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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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01 Jan 2011
TL;DR: In this paper, the authors derive numerical theory results characterizing the properties of the nested fixed point algorithm used to evaluate the objective function of BLP's estimator and recast estimation as a mathematical program with equilibrium constraints, which can be faster and which avoids the numerical issues associated with nested inner loops.
Abstract: The widely-used estimator of Berry, Levinsohn and Pakes (1995) produces estimates of consumer preferences from a discrete-choice demand model with random coecients, market-level demand shocks and endogenous prices. We derive numerical theory results characterizing the properties of the nested fixed point algorithm used to evaluate the objective function of BLP’s estimator. We discuss problems with typical implementations, including cases that can lead to incorrect parameter estimates. As a solution, we recast estimation as a mathematical program with equilibrium constraints, which can be faster and which avoids the numerical issues associated with nested inner loops. The advantages are even more pronounced for forward-looking demand models where the Bellman equation must also be solved repeatedly. Several Monte Carlo and real-data experiments support our numerical concerns about the nested fixed point approach and the advantages of constrained optimization. For static BLP, the constrained optimization approach can be as much as ten to forty times faster for large-dimensional problems with many markets.
42 citations
TL;DR: In this paper, the authors considered degenerate parabolic and elliptic fully nonlinear Bellman equations with Lipschitz coefficients and obtained error bounds of order h 1/2 in the sup norm for certain types of finite-difference schemes.
Abstract: We consider degenerate parabolic and elliptic fully nonlinear Bellman equations with Lipschitz coefficients in domains Error bounds of order h1/2 in the sup norm for certain types of finite-difference schemes are obtained
42 citations
Posted Content•
TL;DR: In this article, a continuous variable extension of the algebraic decision diagram (ADD) is proposed to model real-world decision-theoretic planning problems with discrete and continuous state Markov decision processes (DCMDPs).
Abstract: Many real-world decision-theoretic planning problems can be naturally modeled with discrete and continuous state Markov decision processes (DC-MDPs). While previous work has addressed automated decision-theoretic planning for DCMDPs, optimal solutions have only been defined so far for limited settings, e.g., DC-MDPs having hyper-rectangular piecewise linear value functions. In this work, we extend symbolic dynamic programming (SDP) techniques to provide optimal solutions for a vastly expanded class of DCMDPs. To address the inherent combinatorial aspects of SDP, we introduce the XADD - a continuous variable extension of the algebraic decision diagram (ADD) - that maintains compact representations of the exact value function. Empirically, we demonstrate an implementation of SDP with XADDs on various DC-MDPs, showing the first optimal automated solutions to DCMDPs with linear and nonlinear piecewise partitioned value functions and showing the advantages of constraint-based pruning for XADDs.
41 citations
TL;DR: This paper studies the optimal reinsurance-investment problems in a financial market with jump-diffusion risky asset, where the insurance risk model is modulated by a compound Poisson process, and the two jump number processes are correlated by a common shock.
Abstract: In this paper, we study the optimal reinsurance-investment problems in a financial market with jump-diffusion risky asset, where the insurance risk model is modulated by a compound Poisson process, and the two jump number processes are correlated by a common shock. Moreover, we remove the assumption of nonnegativity on the expected value of the jump size in the stock market, which is more economic reasonable since the jump sizes are always negative in the real financial market. Under the criterion of mean–variance, based on the stochastic linear–quadratic control theory, we derive the explicit expressions of the optimal strategies and value function which is a viscosity solution of the corresponding Hamilton–Jacobi–Bellman equation. Furthermore, we extend the results in the linear–quadratic setting to the original mean–variance problem, and obtain the solutions of efficient strategy and efficient frontier explicitly. Some numerical examples are given to show the impact of model parameters on the efficient frontier.
41 citations
01 Jan 1985
TL;DR: In the analysis of parametric optimization problems, it is of great interest to explore certain stability properties of the optimal value function and the optimal set mapping (or some selection function of this mapping): continuity, smoothness, directional differentiability, Lipschitz continuity and the like as mentioned in this paper.
Abstract: In the analysis of parametric optimization problems it is of great interest to explore certain stability properties of the optimal value function and of the optimal set mapping (or some selection function of this mapping): continuity, smoothness, directional differentiability, Lipschitz continuity and the like. For a survey of this field we refer to comprehensive treatments of various aspects of such questions in the recent works of Fiacco (1983), Bank et al. (1982) and Rockafellar (1982).
41 citations