Small-noise asymptotics of Hamilton–Jacobi–Bellman equations and bifurcations of stochastic optimal control problems

Dieter Grass1, Tatiana Kiseleva2, Florian Wagener3
1Institute for Mathematical Methods in Economics, Vienna University of Technology, Vienna, Austria
2Department of Spatial Economics, Free University Amsterdam, Amsterdam, The Netherlands
3Department of Economics and Econometrics, University of Amsterdam and Tinbergen Institute, Amsterdam, The Netherlands

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Arnold, 1998 Brock, 2003, Managing systems with non-convex positive feedback, Environ Resour Econ, 26, 575, 10.1023/B:EARE.0000007350.11040.e3 Bultmann, 2010, Stochastic skiba sets: an example from models of illicit drug consumption, 239 Caulkins, 2001, Why politics makes strange bedfellows: dynamic model with DNS curves, J Optim Theory Appl, 111, 237, 10.1023/A:1011925332503 Crepin, 2007, Using fast and slow processes to manage resources with thresholds, Environ Resour Econ, 36, 191, 10.1007/s10640-006-9029-8 2004 Dawid, 2005, On the efficiency-effects of private (dis-)trust in the government, J Econ Behav Organiz, 57, 530, 10.1016/j.jebo.2005.04.005 Dechert, 1983, A complete characterization of optimal growth paths in an aggregated model with a non-concave production function, J Econ Theory, 31, 332, 10.1016/0022-0531(83)90081-9 Dechert, 2006, The stochastic lake game: a numerical solution, J Econ Dyn Control, 30, 1569, 10.1016/j.jedc.2005.10.010 Diks, 2008, A bifurcation theory for a class of discrete time Markovian stochastic systems, Phys D, 237, 3297, 10.1016/j.physd.2008.07.021 Diks, 2011, Phenomenological and ratio bifurcations of a class of discrete time stochastic processes, Indagationes Math, 22, 207, 10.1016/j.indag.2011.09.007 Eckhaus W. Matched asymptotic expansions and singular perturbations. Number 6 in North Holland mathematical studies. North Holland, 1973. Fleming, 1971, Stochastic control for small noise intensities, SIAM J Control Optim, 9, 473, 10.1137/0309035 Fleming, 1986, Asymptotic series and the method of vanishing viscosity, Indiana Univ Math J, 35, 425, 10.1512/iumj.1986.35.35026 Gardiner, 2010 Golubitsky, 1973, vol. 14 Guckenheimer, 1983 Haunschmied, 2003, A DNS-curve in a two-state capital accumulation model: a numerical analysis, J Econ Dyn Control, 27, 701, 10.1016/S0165-1889(01)00070-7 Heijdra, 2013, Environmental Abatement and the Macroeconomy in the Presence of Ecological Thresholds, Environ Res Econ, 55, 47, 10.1007/s10640-012-9613-z Heijnen, 2013, Avoiding an ecological regime shift is sound economic policy, J Econ Dyn Control, 37, 1322, 10.1016/j.jedc.2013.03.003 Holmes, 1995 Horsthemke, 2006 Judd, 1998 Kamihigashi, 2001, Necessity of transversality conditions for infinite horizon problems, Econometrica, 69, 995, 10.1111/1468-0262.00227 Kiseleva, 2010, Bifurcations of optimal vector fields in the shallow lake model, J Econ Dyn Control, 34, 825, 10.1016/j.jedc.2009.11.008 Kiseleva, 2014, Bifurcations of optimal vector fields, Math Oper Res. Kossioris, 2008, Feedback Nash equilibria for non-linear differential games in pollution control, J Econ Dyn Control, 32, 1312, 10.1016/j.jedc.2007.05.008 Kushner, 1967, Optimal discounted stochastic control for diffusion processes, SIAM J Control Optim, 5, 520, 10.1137/0305032 Lenton, 2011, Early warning of climate tipping points, Nat Clim Change, 1, 201, 10.1038/nclimate1143 Lenton, 2008, Tipping elements in the earth’s climate system, Proc Natl Acad Sci, 105, 1786, 10.1073/pnas.0705414105 Mäler, 2003, The economics of shallow lakes, Environ Resour Econ, 26, 603, 10.1023/B:EARE.0000007351.99227.42 O’Malley, 1994, On singular perturbations, especially matching, SIAM Rev, 36, 413, 10.1137/1036096 Salerno G, Beard R, McDonald S. Rent seeking behavior and optimal taxation of pollution in shallow lakes. MPRA Working paper. 2007. Scheffer, 2009, Early-warning signals for critical transitions, Nature, 461, 53, 10.1038/nature08227 Sethi, 1977, Nearest feasible paths in optimal control problems: theory, examples, and counterexamples, J Optim Theory Appl, 23, 563, 10.1007/BF00933297 Sethi, 2000 Skiba, 1978, Optimal growth with a convex–concave production function, Econometrica, 46, 527, 10.2307/1914229 Tahvonen, 1996, Nonconvexities in optimal pollution accumulation, J Environ Econ Manage, 31, 160, 10.1006/jeem.1996.0038 Tahvonen, 1996, Optimality of irreversible pollution accumulation, J Econ Dyn Control, 20, 1775, 10.1016/0165-1889(95)00909-4 Treadway, 1969, On rational entrepreneurial behaviour and the demand for investment, Rev Econ Stud, 36, 227, 10.2307/2296839 Van Dyke, 1994, Nineteenth-century roots of the boundary-layer idea, SIAM Rev, 36, 415, 10.1137/1036097 Vasilieva, 1994, On the development of singular perturbation theory at Moscow state university and elsewhere, SIAM Rev, 36, 440, 10.1137/1036100 Verhulst, 2005 Wagener, 2003, Skiba points and heteroclinic bifurcations, with applications to the shallow lake system, J Econ Dyn Control, 27, 1533, 10.1016/S0165-1889(02)00070-2 Wagenmakers, 2005, Transformation invariant stochastic catastrophe theory, Phys D Nonlinear Phenom, 211, 263, 10.1016/j.physd.2005.08.014 Zeeman, 1988, Stability of dynamical systems, Nonlinearity, 1, 115, 10.1088/0951-7715/1/1/005