Krzysztof Szajowski

Markov stopping games with random priority

In the paper a construction of Nash equilibria for a random priority finite horizon two-person non-zero sum game with stopping of Markov process is given. The method is used to solve the two-person non-zero-sum game version of the secretary problem. Each player can choose only one applicant. If both players would like to select the same one, then the lottery chooses the player. The aim of the players is to choose the best candidate. An analysis of the solutions for different lotteries is given. Some lotteries admit equilibria with equal Nash values for the players.

Immunohistochemically stained markers (p53, PCNA, bcl-2) in dysplastic lesions of the larynx.

The percentage of malignant transformation of laryngeal dysplastic lesions is difficult to estimate. There is a need for new histological markers which could enable more objective assessment of the premalignant stages of the larynx and help in estimation of the potential of future neoplastic progression. We performed a retrospective study to determine whether immunohistochemical staining for the proliferating cell nuclear antigen (PCNA), tumour suppressor gene protein p53 and antiapoptotic protein bcl-2 may be prognostic factors in laryngeal epithelial lesions. Staining was performed on 57 paraffin-embedded biopsies from patients with clinically detected precancerous stages of the larynx. Histopathologic examination revealed normal epithelium in six cases, mild dysplasia in 20 cases, moderate dysplasia in 18 cases, severe dysplasia in seven cases, CA in situ in four cases, papilloma in one case and CA invasivum in one case. The p53 count in mild and moderate dysplasia was 26.8 and 38.6%, respectively. This difference was statistically significant. There was significant correlation between PCNA and p53 scores. There was also a relationship between the scores of these markers and bcl-2 expression. In ten out of 45 cases of dysplastic lesions the invasive cancer developed in 4 years of follow-up. The correlation between PCNA score and malignant progression of the dysplastic lesions was on the statistical borderline. There was significant relationship between malignant transformation and age of the patients.

Voting Procedure on Stopping Games of Markov Chain

The paper deals with a p person, non-cooperative game related to the observation of a Markov chain. The players observe the process up to a random moment defined by a monotonic logical function based on an individual players’ decision. The concept of Nash equilibrium is used. The solution of the game for finite and infinite horizon problems is derived. A simple example is presented.

Double optimal stopping of a risk process

The following problem in risk theory is considered. An insurance company, endowed with an initial capital a>0, receives insurance premiums and pays out successive claims. The losses occur according to renewal process. At any moment, the company may broaden or narrow down the offer, what entails the change of the parameters. This change concerns the rate of income, the intensity of renewal process and the distribution of claims. After the change, the management wants to know the moment of the maximal value of the capital assets. Therefore, our goal is finding two optimal stopping times: the best moment of change the parameters and the moment of maximal value of the capital assets. We will use a dynamic programming method to calculate the expected capital at that times.

Double optimal stopping times and dynamic pricing problem: description of the mathematical model

In many industries, managers face the problem of selling a given stock of items by a deadline. We investigate the problem of dynamically pricing such inventories when demand is price sensitive and stochastic and the firm’s objective is to maximize expected revenues. Examples that fit this framework include retailers selling fashion and seasonal goods and the travel and leisure industry, which markets space such as seats on airline flights, cabins on vacation cruises, hotels renting rooms before midnight and theaters selling seats before curtain time that become worthless if not sold by a specific time. Given a fixed number of seats, rooms, or coats, the objective for these industries is to maximize revenues in excess of salvage value. When demand is price sensitive and stochastic, pricing is an effective tool to maximize revenues. In this paper, we address the problem of deciding the optimal timing of a double price changes from a given initial price to given lower or higher prices. Under mild conditions, it is shown that it is optimal to decrease the initial price as soon as the time-to-go falls below a time threshold and increase the price if time-to-go is longer than adequate time threshold. These thresholds depend on the number of yet unsold items.

Optimal detection of transition probability change in random sequence

In this paper, we register a random sequence constructed based on Markov processes by switching between them. At unobservable random moment, a change in distribution of observed sequence takes place. Using probability maximizing approach, the optimal stopping rule for detecting the disorder is identified. Some explicit solution for example is also obtained. The result is the generalization of Bojdeckis model, where before and after the change independent processes are observed.We register a random sequence constructed based on Markov processes by switching between them. At unobservable random moment a change in distribution of observed sequence takes place. Using probability maximizing approach the optimal stopping rule for detecting the disorder is identified. Some explicit solution for example is also obtained. The result is generalization of Bojdecki’s model where before and after the change independent processes are observed. Keywords. Disorder problem, sequential detection, optimal stopping, Markov process, change point.

Bayesian optimal control for a non-autonomous stochastic discrete time system

The main objective of this article is to develop Bayesian optimal control for a class of non-autonomous linear stochastic discrete time systems. By taking into consideration that the disturbances in the system are given by a random vector with components belonging to an exponential family with a natural parameter, we prove that the Bayes control is the solution of a linear system of algebraic equations. For the case that this linear system is singular, we apply optimization techniques to gain the Bayesian optimal control. Furthermore, we extend these results to generalized linear stochastic systems of difference equations and provide the Bayesian optimal control for the case where the coefficients of this type of systems are non-square matrices.

Multi-variate quickest detection of significant change process

The paper deals with a mathematical model of a surveillance system based on a net of sensors. The signals acquired by each node of the net are Markovian process, have two different transition probabilities, which depends on the presence or absence of a intruder nearby. The detection of the transition probability change at one node should be confirmed by a detection of similar change at some other sensors. Based on a simple game the model of a fusion center is then constructed. The aggregate function defined on the net is the background of the definition of a non-cooperative stopping game which is a model of the multivariate disorder detection.

Shelf life of candidates in the generalized secretary problem

The study presents a version of the secretary problem called the duration problem in which the objective is to maximize the time of possession of the relatively best or the second best objects. It is shown that in this duration problem there are threshold numbers such that the optimal strategy is determined by them. When the number of objects tends to infinity the thresholds values are ź 0.120381 N ź and ź 0.417188 N ź , respectively, and the asymptotic mean time of shelf life is 0.403827 N .

Duration problem with multiple exchanges

We treat a version of the multiple-choice secretary problem called the multiple-choice duration problem, in which the objective is to maximize the time of possession of relatively best objects. It is shown that, for the