Maciej Romaniuk

Systems Research Institute | IBSPAN · Department of Stochastic Methods

The precise and large dataset concerning catastrophic events is very important for insurers. To improve the quality of such data three methods based on the bootstrap, bootknife, and GAN algorithms are proposed. Using numerical experiments and real-life data, simulated outputs for these approaches are compared based on the mean squared (MSE) and mea...

Epistemic bootstrap is a resampling algorithm that generates bootstrap real-valued samples based on some epistemic fuzzy data input. We apply this method as a universal basis for various statistical tests which can be then directly used for fuzzy random variables. Two classical goodness-of-fit tests are considered as an example to examine the sugge...

Computer simulations are a powerful tool in many fields of research. This also applies to the broadly understood analysis of experimental data, which are frequently burdened with multiple imperfections. Often the underlying imprecision or vagueness can be suitably described in terms of fuzzy numbers which enable also the capture of subjectivity. On...

The classical bootstrap has proven its usefulness in many areas of statistical inference. However , some shortcomings of this method are also known. Therefore, various bootstrap modifications and other resampling algorithms have been introduced, especially for real-valued data. Recently, bootstrap methods have become popular in statistical reasonin...

The DE-MC algorithm joins two approaches: the differential evolution and the theory of the Markov chains. This population MCMC method aims to improve the numerical effectiveness and the convergence speed of the Metropolis-Hastings algorithm. In this paper, we equip this standard approach with different unsupervised and automated methods for outlier...

A typical two-sample problem we deal with in statistical reasoning is to conclude whether the distributions of these samples differ significantly. In other words, we want to know if the given two samples come from the same distribution. To cope with this task one can utilize the Kolmogorov-Smirnov test which is probably the most popular goodness-of...

The bootstrap has been recognized as a universal method supporting statistical inference in situations where other approaches fail. In particular, the bootstrap appears also useful in fuzzy data analysis. In this paper, we make a review of the bootstrap methods oriented on statistical reasoning with random fuzzy samples. Moreover, we suggest a new...

Fuzzy data applied for modeling imprecise observations cause many problems in statistical reasoning and data analysis. To handle better such observations a new bootstrap technique designed for epistemic fuzzy data is proposed. Our new method is conceptually simple and is not hard computation-ally. Some simulation results reported in the paper show...

Several new resampling methods for generating bootstrap samples of fuzzy numbers are proposed. To avoid undesired repetitions in the secondary samples we do not draw randomly directly observations from the primary samples but construct them allowing for some modifications in their membership functions, however only such which do not disturb the can...

In this paper, a new methodology for simulating bootstrap samples of fuzzy numbers is proposed. Unlike the classical bootstrap, it allows enriching a resampling scheme with values from outside the initial sample. Although a secondary sample may contain results beyond members of the primary set, they are generated smartly so that the crucial charact...

In this paper, imprecise approaches to model the risk reserve process of an insurer’s portfolio, which consists of a catastrophe bond and external help, and with a special penalty function in the case of a bankruptcy event, are presented. Apart from the general framework, two special cases, when parameters of the portfolio are described by L-R fuzz...

In this paper, we discuss imprecise settings for an evaluation of the maintenance costs of a water distribution system (WDS). Moments of failures of pipes are modelled using a newly proposed three-piece convex hazard rate function (HRF) for which number of previous failures is taken into account, too. Both fuzzy sets and shadowed sets are used to m...

In this paper, we propose two new resampling algorithms for the simulation of bootstrap-like samples of interval-valued fuzzy numbers (IVFNs). These methods (namely, the d-method and the s-method) re-use a primary sample (an initial set) of IVFNs to generate a secondary sample, which also consists of this type of fuzzy numbers, and simultaneously u...

In this paper, we discuss two practically oriented applications of computer-aided simulations, which are conducted in a fuzzy setting. The first example is related to the Monte Carlo (MC) approach, which is applied to estimate the present value of maintenance costs of a water distribution system (WDS). Some parameters of this WDS are given by fuzzy...

A new resampling approach for simulating bootstrapped samples of fuzzy numbers is proposed. The secondary samples consist of fuzzy numbers which preserve the canonical representation (i.e., the value and ambiguity) of fuzzy numbers belonging to the primary sample, although may differ from the initial ones. This way the resulting bootstrap distribut...

In this paper, we propose two new nonparametric resampling methods for the simulation of bootstrap-like samples of fuzzy numbers. The generated secondary samples are based on an input set (i.e., a primary sample) consisting of left–right fuzzy numbers. The proposed approaches utilize random simulations in a way which, to some extent, resembles a bo...

Celem niniejszego podręcznika jest zapoznanie czytelników z tematyką statystycznych symulacji komputerowych ze szczególnym uwzględnieniem metod Monte Carlo (MC) i Markov chain Monte Carlo (MCMC). Omówiono w nim m.in. podstawowe pojęcia związane z generowaniem liczb (pseudo)losowych, metody i algorytmy służące do generowania zmiennych z różnych, uży...

