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The paper considers the problem of optimal control in models with disorder and provides the computational scheme of its solution. The proposed method is based on replacing the disorder with its estimate by the stopping time. The calculations take place on the binary tree with nodes divided into two classes. The division of tree nodes into two classes is carried out using a decision rule associated either with testing two hypothesis, or with the optimal stopping of a Markov process developing on a binary tree. The estimate of the disorder is the moment at which the walk on the tree is placed in a node of the first class for the first time. The example of the optimal control problem is related to the problem of quantile hedging of a payoff. [ABSTRACT FROM AUTHOR]

We propose a two-stage sequential method for obtaining tandem-width confidence intervals for a Bernoulli proportion p. The term "tandem-width" refers to the fact that the half-width of the confidence interval is not fixed beforehand; it is instead required to satisfy two different half-width upper bounds, h0 and h1, depending on the (unknown) values of p. To tackle this problem, we first propose a simple but useful sequential method for obtaining fixed-width confidence intervals for p, whose stopping rule is based on the minimax estimator of p. We observe Bernoulli(p) trials sequentially, and for some fixed half-width h = h0 or h1, we develop a stopping time T such that the resulting confidence interval for p, [ ], covers the parameter with confidence at least where is the maximum likelihood estimator of p at time T. Furthermore, we derive theoretical properties of our proposed fixed-width and tandem-width methods and compare their performances with existing alternative sequential schemes. The proposed minimax-based fixed-width method performs similarly to alternative fixed-width methods, while being easier to implement in practice. In addition, the proposed tandem-width method produces effective savings in sample size compared to the fixed-width counterpart and provides excellent results for scientists to use when no prior knowledge of p is available. [ABSTRACT FROM AUTHOR]

. We start with the definition of stopping time according to [4], p.283. We prove, that different definitions for stopping time can coincide. We give examples of stopping time using constant-functions or functions defined with the operator max or min (defined in [6], pp.37-38). Finally we give an example with some given filtration. Stopping time is very important for stochastic finance. A stopping time is the moment, where a certain event occurs ([7], p.372) and can be used together with stochastic processes ([4], p.283). Look at the following example: we install a function ST: {1,2,3,4} → {0, 1, 2} ∪ {+∞}, we define:. a. ST(1)=1, ST(2)=1, ST(3)=2, ST(4)=2. b. The set {0,1,2} consists of time points: 0=now, 1=tomorrow, 2=the day after tomorrow. We can prove: c. {w, where w is Element of Ω: ST.w=0}=ø & {w, where w is Element of Ω: ST.w=1}={1,2} & {w, where w is Element of Ω: ST.w=2}={3,4} and ST is a stopping time. We use a function Filt as Filtration of {0,1,2}, Σ where Filt(0)=Ωnow, Filt(1)=Ωfut1 and Filt(2)=Ωfut2. From a.,b. and c. we know that: d. {w, where w is Element of Ω: ST.w=0} in Ωnow and {w, where w is Element of Ω: ST.w=1} in Ωfut1 and {w, where w is Element of Ω: ST.w=2} in Ωfut2. The sets in d. are events, which occur at the time points 0(=now), 1(=tomorrow) or 2(=the day after tomorrow), see also [7], p.371. Suppose we have ST(1)=+∞, then this means that for 1 the corresponding event never occurs. As an interpretation for our installed functions consider the given adapted stochastic process in the article [5]. ST(1)=1 means, that the given element 1 in {1,2,3,4} is stopped in 1 (=tomorrow). That tells us, that we have to look at the value f2(1) which is equal to 80. The same argumentation can be applied for the element 2 in {1,2,3,4}. ST(3)=2 means, that the given element 3 in {1,2,3,4} is stopped in 2 (=the day after tomorrow). That tells us, that we have to look at the value f3(3) which is equal to 100. ST(4)=2 means, that the given element 4 in {1,2,3,4} is stopped in 2 (=the day after tomorrow). That tells us, that we have to look at the value f3(4) which is equal to 120. In the real world, these functions can be used for questions like: when does the share price exceed a certain limit? (see [7], p.372). [ABSTRACT FROM AUTHOR]

