Panagiotis Vassiliou

Panagiotis Vassiliou
University College London | UCL · Department of Statistical Science

Doctor of Philosophy

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97
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Publications

Publications (97)
Article
In the present we establish a law of large numbers for non-homogeneous Markov systems (NHMS), for which the inherent non-homogeneous Markov chain has arbitrary transition probability matrices which are given in chronological order. We start by providing conditions under which we get Cesaro convergence for a non-homogeneous Markov chain for which tr...
Article
Full-text available
The stochastic process non-homogeneous Markov system in a stochastic environment in continuous time (S-NHMSC) is introduced in the present paper. The ordinary non-homogeneous Markov process is a very special case of an S-NHMSC. I studied the expected population structure of the S-NHMSC, the first central classical problem of finding the conditions...
Article
Probability resembles the ancient Roman God Janus since, like Janus, probability also has a face with two different sides, which correspond to the metaphorical gateways and transitions between the past and the future [...]
Article
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A more realistic way to describe a model is the use of intervals which contain the required values of the parameters. In practice we estimate the parameters from a set of data and it is natural that they will be in confidence intervals. In the present study, we study Non-Homogeneous Markov Systems (NHMS) processes for which the required basic param...
Article
Full-text available
For a G-inhomogeneous semi-Markov chain and G-inhomogeneous Markov renewal processes, we study the change from real probability measure into a forward probability measure. We find the values of risky bonds using the forward probabilities that the bond will not default up to maturity time for both processes. It is established in the form of a theore...
Article
Full-text available
In the present we establish Laws of Large Numbers for Non-Homogeneous Markov Systems and Cyclic Non-homogeneous Markov systems. We start with a theorem, where we establish, that for a NHMS under certain conditions, the fraction of time that a membership is in a certain state, asymptotically converges in mean square to the limit of the relative popu...
Article
Full-text available
In this article we study the asymptotic behaviour of the expected population structure of a Markov system that lives in a general state space (MSGS) and its rate of convergence. We continue with the study of the asymptotic periodicity of the expected population structure. We conclude with the study of total variability from the invariant measure in...
Article
We firstly recall the concept of an NHMS and introduce concepts and known results necessary for the study of asymptotic periodicity of the vector of expectations, variances and covariances of the state sizes. This sequence of vectors is called the variability of population structures. We then proceed to prove some useful propositions and lemmas for...
Article
In the present the idea of stochastic Market environment comes into play to express the changes in general economy, which affects any industry in small or great amounts of turbulence. We model the evolution of the Market among its possible ν -states as an FF-inhomogeneous semi-Markov process. This idea leads us to modeling the migration process of...
Article
In the present article we study the asymptotic behaviour of the sequence of vectors with components expectations, variances, and covariances of the state sizes of a semi-Markov system. In this respect, we transform the semi-Markov system into a Markov system with a different though equivalent state space and relate the sequence of the transition pr...
Article
In this article, we introduce and study Markov systems on general spaces (MSGS) as a first step of an entire theory on the subject. Also, all the concepts and basic results needed for this scope are given and analyzed. This could be thought of as an extension of the theory of a non homogeneous Markov system (NHMS) and that of a non homogeneous semi...
Article
In this article we study what we chose to call exotic properties of NHMS and NHSMS. The interplay between stochastic theory of NHMS and NHSMS and other branches of probability, stochastic processes and mathematics, we believe is a fascinating one apart from being important. In many cases the information needed for the evolution of a NHMS is a large...
Article
We explore the rating system used by credit agencies with a focus on problems that justify the use of fuzzy set theory. We prove that a fuzzy market is viable if and only if an equivalent martingale measure exists, from which we construct the forward probability measure and under which the discounted price of a default-free bond is a martingale. We...
Article
We start with the stochastic foundation of the general discrete-time Market of defaultable bonds. We prove that the above market is viable, if and only if there exists an equivalent martingale measure, from which we construct the forward probability measure and under which the discounted default free bond price is a martingale. Assuming that the mi...
