Many important probabilistic models in queuing theory, insurance and finance deal with partial su... more Many important probabilistic models in queuing theory, insurance and finance deal with partial sums of a negative mean stationary process (a negative drift random walk), and the law of the supremum of such a process is used to calculate, depending on the context, the ruin probability, the steady state distribution of the number of customers in the system or the value at risk. When the stationary process is heavy-tailed, the corresponding ruin probabilities are high and the stationary distributions are heavy-tailed as well. If the steps of the random walk are independent, then the exact asymptotic behavior of such probability tails was described by Embrechts and Veraverbeke. We show that this asymptotic behavior may be different if the steps of the random walk are not independent, and the dependence affects the joint probability tails of the stationary process. Such type of dependence can be modeled, for example, by a linear process.
In 1713, the noted mathematician Jacob Bernoulli's paper Ars Conjectandi (Latin for Art of Conjec... more In 1713, the noted mathematician Jacob Bernoulli's paper Ars Conjectandi (Latin for Art of Conjecturing) was published, eight years after his death. It is unlikely that Bernoulli, or anyone for that matter, could have imagined the impact this article would have had, not only on the development of probably and statistics, but science at large. This paper would lay out many of the building blocks and foundational results that are taught in any first year course in statistics; from the simple distribution of the number of successes in n independent trials, which bears his name, to the law of large numbers. This law provides the basis for the frequentist view of statistics and underpins modern probability theory. But beyond this scientific framework, the law of large numbers "holds the world together in its inmost folds". It is only natural that the Bernoulli Society would devote a special issue of its flagship journal, Bernoulli, to help commemorate the 300th year anniversary of the publication of Ars Conjectandi. In this issue, we have tried to capture a snapshot of current and future research directions in probability and statistics from leading experts in the field. As you can see from the contents, the field has evolved into many exciting and important directions well beyond what Bernoulli could have imagined. We are indebted to the authors of this issue for their contributions and for sharing their thoughts with the scientific community. It is our hope these articles will inspire all of us to be faithful to the continuing development of probability and statistics that meets the challenges faced in science, technology, and society.
A pathwise approach to stochastic integral equations is advocated. Linear extended Riemann±Stielt... more A pathwise approach to stochastic integral equations is advocated. Linear extended Riemann±Stieltjes integral equations driven by certain stochastic processes are solved. Boundedness of the p-variation for some 0 , p , 2 is the only condition on the driving stochastic process. Typical examples of such processes are in®nite-variance stable Le Âvy motion, hyperbolic Le Âvy motion, normal inverse Gaussian processes, and fractional Brownian motion. The approach used in the paper is based on a chain rule for the composition of a smooth function and a function of bounded p-variation with 0 , p , 2.
this article will not focus on these more classical topics but mainly on the approach which was s... more this article will not focus on these more classical topics but mainly on the approach which was started by Fisher Black (1938-1995), Myron Scholes [6] and Robert Merton [25] in 1973. What was so entirely new that Scholes and Merton were awarded the Nobel prize for economics in 1997?
By means of a detailed analysis of the returns of the Standard & Poors 500 (S&P 500) composite st... more By means of a detailed analysis of the returns of the Standard & Poors 500 (S&P 500) composite stock index over the last fifty years we show how theoretical results and methodological recommendations from the statistical theory of non-parametric curve inference allow one to consistently estimate expected return and volatility. In this approach we do not postulate an a priori relationship risk-return nor do we specify the evolution of the first two moments through covariates. Our analysis gives statistical evidence that the expected return of the S&P 500 index as well as the market price of risk (the ratio expected return minus risk free interest rate over volatility) vary significantly through time both in size and sign. In particular, the periods of negative (positive) estimated expected return and market price of risk coincide with the bear (bull) markets of the index as defined in the literature. A complex relationship between risk and expected return emerges which is far from the common assumption of a positive linear time-invariant relation.
The asymptotic theory for the sample autocorrelations and extremes of a GARCH 1 1 process is prov... more The asymptotic theory for the sample autocorrelations and extremes of a GARCH 1 1 process is provided. Special attention is given to the case when the sum of the ARCH and GARCH parameters is close to 1, that is, when one is close to an infinite variance marginal distribution. This situation has been observed for various financial log-return series and led to the introduction of the IGARCH model. In such a situation, the sample autocorrelations are unreliable estimators of their deterministic counterparts for the time series and its absolute values, and the sample autocorrelations of the squared time series have nondegenerate limit distributions. We discuss the consequences for a foreign exchange rate series.
