[Zurück]

F. Wagener, T. Kiseleva, D. Grass:
"Small-noise asymptotics of Hamilton-Jacobi-Bellman equations and bifurcations of stochastic optimal control problems";
Communications in Nonlinear Science and Numerical Simulation, 22 (2015), 1-3; S. 38 - 54.

We derive small-noise approximations of the value function of stochastic optimal control problems over an unbounded domain and use these to perform a bifurcation analysis of these problems. The corresponding zero-noise problems may feature indifference (shock, Skiba) points, that is, points of non-differentiability of the value function. Small-noise expansions are obtained in regions of regularity by a singular perturbation analysis of the stochastic Hamilton-Jacobi-Bellman equation; the expansions are matched at the boundaries of these regions to obtain an approximation over the whole state space. From this approximation, a functional geometric invariant is computed: in the presence of zero-noise indifference points, this invariant is multimodal. Regime switching thresholds of the optimally controlled dynamics are defined as those critical points where the invariant takes a local minimum. A change in the number of thresholds is a bifurcation of the dynamics. The concepts are applied to analyse the stochastic lake model.

Small noise asymptotics; Stochastic control; Regime shifts; Bifurcations

http://dx.doi.org/10.1016/j.cnsns.2014.09.029