Infinite Dimensions ◽  
The Value Function ◽  
Hamilton Jacobi Equation

AbstractWe study the averaging of the Hamilton-Jacobi equation with fast variables in the viscosity solution sense in infinite dimensions. We prove that the viscosity solution of the original equation converges to the viscosity solution of the averaged equation and apply this result to the limit problem of the value function for an optimal control problem with fast variables.

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