We consider causality relationships between σ-fields (filtrations) associated by stopping times, which can be applied to the stopped processes. These results are motivated by the causality relationship between filtrations '( $$ \mathcal{G} $$ t) is a cause of (ℋ t) within (ℱ t)' and which is based on Granger's definition of causality. Then we give some basic properties of causality up to some stopping time. The given concept of causality associated to stopping times is equivalent with the preservation of the martingale property for the stopped processes when the filtration is getting larger. [ABSTRACT FROM AUTHOR]

The family Q = (Q(n),Fn)n=0∞ of probability measures Q(n) defined on Fn is considered. It is called a deformation of the first kind if for all n ∊ {0, 1, 2,... }, Q(n+1)1)∣Fn ⩽ Q(n), and a deformation of the second kind if Q(n+1)∣Fn ⩽ Q(n). For finite stopping times τ measures Q(τ) are introduced by the formula Q(t)(A) = Σ i∞ =0 Q(i)(A{t = i}), where A ϵ Ft . With the help of these measures a generalization of Doob's optional sampling theorem is formulated and proved for deformed submartingales of the first kind in the case of neighboring stopping times, and of the second kind in the case of boundedly spaced stopping times. [ABSTRACT FROM AUTHOR]

A class of sequential estimation procedures is considered in the case when relevant data may become available only at random times. The exact distributions of the optimal stopping time and the number of observations at the moment of stopping are derived in some sequential procedures. The results obtained in an explicit form are applied to derive the expected time of observing the process, the average number of observations and the expected loss of sequential estimation procedures based on delayed observations. The use of the results is illustrated in a special model of normally distributed observations and the Weibull distributed lifetimes. The probabilistic characteristics are also derived for an adaptive sequential procedures and the behavior of the adaptive procedure is compared with the corresponding optimal sequential procedure. [ABSTRACT FROM AUTHOR]

We consider the optimal stopping problem with a possible compensated refusal of reward. We discuss functionals of exponential Brownian motion. The optimal stopping time is defined on the set of all finite stopping times. The functionals under consideration correspond to payments for standard American options. [ABSTRACT FROM AUTHOR]

The optimal stopping problem with incomplete data for the Kalman-Bucy scheme is reduced to the optimal stopping problem with complete data. In the case of a continuous gain function it is proven that price convergence has an order (ϵ1 +ϵ2) when small perturbation parameters ϵ1, ϵ2 of an observable process tend to zero. [ABSTRACT FROM AUTHOR]

Abstract: Shepp’s urn model is a useful tool for analyzing the stopping-rule problems in economics and finance. In [R.W. Chen, A. Zame, C.T. Lin, H. Wu, A random version of Shepp’s urn scheme, SIAM J. Discrete Math. 19 (1) (2005) 149–164], Chen et al. considered a random version of Shepp’s urn scheme and showed that a simple drawing policy (called “the in the hole policy”) can asymptotically maximize the expected value of the game. By extending the work done by Chen et al., this note considers a more general urn scheme that is better suited to real-life price models in which the short-term value might not fluctuate. Further, “the in the hole policy” is shown to be asymptotically optimal for this new urn scheme. [Copyright &y& Elsevier]

Let Sn be the partial sum of i.i.d. random variables X1,X2,..., and let N be the usual CUSUM stopping time based on Sn. Under suitable conditions we determine ψ(α,β) = E exp(α SN - β N), where β > 0 and α is a suitable number. The given formula can be used to study the distributional properties of N, SN, and SN - h. Because the CUSUM based on maxima is reducible to N, the formula can be used to obtain the distributional properties of the maximal process as well. Several examples are discussed, and certain applications are shown in the so called trading securities. The formulas can also be used to study the distributional properties of a symmetric two-sided CUSUM. [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]

Abstract: Consider a finite population of large but unknown size of hidden objects. Consider searching for these objects for a period of time, at a certain cost, and receiving a reward depending on the sizes of the objects found. Suppose that the size and discovery time of the objects both have unknown distributions, but the conditional distribution of time given size is exponential with an unknown non-negative and non-decreasing function of the size as failure rate. The goal is to find an optimal way to stop the discovery process. Assuming that the above parameters are known, an optimal stopping time is derived and its asymptotic properties are studied. Then, an adaptive rule based on order restricted estimates of the distributions from truncated data is presented. This adaptive rule is shown to perform nearly as well as the optimal stopping time for large population size. [Copyright &y& Elsevier]