In this paper, important properties of an insurer’s portfolio, which consists of a catastrophe bond and a reinsurance contract, are numerically analysed. Because of stochastic nature of considered processes, simulations and the Monte Carlo (MC) methods are applied. Special attention is paid to estimation and optimization of an average value of the...

In this paper I focus on an evaluation of maintenance costs of a water distribution system (WDS), if a concept of a value of money in time is taken into account. Contrary to more classical approaches, instead of a constant yield, a strictly stochastic process (i.e., the one-factor Vasicek model) of an interest rate is assumed. Such an assumption pr...

In this paper, an integrated insurer's portfolio, which consists of a few layers of insurance and financial instruments, is numerically analysed. A future behaviour of such a portfolio is related to stochastic processes (like a random interest rate yield and uncertain catastrophic losses), therefore the Monte Carlo (MC) approach is applied. A speci...

In this paper, a behavior of an insurer’s portfolio, which consists of two layers: a classical risk process and a special financial instrument, which is known as a catastrophe bond, is analyzed. Especially, a probability of a ruin for such a portfolio is estimated using the Monte Carlo simulations. A special attention is given to a problem of an in...

Catastrophe bonds are financial instruments, which enable to transfer the natural catastrophe risk to financial markets. This paper is a continuation of our earlier research concerning catastrophe bond pricing. We assume the absence of arbitrage and neutral attitude of investors toward catastrophe risk. The interest rate behavior is described by th...

In this paper we propose a model for evaluation of maintenance costs of a water distribution system (WDS). The set of possible states of each connection (i.e. a pipeline in the WDS) is related to various possible degrees of quality of the pipe and types of its malfunctions. The process of transitions between these states forms a semi-Markov process...

Natural catastrophes lead to problems of insurance and reinsurance industry. Classic insurance mechanisms are often inadequate for dealing with consequences of catastrophic events. Therefore, new financial instruments, including catastrophe bonds (cat bonds), were developed. In this paper we price the catastrophe bonds with a generalized payoff str...

In this paper we assume that the behaviour of each section of water pipes network may be modelled using Markov chain. The states of this Markov chain are related to various types of maintenance services and probabilities of failures. Taking into account the related intensity functions and the derived matrix of transition probabilities, the behaviou...

The number and amount of losses caused by natural catastrophes are important problems for insurance industry. New financial instruments were introduced to transfer risks from insurance to financial market. In this paper we consider the problem of pricing such instruments, called the catastrophe bonds (CAT bonds). We derive valuation formulas using...

In this paper we consider the European option valuation problem. We assume that the underlying asset follows a geometric Levy process. The log-price is a sum of a Brownian motion with drift and a linear combination of Poisson processes describing jumps in price. In our approach we use martingale method and theory of fuzzy sets. To obtain the Europe...

The number of natural catastrophes and losses caused by them increase in time. The damages caused by natural disasters are difficult to handle for insurers. Therefore catastrophe bonds were introduced to transfer the catastrophic risk to financial markets. In this paper we continue our research concerning catastrophe bond pricing. In our approach w...

In this paper the problem of European option valuation in a Levy process setting is analysed. In our model the underlying asset follows a geometric Levy process. The jump part of the log-price process, which is a linear combination of Poisson processes, describes upward and downward jumps in price. The proposed pricing method is based on stochastic...

The increasing number of natural catastrophes like floods, hurricanes, and earthquakes not only causes many victims, but also leads to severe production, infrastructure, and individual property losses. Classic insurance mechanisms may be inadequate for dealing with such losses because of the dependencies that exist, inter alia, between the sources...

In the paper we consider catastrophe bonds with a stepwise payoff structure.We use the martingale method to price it under the condition of no arbitrage. We assume a stochastic form of the spot interest rate, replicability of interest rate changes by financial instruments existing in the market as well as independence between a catastrophe occurren...

The increasing number of natural catastrophes leads to severe losses for production, in infrastructure and individual property. Classical insurance mechanisms may not be sufficient in dealing with such losses because of dependencies among sources of losses, huge values of damages, problems with adverse selection and moral hazard. To cope with drama...

The increasing number and value of losses of natural catastrophes leads to problems with financial reserves for many insurers. The classical insurance mechanisms are not suitable for losses caused by natural catastrophes. Therefore new financial mechanisms like catastrophe bonds (in abbreviation cat bond) or options may be used to cope with consequ...

Klasyczny model ubezpieczeniowy związany jest z występowaniem częstych, ale niewielkich i niezależnych szkód. Tymczasem katastrofy naturalne powodują szkody rzadko występujące, o dużych wartościach, zależne geograficznie i czasowo. Dlatego coraz częstsze katastrofy naturalne skutkują problemami ze stabilnością finansową ubezpieczycieli. W związku z...

