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Bellman equation
About: Bellman equation is a research topic. Over the lifetime, 5884 publications have been published within this topic receiving 135589 citations.
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TL;DR: In this paper, the authors considered an optimal stochastic control problem, where the diffusion coefficient also depends on the control and is possibly degenerate, and the maximum principle, dynamic programming and their connections were established within a unified framework of viscosity solution.
Abstract: An optimal stochastic control problem is considered in this paper, where the diffusion coefficient also depends on the control and is possibly degenerate. In addition to the usual adjoint process, a second-order adjoint process is introduced. Some relationships between the value function and the adjoint processes are presented via the “super- and sub-differential” which is related to the viscosity solution, without assuming the smoothness of the value function. The maximum principle, dynamic programming and their connections are then established within a unified framework of viscosity solution
78 citations
TL;DR: In this article, the equivalence of L 2 -stabilizability for linear mean-field stochastic difference equations with random initial value has been proved, and it has been shown that the optimal control of a mean field linear-quadratic optimal control with an infinite time horizon uniquely exists, and the optimal controller can be expressed as a linear state feedback involving the state and its mean via the minimal nonnegative definite solution of two coupled algebraic Riccati equations.
78 citations
TL;DR: In this article, the boundary behavior and optimal portfolio rules for cases when marginal utility at zero consumption is finite are discussed. But they do not satisfy the Hamilton-Jacobi Bellman equations and do not represent appropriate value functions because the boundary behaviour near zero wealth is not satisfactorily dealt with.
78 citations
TL;DR: An approach to solve finite time horizon suboptimal feedback control problems for partial differential equations is proposed by solving dynamic programming equations on adaptive sparse grids with semi-discrete optimal control problem and the feedback control is derived from the corresponding value function.
Abstract: An approach to solve finite time horizon suboptimal feedback control problems for partial differential equations is proposed by solving dynamic programming equations on adaptive sparse grids. A semi-discrete optimal control problem is introduced and the feedback control is derived from the corresponding value function. The value function can be characterized as the solution of an evolutionary Hamilton---Jacobi Bellman (HJB) equation which is defined over a state space whose dimension is equal to the dimension of the underlying semi-discrete system. Besides a low dimensional semi-discretization it is important to solve the HJB equation efficiently to address the curse of dimensionality. We propose to apply a semi-Lagrangian scheme using spatially adaptive sparse grids. Sparse grids allow the discretization of the value functions in (higher) space dimensions since the curse of dimensionality of full grid methods arises to a much smaller extent. For additional efficiency an adaptive grid refinement procedure is explored. The approach is illustrated for the wave equation and an extension to equations of Schrodinger type is indicated. We present several numerical examples studying the effect the parameters characterizing the sparse grid have on the accuracy of the value function and the optimal trajectory.
77 citations
TL;DR: In this article, the authors 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.
Abstract: 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.
77 citations