Article
Stochastic finance and financial engineering have been rapidly expanding fields of science over the past four decades, mainly due to the success of sophisticated quantitative methodologies in helping professionals manage financial risks. In recent years, we have witnessed a tremendous acceleration in research efforts aimed at better comprehending,...
Chapter
The construction of stochastic models for the pricing of financial instruments, especially in continuous time, often involves very sophisticated stochastic and mathematical concepts. This chapter first discusses the bonds and basic interest rates, and then describes forward contracts. A futures contract is similar to a forward contract in the sense...
Chapter
The theory of martingales relies on the theory of conditional expectation. This chapter discusses the theory of martingales. Option pricing, and the Black and Scholes formula are all based on martingales and stochastic calculus. Stochastic finance and martingale theory are a very closely related, however, stochastic finance has also posed some prob...
Chapter
This chapter explains what is termed by many authors as fixed income markets, that is, the sector of the global financial market on which various interest rate securities, such as bonds, swaps, and others are traded. The simplest fixed income asset is a zero coupon bond, a bond that pays a specified amount called its face value or par value at a sp...
Chapter
This introductory chapter of the book presents the gist of probability theory. It first discusses the probability space, and the conditional probability and independence. The chapter then introduces the concept of the random variable with an example. A random variable that takes integer values is called a discrete random variable. A random variable...
Chapter
This chapter starts by presenting the definition of an American call option. It then presents the algorithm for pricing an American put option with maturity time T = 3. This naturally will serve as a model for building an algorithm for pricing an American option for any T. The chapter establishes the hedging strategy for the writer of an American p...
Chapter
This chapter starts by providing the definition of two equivalent probability measures, and then discusses the Randon-Nikodým derivative process. It explains the generalization of our market and consequently generalizes some of the results already obtained. The also chapter considers the finite general markets, i.e. discrete time models of financia...
Chapter
Introductory thoughts Genesis The Decisive Steps A brief glance towards the flow of research paths
Chapter
To deal with dynamic situations that include a large amount of randomness that evolve with time the person needs more sophisticated instruments. These are part of the theory of stochastic process, and for finance especially, the parts of Markov processes and martingales. This chapter starts with a discussion of a basic knowledge for conditional exp...
Chapter
Duffie and Singleton (2003) very correctly classified the risks involved for a financial institution operating in various markets into Market risk, Credit risk, Liquidity risk, Operational risk, and Systematic risk. This chapter mainly discusses the market of corporate bonds of all maturities where various kinds of credit derivatives are written. T...
Chapter
This chapter discusses the discrete time approximation of the continuous time model of Heath, Jarrow and Morton (HJM) (1992, 1990). It presents the HJM model in four steps. The first step is the study of the evolution of forward rate process. The second step is the study of evolution of the savings account and the short-term interest rate process....
Chapter
This chapter introduces the no-arbitrage binomial pricing model and discusses arbitrage pricing and hedging of derivative securities. Specifically, it uses this model for the pricing and hedging of European options, which yield important insights into the pricing and hedging of other derivative securities. The chapter first explains this model and...
Chapter
Introduction The main theorem
Article
Full-text available
In the present, we introduce and study the G-inhomogeneous Markov system of high order, which is a more general in many respects stochastic process than the known inhomogeneous Markov system. We define the inhomogeneous superficial razor cut mixture transition distribution model extending for the homogeneous case the idea of the mixture transition...
Conference Paper
In the present, we present a study of variability and asymptotic variability of G−inhomogeneous Markov systems of high order. These are modelled by the superficial razor cut mixture transition model and it is established that asymptotically the variability of the two processes converge to the same limit.
Article
Full-text available
In this paper the periodicity of a perturbed non homogeneous Markov system (P-NHMS) is studied. More specifically, the concept of a periodic P-NHMS is introduced, when the sequence of the total transition matrices {Q(t)} ∞ t=0 does not converge, but oscillates among several different matrices, which are rather close to a periodic matrix Q, with per...
Article
We model the evolution of the credit migration of a corporate bond as an inhomogeneous semi-Markov chain. The valuation of a defaultable bond is done with the use of the forward probability of no default up to maturity time. It is proved that, under the forward probability measure, the semi-Markov property is maintained. We find the functional rela...
