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Let X = (Xt)t≥0 be a strong Markov process, and let G¹, G², and G³ be continuous functions satisfying G¹ ≤ G² ≤ G² and Ex supt ∣Gi(Xt)∣ < ∞ for i = 1, 2, 3. Consider the optimal stopping game where the sup-player chooses a stopping time τ to maximize, and the inf-player chooses a stopping time s to minimize, the expected payoff Mx(τ,σ) = Ex[G1(Xτ) I(τ < σ) + G2(Xσ) I(σ< σ) + G3(Xτ ) I(τ = σ)], where X0 = x under Px. Define the upper value and the lower value of the game by V *(x) = infσ supτ Mx(τ, σ) and V*(x) = supτ infσ Mx(τ, σ), respectively, where the horizon T (the upper bound for τ and σ above) may be either finite or infinite (it is assumed that G1(XT) = G2(XT ) if T is finite and lim inft→∞ G2(Xt) ≤ lim supt→∞ G1(Xt) if T is infinite). If X is right-continuous, then the Stackelberg equilibrium holds, in the sense that V *(x) = V*(x) for all x with V := V * = V* defining a measurable function. If X is right-continuous and left-continuous over stopping times (quasi-left-continuous), then the Nash equilibrium holds, in the sense that there exist stopping times τ* and σ* such that Mx(τ, σ*) ≤ Mx(τ*, σ*) ≤ Mx(τ*, s) for all stopping times τ and s, implying also that V (x) = Mx(τ*, σ*) for all x. Further properties of the value function V and the optimal stopping times τ* and σ* are exhibited in the proof. [ABSTRACT FROM AUTHOR]

We consider the development of a competing risks system following a two-stage stopping rule. This stopping procedure takes into account a desired probability of successfully completing an assigned task. Upon termination, the asymptotic properties of the estimator of the system reliability, as well as the stopping time variable, are examined. Moreover, the asymptotic risk associated with the two-stage procedure is presented. This risk is anchored on a loss function that considers losses incurred in not attaining the desired reliability and that of excessive testing. [ABSTRACT FROM AUTHOR]

We establish sharp weak type and logarithmic estimates for the diameter of the stopped Brownian motion. Then, using standard embedding theorems, we extend the results to the case of general real-valued continuous-path martingales. The proof rests on finding of the solutions to the corresponding three-dimensional optimal stopping problems. [ABSTRACT FROM PUBLISHER]

In sequential analysis and starting with the seminal treatment of the optimality of the SPRT due to Wald and Wolfowitz [Wald, A.; Wolfowitz, J. Bayes solutions of sequential decision problems. Ann. Math. Statist. 1950, 21, 82–99] various problems of optimal stopping with linear costs of observation arise. Here, we shall describe a new method for solving such problems. For a payoff g(Xt) − ct we propose a linear representation of the form where Mt is a local martingale, and the function h and the local martingale depend on a parameterλ. In the case of a diffusion process Xt we shall show that, for a proper choice ofλ, the boundary points of the optimal stopping region can be obtained from those points of the state space where the maxima of h are located. This method is inspired by a method of Beibel and Lerche [Beibel, M.; Lerche, H.R. A new look at optimal stopping problems related to mathematical finance. Statistica Sinica 1997, 7, 93–108; Beibel, M.; Lerche, H.R. Optimal stopping of regular diffusions under random discounting. Theory Probab. Appl. 2001, 45 (4), 547–557] who, for optimal stopping problems with discounted payoff, used a multiplicative representation e−rtg(Xt) = h(Xt)Mt with a suitable martingale. [ABSTRACT FROM AUTHOR]

Common software release procedures based on statistical techniques try to optimize the trade-off between further testing costs and costs due to remaining errors. We propose new software release procedures where the aim is to certify with a certain confidence level that the software does not contain errors. The underlying model is a discrete time model similar to the geometric Moranda model. The decisions are based on a mix of classical and Bayesian approaches to sequential testing and do not require any assumption on the initial number of errors. [ABSTRACT FROM AUTHOR]