Abstract. In this paper we propose a goodness of fit test that checks the resemblance of the spec... more Abstract. In this paper we propose a goodness of fit test that checks the resemblance of the spectral density of a GARCH process to that of the log-returns. The asymptotic behavior of the test statistics are given by a functional central limit theorem for the integrated periodogram of the data. ...
For a random walk with negative drift we study the exceedance probability (ruin probability) of a... more For a random walk with negative drift we study the exceedance probability (ruin probability) of a high threshold. The steps of this walk (claim sizes) constitute a stationary ergodic stable process. We study how ruin occurs in this situation and evaluate the asymptotic behavior of the ruin probability for a large variety of stationary ergodic stable processes. Our findings show that the order of magnitude of the ruin probability varies significantly from one model to another. In particular, ruin becomes much more likely when the claim sizes exhibit long-range dependence. The proofs exploit large deviation techniques for sums of dependent stable random variables and the series representation of a stable process as a function of a Poisson process.
In this paper we propagate a large deviations approach for proving limit theory for (generally) m... more In this paper we propagate a large deviations approach for proving limit theory for (generally) multivariate time series with heavy tails. We make this notion precise by introducing regularly varying time series. We provide general large deviation results for functionals acting on a sample path and vanishing in some neighborhood of the origin. We study a variety of such functionals, including large deviations of random walks, their suprema, the ruin functional, and further derive weak limit theory for maxima, point processes, cluster functionals and the tail empirical process. One of the main results of this paper concerns bounds for the ruin probability in various heavy-tailed models including GARCH, stochastic volatility models and solutions to stochastic recurrence equations.
In this paper we work in the framework of a k-dimensional vector of loglinear risks. Under weak c... more In this paper we work in the framework of a k-dimensional vector of loglinear risks. Under weak conditions on the marginal tails and the dependence structure of a vector of positive risks we derive the asymptotic tail behaviour of the aggregated risk and present an application concerning log-normal risks with stochastic volatility.
We prove large deviation results for the random sum , , where are non-negative integer-valued ran... more We prove large deviation results for the random sum , , where are non-negative integer-valued random variables and are i.i.d. non-negative random variables with common distribution function F, independent of . Special attention is paid to the compound Poisson process and its ramifications. The right tail of the distribution function F is supposed to be of Pareto type (regularly or extended regularly varying). The large deviation results are applied to certain problems in insurance and finance which are related to large claims.
Many important probabilistic models in queuing theory, insurance and finance deal with partial su... more Many important probabilistic models in queuing theory, insurance and finance deal with partial sums of a negative mean stationary process (a negative drift random walk), and the law of the supremum of such a process is used to calculate, depending on the context, the ruin probability, the steady state distribution of the number of customers in the system or the value at risk. When the stationary process is heavy-tailed, the corresponding ruin probabilities are high and the stationary distributions are heavy-tailed as well. If the steps of the random walk are independent, then the exact asymptotic behavior of such probability tails was described by Embrechts and Veraverbeke. We show that this asymptotic behavior may be different if the steps of the random walk are not independent, and the dependence affects the joint probability tails of the stationary process. Such type of dependence can be modeled, for example, by a linear process.
In 1713, the noted mathematician Jacob Bernoulli's paper Ars Conjectandi (Latin for Art of Conjec... more In 1713, the noted mathematician Jacob Bernoulli's paper Ars Conjectandi (Latin for Art of Conjecturing) was published, eight years after his death. It is unlikely that Bernoulli, or anyone for that matter, could have imagined the impact this article would have had, not only on the development of probably and statistics, but science at large. This paper would lay out many of the building blocks and foundational results that are taught in any first year course in statistics; from the simple distribution of the number of successes in n independent trials, which bears his name, to the law of large numbers. This law provides the basis for the frequentist view of statistics and underpins modern probability theory. But beyond this scientific framework, the law of large numbers "holds the world together in its inmost folds". It is only natural that the Bernoulli Society would devote a special issue of its flagship journal, Bernoulli, to help commemorate the 300th year anniversary of the publication of Ars Conjectandi. In this issue, we have tried to capture a snapshot of current and future research directions in probability and statistics from leading experts in the field. As you can see from the contents, the field has evolved into many exciting and important directions well beyond what Bernoulli could have imagined. We are indebted to the authors of this issue for their contributions and for sharing their thoughts with the scientific community. It is our hope these articles will inspire all of us to be faithful to the continuing development of probability and statistics that meets the challenges faced in science, technology, and society.