Article
We model the evolution of the credit migration of a corporate bond as an inhomogeneous semi-Markov chain. The valuation of a defaultable bond is done with the use of the forward probability of no default up to maturity time. It is proved that, under the forward probability measure, the semi-Markov property is maintained. We find the functional rela...
Article
Full-text available
In the present we study the asymptotic behavior of the sequence of the first passage probability matrices in a Perturbed Non-Homogeneous Semi-Markov System (P-NHSMS). We study initially triple sequences of matrices and we establish a basic result which provides the conditions under which the triple limit of a sequence coinsides. Then we provide som...
Chapter
In the present we study the evolutions of various population structures in a Perturbed Non-Homogeneous Semi-Markov System (P-NHSMS) with different goals in mind. Firstly we start with the expected population structure with respect to the first passage time probabilities and we follow with the study of the expected population structures with respect...
Article
In this paper we introduce for the first time the concept of a perturbed nonhomogeneous Markov system (P-NHMS). The expected population structure is found and its asymptotic behavior is provided under more realistic assumptions than previous studies, by relaxing the assumption that the imbedded nonhomogeneous Markov chain of a NHMS is converging to...
Article
In the present we introduce and define the non-homogeneous semi-Markov system in continuous time. We study the problem of finding the expected population structure in closed analytic form, in relation with the basic sequences of the system. Moreover, the problem of the asymptotic behavior of the system is studied under certain conditions. We first...
Chapter
In the present we introduce and define for the first time the concept of a perturbed non-homogeneous semi-Markov system (P-NHSMS). We study the problem of the expected population structure as a function of the basic parameters of the system.
Article
A review of the evolution of the theory of non-homogeneous Markov systems (NHMSs) is given. It starts with the definition of the stochastic process NHMS, discusses the variability of the process and provides as examples some real applications of the theory. Then, important novel ideas and basic theorems are presented on: (i) the asymptotic behaviou...
Article
In the present paper, we present the concept of a perturbed non-homogeneous Markov system in continuous time. The expected population structure of the system is found and its asymptotic behaviour is provided under more realistic assumptions than in previous studies, by relaxing the assumption that the imbedded non-homogeneous Markov chain of the NH...
Article
We study the problem of introducing the phases through which a nonhomogeneous Markov system (NHMS) passes, as time goes to infinity. For each phase an appropriate objective function is introduced and a cost problem is formulated. Appropriate input policies subject to the cost objective functions which are established on the Markov manpower system a...
Article
We introduce and define for the first time the concept of a non-homogeneous semi-Markov system in a stochastic environment. We study the problem of finding the expected population structure as a function of the basic parameters of the system. Important properties are established among the basic parameters of a non-homogeneous semi-Markov system in...
Article
We study the asymptotic behavior of a nonhomogeneous semi-Markov system (population) in discrete time. After a series of definitions, lemmas, and theorems, we firstly establish the conditions under which the ergodic behavior of a nonhomogeneous semi-Markov chain exists and then find the limit of the basic matrix of the chain ℚ(n,s) in closed form....
Article
We introduce and define for the first time the concept of a non- homogeneous Markov system in a stochastic environment (S-NHMS). The problem of finding the expected population structure in an S-NHMS is studied, and important properties among the basic parameters of the S- NHMS are established. Moreover, we study the problem of maintaining the relat...
Article
We introduce and define for the first time the concept of a non-homogeneous Markov system in a stochastic environment (S-NHMS). The problem of finding the expected population structure in an S-NHMS is studied, and important properties among the basic parameters of the S-NHMS are established. Moreover, we study the problem of maintaining the relativ...
Article
The problem of periodicity for the infinite products of a class of matrices, the V-matrices, with some negative elements and row sums equal to one is studied. After a series of lemmas, propositions, and theorems, a basic theorem is proved: that an infinite product of V-matrices under certain conditions splits into a number of subsequences which con...
Article
We find the sets of d-periodic asymptotically attainable structures, and we establish the periodicities that exist between these structures, for a nonhomogeneous Markov system in the case where the imbedded nonhomogeneous Markov chain is periodic with period d. Also, it is proved that under certain conditions each converging subsequence of the sequ...