In this paper we discuss the example of catastrophe bond with linear payment function. The approach based on neutral martingale method is used. In order to price the catastrophe bond we use fuzzy parameters and apply Vasicek model under assumption of independence between catastrophe occurrence and behaviour of financial market. Then the Monte Carlo...

In the following paper we propose the method for option pricing based on application of stochastic analysis and theory of fuzzy numbers. The process of underlying asset trajectory belongs to a subclass of Levy processes with jumps. From practical point of view some parameters of such trajectory cannot be precisely described. Therefore, some degree...

The increasing number of natural catastrophes leads to severe losses. Classical insurance mechanisms based on traditional insurance approach may not be sufficient in dealing with. Even one, single catastrophe could cause problems with reserves for many insurers. Therefore to cope with dramatic consequences of such events integrated policy is requir...

Klasyczny model ubezpieczeniowy związany jest z występowaniem częstych, ale niewielkich i niezależnych szkód. Tymczasem katastrofy naturalne powodują szkody rzadko występujące, o dużych wartościach, zależne geograficznie i czasowo. Dlate-go coraz częstsze katastrofy naturalne powodują problemy ze stabilnością finansową ubezpieczycieli. W związku z...

In this paper we evaluate the model of portfolio which consists of a few layers of insurance and financial instruments. The approach based on neutral martingale method and Monte Carlo simulations is used. In order to price the catastrophe bond we use fuzzy parameters and apply Vasicek model under assumption of independence between catastrophe occur...

The emission trading is proposed in the Kyoto Protocol. An appropriate market and the market of financial derivatives for allowances will be established. Using the neutral martingale method and Monte Carlo simulations, we propose a stochastic model with a pricing formula, which may be useful for an evaluation of derivatives inspired by the Kyoto Pr...

1. Wartość pieniądza w czasie 1.1 Podstawowe pojęcia 1.2 Procent prosty i złożony 1.3 Stopy procentowe 1.3.1 Stopa nominalna 1.3.2 Stopa efektywna 1.3.3 Warunek zgodności 1.3.4 Chwilowa stopa zwrotu 1.3.5 Dyskontowanie. Stopa dyskontowa 1.3.6 Niektóre realne stopy procentowe 1.4 Strumień pieniądza w czasie 1.4.1 Strumień dyskretny 1.4.2 Strumień ci...

MCMC setups are one of the best known methods for conducting computer simulations useful in such areas as statistics, physics, biology, etc. However, to obtain appropriate solutions, the additional convergence diagnosis must be applied for Markov Chain trajectory generated by the algorithm. We present the method for dealing with this problem based...

In the following paper we propose the approach to option pricing problem based on of stochastic analysis and theory of fuzzy numbers. The process of underlying asset trajectory belongs to a subclass of Levy processes with jumps. From practical point of view some parameters of such trajectory cannot be precisely described because of some uncertainti...

In the following paper we propose an application of fuzzy number theory for stochastic model of underlying asset trajectory based on some subclass of processes with stationary and independent increments. This approach is validated because of the fluctuation of financial market from time to time, some input parameters of the financial models cannot...

The natural, weather-related catastrophes lead to severe losses for agricultural production and decrease of farmers’ income. Classical insurance mechanisms may not be sufficient in such cases. They are often criticized because of serious problems with adverse selection and moral hazard. In this chapter we discuss an integrated catastrophic risk man...

MCMC setups are among the best known methods for conducting computer simulations necessary in statistics, physics, biology, etc. However, to obtain appropriate solutions, additional convergence diagnosis must be applied for trajectory generated by Markov Chain. In the paper we present the method for dealing with this problem, based on features of s...

The natural catastrophes lead to severe losses for agricultural production and decrease of farmers’ in come. These problems are especially important for poor and underdeveloped countries, which also face huge disasters like hurricanes, droughts, floods and earthquakes. Classical mechanisms of insurance may not be sufficient or not enough developed...

The aim of this paper is to discuss a spatially explicit integrated framework for management of catastrophic risks. Typical characteristics of catastrophes are highly mutually dependent and heterogeneous losses at locations and the lack of historical information for prediction of catastrophes. This calls for the development of catastrophe generator...

In this paper we discuss the introduction of continuous time assumption in automobile insurance premium calculation system for quadratic and LINEX loss functions. Such assumption corresponds to the situation when information and the premiums are continually flowing to the insurer. A short discussion on "hunger for bonus effect" for a model with the...

The article is devoted to finding the present value of catastrophe bonds using a combination of Monte Carlo and Iterative Stochastic Equation methods. Apart from general methodology, three practical examples of catastrophe bonds connected with earthquakes are also considered. For these examples algorithms in pseudocode with procedures originated fr...