scholarly journals On the Disorder Problem for a Negative Binomial Process

2015 ◽  
Vol 52 (1) ◽  
pp. 167-179 ◽  
Author(s):  
Bruno Buonaguidi ◽  
Pietro Muliere
Keyword(s):  
Free Boundary ◽  
Optimal Stopping ◽  
Negative Binomial ◽  
Stopping Time ◽  
Careful Analysis ◽  
Optimal Stopping Problem ◽  
Boundary Equation ◽  
Negative Binomial Process ◽  
Probabilistic Nature ◽  
Binomial Process

We study the Bayesian disorder problem for a negative binomial process. The aim is to determine a stopping time which is as close as possible to the random and unknown moment at which a sequentially observed negative binomial process changes the value of its characterizing parameter p ∈ (0, 1). The solution to this problem is explicitly derived through the reduction of the original optimal stopping problem to an integro-differential free-boundary problem. A careful analysis of the free-boundary equation and of the probabilistic nature of the boundary point allows us to specify when the smooth fit principle holds and when it breaks down in favour of the continuous fit principle.

On the Disorder Problem for a Negative Binomial Process

2015 ◽  
Vol 52 (01) ◽  
pp. 167-179 ◽  
Author(s):  
Bruno Buonaguidi ◽  
Pietro Muliere
Keyword(s):  
Free Boundary ◽  
Optimal Stopping ◽  
Negative Binomial ◽  
Stopping Time ◽  
Careful Analysis ◽  
Optimal Stopping Problem ◽  
Boundary Equation ◽  
Negative Binomial Process ◽  
Probabilistic Nature ◽  
Binomial Process

We study the Bayesian disorder problem for a negative binomial process. The aim is to determine a stopping time which is as close as possible to the random and unknown moment at which a sequentially observed negative binomial process changes the value of its characterizing parameter p ∈ (0, 1). The solution to this problem is explicitly derived through the reduction of the original optimal stopping problem to an integro-differential free-boundary problem. A careful analysis of the free-boundary equation and of the probabilistic nature of the boundary point allows us to specify when the smooth fit principle holds and when it breaks down in favour of the continuous fit principle.

scholarly journals Fractional negative binomial and Pólya processes

2018 ◽  
Vol 38 (1) ◽  
pp. 77-101
Author(s):  
Palaniappan Vellai Samy ◽  
Aditya Maheshwari
Keyword(s):  
Poisson Process ◽  
Negative Binomial ◽  
Random Variable ◽  
Infinitely Divisible ◽  
One Dimensional ◽  
Fractional Poisson Process ◽  
Negative Binomial Process ◽  
The One ◽  
Definition Of ◽  
Binomial Process

In this paper, we define a fractional negative binomial process FNBP by replacing the Poisson process by a fractional Poisson process FPP in the gamma subordinated form of the negative binomial process. It is shown that the one-dimensional distributions of the FPP and the FNBP are not infinitely divisible. Also, the space fractional Pólya process SFPP is defined by replacing the rate parameter λ by a gamma random variable in the definition of the space fractional Poisson process. The properties of the FNBP and the SFPP and the connections to PDEs governing the density of the FNBP and the SFPP are also investigated.

scholarly journals A Forward Algorithm for Solving Optimal Stopping Problems

2006 ◽  
Vol 43 (01) ◽  
pp. 102-113
Author(s):  
Albrecht Irle
Keyword(s):  
State Space ◽  
Optimal Stopping ◽  
Stopping Time ◽  
The State ◽  
Optimal Stopping Problem ◽  
Forward Algorithm ◽  
Optimal Stopping Time ◽  
Entrance Time ◽  
Markov Sequence ◽  
Optimal Stopping Problems

We consider the optimal stopping problem for g(Z n ), where Z n , n = 1, 2, …, is a homogeneous Markov sequence. An algorithm, called forward improvement iteration, is presented by which an optimal stopping time can be computed. Using an iterative step, this algorithm computes a sequence B 0 ⊇ B 1 ⊇ B 2 ⊇ · · · of subsets of the state space such that the first entrance time into the intersection F of these sets is an optimal stopping time. Various applications are given.

scholarly journals Multiobjective Stopping Problem for Discrete-Time Markov Processes: Convex Analytic Approach

2010 ◽  
Vol 47 (04) ◽  
pp. 947-966 ◽  
Author(s):  
F. Dufour ◽  
A. B. Piunovskiy
Keyword(s):  
Optimal Stopping ◽  
Linear Problem ◽  
Stopping Time ◽  
Optimal Solution ◽  
Analytic Approach ◽  
General State ◽  
Occupation Measure ◽  
Optimal Stopping Problem ◽  
Weak Hypothesis ◽  
General State Space

The purpose of this paper is to study an optimal stopping problem with constraints for a Markov chain with general state space by using the convex analytic approach. The costs are assumed to be nonnegative. Our model is not assumed to be transient or absorbing and the stopping time does not necessarily have a finite expectation. As a consequence, the occupation measure is not necessarily finite, which poses some difficulties in the analysis of the associated linear program. Under a very weak hypothesis, it is shown that the linear problem admits an optimal solution, guaranteeing the existence of an optimal stopping strategy for the optimal stopping problem with constraints.

Approximations of the optimal stopping problem in partial observation

1986 ◽  
Vol 23 (2) ◽  
pp. 341-354 ◽  
Author(s):  
G. Mazziotto
Keyword(s):  
Optimal Stopping ◽  
Measure Space ◽  
Stopping Time ◽  
Optimal Stopping Problem ◽  
Snell Envelope ◽  
Infinite Dimensional ◽  
Markov State ◽  
State Process ◽  
Partially Observed ◽  
True State

The resolution of the optimal stopping problem for a partially observed Markov state process reduces to the computation of a function — the Snell envelope — defined on a measure space which is in general infinite-dimensional. To avoid these computational difficulties, we propose in this paper to approximate the optimal stopping time as the limit of times associated to similar problems for a sequence of processes converging towards the true state. We show on two examples that these approximating states can be chosen such that the Snell envelopes can be explicitly computed.

Using EWMA control schemes for monitoring wafer quality in negative binomial process

2011 ◽  
Vol 51 (2) ◽  
pp. 400-405 ◽  
Author(s):  
Fong-Jung Yu ◽  
Yung-Yu Yang ◽  
Ming-Jaan Wang ◽  
Zhang Wu
Keyword(s):  
Negative Binomial ◽  
Control Schemes ◽  
Negative Binomial Process ◽  
Binomial Process

Negative binomial processes

1969 ◽  
Vol 6 (03) ◽  
pp. 633-647 ◽  
Author(s):  
Ole Barndorff-Nielsen ◽  
G. F. Yeo
Keyword(s):  
Finite Number ◽  
Compact Subset ◽  
Gaussian Processes ◽  
Negative Binomial ◽  
Sum Of Squares ◽  
Poisson Processes ◽  
Intensity Parameter ◽  
Negative Binomial Process ◽  
Binomial Process

Summary This paper is concerned with negative binomial processes which are essentially mixed Poisson processes whose intensity parameter is given by the sum of squares of a finite number of independently and identically distributed Gaussian processes. A study is made of the distribution of the number of points of a k-dimensional negative binomial process in a compact subset of Rk , and in particular in the case where the underlying Gaussian processes are independent Ornstein-Uhlenbeck processes when more detailed results may be obtained.

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