Article
In this paper we introduce and define for the first time the concept of a non-homogeneous semi-Markov system (NHSMS). The problem of finding the expected population stucture is studied and a method is provided in order to find it in closed analytic form with the basic parameters of the system. Moreover, the problem of the expected duration structur...
Article
In this paper we introduce and define for the first time the concept of a non-homogeneous semi-Markov system (NHSMS). The problem of finding the expected population stucture is studied and a method is provided in order to find it in closed analytic form with the basic parameters of the system. Moreover, the problem of the expected duration structur...
Article
In this paper we provide two basic results. First, we find the set of all the limiting vectors of expectations, variances and covariances in an NHMS which are possible provided that we control the limit vector of the sequence of vectors of input probabilities. Secondly, under certain conditions easily met in practice we find the distribution of the...
Article
In this paper we provide two basic results. First, we find the set of all the limiting vectors of expectations, variances and covariances in an NHMS which are possible provided that we control the limit vector of the sequence of vectors of input probabilities. Secondly, under certain conditions easily met in practice we find the distribution of the...
Article
The aims of this paper are twofold. The first is to study the rate of convergence of the sequence of vectors of expectations, variances and covariances in NHMS. It is proved that under certain conditions easily met in practice the rate is geometric. The second is to study the rate of convergence of the same sequence when the NHMS is under cyclic be...
Article
The aims of this paper are twofold. The first is to study the rate of convergence of the sequence of vectors of expectations, variances and covariances in NHMS. It is proved that under certain conditions easily met in practice the rate is geometric. The second is to study the rate of convergence of the same sequence when the NHMS is under cyclic be...
Article
In this paper we study the asymptotic periodicity of the sequence of means, variances and covariances of the state sizes of non-homogeneous Markov systems. It is proved that under the assumption that the sequence of the extended stochastic transition matrices converge to a matrix which is an irreducible stochastic matrix of period d, and all the ma...
Article
In this paper we study the asymptotic periodicity of the sequence of means, variances and covariances of the state sizes of non-homogeneous Markov systems. It is proved that under the assumption that the sequence of the extended stochastic transition matrices converge to a matrix which is an irreducible stochastic matrix of period d , and all the m...
Article
In this paper the asymptotic variability of nonhomogeneous Markov systems under cyclic behaviour is studied. It is found that the sequence of vectors of means variances and covariances splits into d subsequences which converge, where d is the period of the cyclic. Finally, an illustration of the present results with data from a manpower system is g...
Article
Full-text available
The asymptotic behaviour of the variances and covariances of the class sizes in closed and open manpower systems is considered. Firstly, the homogeneous case is studied and a theorem is stated which provides the answer to the problem in the most general case for the homogeneous Markov-chain models in manpower systems (open systems) and social mobil...
Article
Full-text available
The asymptotic behaviour of the variances and covariances of the class sizes in closed and open manpower systems is considered. Firstly, the homogeneous case is studied and a theorem is stated which provides the answer to the problem in the most general case for the homogeneous Markov-chain models in manpower systems (open systems) and social mobil...
Article
Full-text available
It is important when studying a manpower system to be able to measure stability of the experience distribution and thus evade possible trauma. This paper develops the use of entropy as such a measure, both for the comparison of different companies in steady state, and as a means of monitoring the changes in tenure distribution in time for a particu...
Article
We study the problem of n-step maintainability of the structure in a nonhomogeneous Markov system (NHMS), which undergoes a cyclic behavior of n steps under input control. The family of n-step maintainable structures is determined as a convex hull of k n points in ℝ k , where k is the number of states of the NHMS. We also provide applications of th...
Article
In this paper we study the cyclic behaviour of non-homogeneous Markov systems, i.e. the behaviour of the system under the assumption of periodic sequences of transition matrices, input probabilities, output probabilities and total numbers in the system. We provide a general theorem for the limiting structure of such a system under the cyclic behavi...
Article
In this paper we study the cyclic behaviour of non-homogeneous Markov systems, i.e. the behaviour of the system under the assumption of periodic sequences of transition matrices, input probabilities, output probabilities and total numbers in the system. We provide a general theorem for the limiting structure of such a system under the cyclic behavi...
