Lyapunov-Like Conditions for Tight Exit Probability Bounds through Comparison Theorems for SDEs. Nilsson, P. & Ames, A. D
abstract   bibtex   
Computing upper bounds on exit probabilities—the probability that a system reaches certain “bad” sets—may assist decision-making in control of stochastic systems. Existing analytical bounds for systems described by stochastic differential equations are quite loose, especially for low-probability events, which limits their applicability in practical situations. In this paper we analyze why existing bounds are loose, and conclude that it is a fundamental issue with the underlying techniques based on martingale inequalities. As an alternative, we give comparison results for stochastic differential equations that via a Lyapunov-like function allow exit probabilities of an ndimensional system to be upper-bounded by an exit probability of a one-dimensional Ornstein-Uhlenbeck process. Even though no closed-form expression is known for the latter, it depends on three or four parameters and can be a priori tabulated for applications. We extend these ideas to the controlled setting and state a stochastic analogue of control barrier functions. The bounds are illustrated on numerical examples and are shown to be much tighter than those based on martingale inequalities.
@article{nilsson_lyapunov-like_nodate,
	title = {Lyapunov-{Like} {Conditions} for {Tight} {Exit} {Probability} {Bounds} through {Comparison} {Theorems} for {SDEs}},
	abstract = {Computing upper bounds on exit probabilities—the probability that a system reaches certain “bad” sets—may assist decision-making in control of stochastic systems. Existing analytical bounds for systems described by stochastic differential equations are quite loose, especially for low-probability events, which limits their applicability in practical situations. In this paper we analyze why existing bounds are loose, and conclude that it is a fundamental issue with the underlying techniques based on martingale inequalities. As an alternative, we give comparison results for stochastic differential equations that via a Lyapunov-like function allow exit probabilities of an ndimensional system to be upper-bounded by an exit probability of a one-dimensional Ornstein-Uhlenbeck process. Even though no closed-form expression is known for the latter, it depends on three or four parameters and can be a priori tabulated for applications. We extend these ideas to the controlled setting and state a stochastic analogue of control barrier functions. The bounds are illustrated on numerical examples and are shown to be much tighter than those based on martingale inequalities.},
	language = {en},
	author = {Nilsson, Petter and Ames, Aaron D},
	pages = {7},
}
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