A pathwise approach to stochastic integral equations is advocated. Linear extended Riemann±Stielt... more A pathwise approach to stochastic integral equations is advocated. Linear extended Riemann±Stieltjes integral equations driven by certain stochastic processes are solved. Boundedness of the p-variation for some 0 , p , 2 is the only condition on the driving stochastic process. Typical examples of such processes are in®nite-variance stable Le Âvy motion, hyperbolic Le Âvy motion, normal inverse Gaussian processes, and fractional Brownian motion. The approach used in the paper is based on a chain rule for the composition of a smooth function and a function of bounded p-variation with 0 , p , 2.
this article will not focus on these more classical topics but mainly on the approach which was s... more this article will not focus on these more classical topics but mainly on the approach which was started by Fisher Black (1938-1995), Myron Scholes [6] and Robert Merton [25] in 1973. What was so entirely new that Scholes and Merton were awarded the Nobel prize for economics in 1997?
By means of a detailed analysis of the returns of the Standard & Poors 500 (S&P 500) composite st... more By means of a detailed analysis of the returns of the Standard & Poors 500 (S&P 500) composite stock index over the last fifty years we show how theoretical results and methodological recommendations from the statistical theory of non-parametric curve inference allow one to consistently estimate expected return and volatility. In this approach we do not postulate an a priori relationship risk-return nor do we specify the evolution of the first two moments through covariates. Our analysis gives statistical evidence that the expected return of the S&P 500 index as well as the market price of risk (the ratio expected return minus risk free interest rate over volatility) vary significantly through time both in size and sign. In particular, the periods of negative (positive) estimated expected return and market price of risk coincide with the bear (bull) markets of the index as defined in the literature. A complex relationship between risk and expected return emerges which is far from the common assumption of a positive linear time-invariant relation.
The asymptotic theory for the sample autocorrelations and extremes of a GARCH 1 1 process is prov... more The asymptotic theory for the sample autocorrelations and extremes of a GARCH 1 1 process is provided. Special attention is given to the case when the sum of the ARCH and GARCH parameters is close to 1, that is, when one is close to an infinite variance marginal distribution. This situation has been observed for various financial log-return series and led to the introduction of the IGARCH model. In such a situation, the sample autocorrelations are unreliable estimators of their deterministic counterparts for the time series and its absolute values, and the sample autocorrelations of the squared time series have nondegenerate limit distributions. We discuss the consequences for a foreign exchange rate series.
Abstract. In this paper we propose a goodness of fit test that checks the resemblance of the spec... more Abstract. In this paper we propose a goodness of fit test that checks the resemblance of the spectral density of a GARCH process to that of the log-returns. The asymptotic behavior of the test statistics are given by a functional central limit theorem for the integrated periodogram of the data. ...
For a random walk with negative drift we study the exceedance probability (ruin probability) of a... more For a random walk with negative drift we study the exceedance probability (ruin probability) of a high threshold. The steps of this walk (claim sizes) constitute a stationary ergodic stable process. We study how ruin occurs in this situation and evaluate the asymptotic behavior of the ruin probability for a large variety of stationary ergodic stable processes. Our findings show that the order of magnitude of the ruin probability varies significantly from one model to another. In particular, ruin becomes much more likely when the claim sizes exhibit long-range dependence. The proofs exploit large deviation techniques for sums of dependent stable random variables and the series representation of a stable process as a function of a Poisson process.
In this paper we propagate a large deviations approach for proving limit theory for (generally) m... more In this paper we propagate a large deviations approach for proving limit theory for (generally) multivariate time series with heavy tails. We make this notion precise by introducing regularly varying time series. We provide general large deviation results for functionals acting on a sample path and vanishing in some neighborhood of the origin. We study a variety of such functionals, including large deviations of random walks, their suprema, the ruin functional, and further derive weak limit theory for maxima, point processes, cluster functionals and the tail empirical process. One of the main results of this paper concerns bounds for the ruin probability in various heavy-tailed models including GARCH, stochastic volatility models and solutions to stochastic recurrence equations.
In this paper we work in the framework of a k-dimensional vector of loglinear risks. Under weak c... more In this paper we work in the framework of a k-dimensional vector of loglinear risks. Under weak conditions on the marginal tails and the dependence structure of a vector of positive risks we derive the asymptotic tail behaviour of the aggregated risk and present an application concerning log-normal risks with stochastic volatility.
We prove large deviation results for the random sum , , where are non-negative integer-valued ran... more We prove large deviation results for the random sum , , where are non-negative integer-valued random variables and are i.i.d. non-negative random variables with common distribution function F, independent of . Special attention is paid to the compound Poisson process and its ramifications. The right tail of the distribution function F is supposed to be of Pareto type (regularly or extended regularly varying). The large deviation results are applied to certain problems in insurance and finance which are related to large claims.
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Papers by T. Mikosch