Article
The asymptotic behavior of Markov systems and especially non-homogeneous Markov systems is studied. It is found that the limiting structure and the relative limiting structure exist under certain conditions. The problem of weak ergodicity in the above non-homogeneous systems is studied. Necessary and sufficient conditions are provided for weak ergo...
Article
In this paper we study the asymptotic behavior of Markov systems and especially non-homogeneous Markov systems. It is found that the limiting structure and the relative limiting structure exist under certain conditions. The problem of weak ergodicity in the above non-homogeneous systems is studied. Necessary and sufficient conditions are provided f...
Article
We study the limiting behaviour of a manpower system where the non-homogeneous Markov chain model proposed by Young and Vassiliou (1974) is applicable. This is done in the cases where the input is a time-homogeneous and time-inhomogeneous Poisson random variable. It is also found that the number in the various grades are asymptotically mutually ind...
Article
We study the limiting behaviour of a manpower system where the non-homogeneous Markov chain model proposed by Young and Vassiliou (1974) is applicable. This is done in the cases where the input is a time-homogeneous and time-inhomogeneous Poisson random variable. It is also found that the number in the various grades are asymptotically mutually ind...
Article
Stochastic models for the prediction and description of the service in the grade distribution in the various grades of a hierarchically structured manpower system are given in the present. An application of the present stochastic models in a large British firm is given. Interesting aspects of the promotion and wastage flows are revealed by the pres...
Article
Necessary and sufficient conditions for stability, imposed firstly on the initial structure and the sequence of recruitment, and secondly on the initial structure and the sequence of expansion are provided in forms of two theorems. Also the limiting behaviour of the expected relative grade sizes is studied if we drop the conditions for stability im...
Article
Necessary and sufficient conditions for stability, imposed firstly on the initial structure and the sequence of recruitment, and secondly on the initial structure and the sequence of expansion are provided in forms of two theorems. Also the limiting behaviour of the expected relative grade sizes is studied if we drop the conditions for stability im...
Article
We study the limiting structure of an expanding manpower system where the nonhomogeneous Markov chain model proposed by Young & Vassiliou (1974) is applicable. The limiting structure and the relative limiting structure exist under certain conditions which are easily met in practical situations. Finally the problem of weak ergodicity in the above ma...
Article
Full-text available
The literature on supply models for manpower planning shows that an important consideration is the size of the discrepancy between the age or length of service distribution of the population and the age distribution which would be reached if present policies were continued indefinitely. In the present paper we study the asymptotic behaviour of the...
Article
Full-text available
The paper provides a generalization of a result given by Feichtinger on the stability in Gani-type models in manpower systems. This generalization on the important question of stability is important from the practical point of view, because it can be used for any Markovian manpower system.
Article
Certain aspects of the simple random walk with one random absorbing barrier and one non-random absorbing barrier are presented in this paper. The probability distributions of the maximum distance upwards run by the barrier at time n are studied. This is done in two ways: (i) difference equations for certain elementary probabilities are derived and...
Article
A high order non-linear Markovian model for a hierarchically graded organization is given in this paper. A stochastic model for promotion is given here based on two relevant features: an ecological principle that promotion rates depend both on the number available in the grade from which they come and on the expected vacancies in the grade to which...
Article
A stochastic model for the description, prediction and control of wastage in hierarchically structured manpower systems is given in this paper. In the model we identify reasons for leaving. When the expansion of the organization slows down people start to leave faster; conversely when expansion increases wastage tends to decrease. Thus the probabil...
Article
As organizations develop, the vacancies in higher grades to be filled by promotion fluctuate from year to year. Thus the probabilities of individuals being promoted are not static. A stochastic model of promotion in such organizations is given here based on an ecological principle that the probability of a principle of inertia which takes account o...
Article
The techniques presented in this paper are applicable to the valuation of general corporate liabilities, corporate loans etc. However, merely for presentation purposes we choose to limit the discussion to defaultable corporate bonds. It is important to note though that most of the present results are of interest in their own right in the theory of...