On Distributions of First Passage Times and Optimal Stopping of AR(1) Sequences

Novikov, Alexander

doi:10.1137/s0040585x97983730

Cited by 13 publications

(12 citation statements)

References 11 publications

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“…Under these technical assumptions and P (z 0 < 0) > 0, the results of [23] ensure that E{τ } < ∞ and (35) holds.…”

Section: Convergence Speedmentioning

confidence: 99%

“…(6)] originally proved in [24, Theorem 3] under slightly stronger assumptions. The results of [23] apply to the problem of the first passage of a given barrier from below, but our problem is that of passing the barrier from above. However, since our barrier is equal to zero, the results of [23] apply to the symmetrical version of our problem with X k replaced by −X k and z k by −z k .…”

Section: Convergence Speedmentioning

confidence: 99%

“…The results of [23] apply to the problem of the first passage of a given barrier from below, but our problem is that of passing the barrier from above. However, since our barrier is equal to zero, the results of [23] apply to the symmetrical version of our problem with X k replaced by −X k and z k by −z k . This leads to (35) after verifying technical assumptions: Since z 0 = l(μ + g T w ) − ξ is bounded (by Assumption 1 and compactness of W) we have…”

Section: Convergence Speedmentioning

confidence: 99%

See 2 more Smart Citations

Stochastic MPC Framework for Controlling the Average Constraint Violation

Korda

Linares

Oldewurtel

et al. 2014

IEEE Trans. Automat. Contr.

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This paper considers linear discrete-time systems with additive, bounded, disturbances subject to hard control input bounds and a stochastic constraint on the amount of state-constraint violation averaged over time. The amount of violations is quantified by a loss function and the averaging can be weighted, corresponding to exponential forgetting of past violations. The freedom in the choice of the loss function makes this formulation highly flexible -for instance, probabilistic constraints or integrated chance constraints can be enforced by an appropriate choice of the loss function. For the type of constraint considered, we develop a recursively feasible receding horizon control scheme exploiting the averaged-over-time nature by explicitly taking into account the amount of past constraint violations when determining the current control input. This leads to a significant reduction in conservatism. As a simple extension of the proposed approach we show how time-varying state-constraints can be handled within our framework. The computational complexity (online as well as offline) is comparable to existing model predictive control schemes. The effectiveness of the proposed methodology is demonstrated by means of a numerical example.

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“…Under these technical assumptions and P (z 0 < 0) > 0, the results of [23] ensure that E{τ } < ∞ and (35) holds.…”

Section: Convergence Speedmentioning

confidence: 99%

Section: Convergence Speedmentioning

confidence: 99%

Section: Convergence Speedmentioning

confidence: 99%

See 1 more Smart Citation

Stochastic MPC Framework for Controlling the Average Constraint Violation

Korda

Linares

Oldewurtel

et al. 2014

IEEE Trans. Automat. Contr.

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show abstract

“…leading to the following result. For the special case µ = 1, this formula was obtained using differential equations based on the generator in [11,Theorem 3].…”

Section: Theorem 32 It Holds Thatmentioning

confidence: 99%

“…Only very few results are known for this problem. In [3] and [11] the innovations are assumed to be exponentially distributed, and in [5] asymptotic results were given for g(x) = x.…”

mentioning

confidence: 99%

Phase-Type Distributions and Optimal Stopping for Autoregressive Processes

Christensen

2012

Journal of Applied Probability

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Autoregressive processes are intensively studied in statistics and other fields of applied stochastics. For many applications, the overshoot and the threshold time are of special interest. When the upward innovations are in the class of phase-type distributions, we determine the joint distribution of these two quantities and apply this result to problems of optimal stopping. Using a principle of continuous fit, this leads to explicit solutions.

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Persistence of One-Dimensional AR(1)-Sequences

2018

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For a class of one-dimensional autoregressive sequences (Xn) we consider the tail behaviour of the stopping time T 0 = min{n ≥ 1 : Xn ≤ 0}. We discuss existing general analytical approaches to this and related problems and propose a new one, which is based on a renewal-type decomposition for the moment generating function of T 0 and on the analytical Fredholm alternative. Using this method, we show that Px(T 0 = n) ∼ V (x)R n 0 for some 0 < R 0 < 1 and a positive R 0 -harmonic function V . Further we prove that our conditions on the tail behaviour of the innovations are sharp in the sense that fatter tails produce non-exponential decay factors.T 0 := min{k ≥ 1 : X k ≤ 0}, 1991 Mathematics Subject Classification. Primary 60J05; Secondary 60G40.

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On Distributions of First Passage Times and Optimal Stopping of AR(1) Sequences

Cited by 13 publications

References 11 publications

Stochastic MPC Framework for Controlling the Average Constraint Violation

Stochastic MPC Framework for Controlling the Average Constraint Violation

Phase-Type Distributions and Optimal Stopping for Autoregressive Processes

Persistence of One-Dimensional AR(1)-Sequences

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