A task with ideal execution time $L$ such as the execution of a computer program or the transmission of a file on a data link may fail, and the task then needs to be restarted. The task is handled by a complex system with features similar to the ones in classical reliability: failures may be mitigated by using server redundancy in parallel or $k$-out-of-$n$ arrangements, standbys may be cold or warm, one or more repairmen may take care of failed components, etc. The total task time $X$ (including restarts and pauses in failed states) is investigated with particular emphasis on the tail $\mathbb{P}(X>x)$. A general alternating Markov renewal model is proposed and an asymptotic exponential form $\mathbb{P}(X>x)\sim C\mathrm{e}^{-\gamma x}$ identified for the case of a deterministic task time $L\equiv \ell$. The rate $\gamma$ is given by equating the spectral radius of a certain matrix to 1, and the asymptotic form of $\gamma=\gamma(\ell)$ as $\ell\to\infty$ is derived, leading to the asymptotics of $\mathbb{P}(X>x)$ for random task times $L$. A main finding is that $X$ is always heavy-tailed if $L$ has unbounded support. The case where the Markov renewal model is derived by lumping in a continuous-time finite Markov process with exponential holding times is given special attention, and the study includes analysis of the effect of processing rates that differ with state or time.
In this talk we will consider some new limit theorems for power variation of $k$th order increments of stationary increments Lévy driven moving average processes. In this infill sampling setting, the asymptotic theory gives very surprising results, which (partially) have no counterpart in the theory of discrete moving average processes. More specifically, we will show that first order limit theorems and the mode of convergence strongly depend on the interplay between the given order of the increments, the considered power $p>0$, the Blumenthal-Getoor index $\beta \in (0,2)$ of the driving pure jump Lévy process $L$ and the behaviour of the kernel function $g$ at $0$ determined by the power $\alpha $. First order asymptotic theory essentially comprise three cases: stable convergence towards a certain infinitely divisible distribution, an ergodic type limit theorem and convergence in probability towards an integrated random process. We also prove the second order limit theorem connected to the ergodic type result. When the driving Lévy motion $L$ is a symmetric $\beta$-stable process we may obtain two different limits: a central limit theorem and convergence in distribution towards a $(1-\alpha )\beta$-stable random variable.
We define an ambit field as a stochastic process in Hilbert space, and show that it may be represented as a series of Lévy semistationary processes. In special cases, ambit fields are solutions of stochastic partial differential equations. Since ambit fields are also Hilbert-space valued Volterra processes, they can be seen as boundary solutions of simple hyperbolic stochastic partial differential equations. Finally, we introduce continuous-time autoregressive Hilbert-space valued processes or order 2 which constitute an interesting class of ambit fields. Examples from forward price modelling in energy markets will be discussed.
Consider a Markov process ${X}$ on $[0,\infty)$ which has only negative jumps and converges to $0$ at infinity a.s. We interpret $X(t)$ as the size of a typical cell at time $t$, and each jump as a birth event. More precisely, if $\Delta {X}(s)=-y<0$, then $s$ is the birth at time of a daughter cell with size $y$ which then evolves independently and according to the same dynamics. We provide a simple criterion on ${X}$ to ensure that the family of cell sizes does not explode locally. The case when ${X}$ is self-similar is discussed in details, as well as connexions with related processes which have appeared in the literature.
We show that the probability distribution function (PDF) of the velocity differences in turbulence consists of a continuous distribution convolved with a discrete one. This is a direct consequence of the similar structure of the invariant measure of turbulence. The continuous part of the PDF corresponds to the Kolmogorov-Obukhov (K-O) '41 theory represented by an infinite-dimensional Gaussian measure, in the invariant measure, whereas the discrete part corresponds to the She-Leveque intermittency corrections in the '62 K-O theory, just as it does in the invariant measure itself. The continuous part of the PDF is the Generalized Hyperbolic Distribution. Remarkably, all of this follows from the Navier-Stokes equations when generic noise is added to it. Then we discuss applications to experiments, simulations and boundary value problem.
Contingent Convertible Bonds, or CoCos, are contingent capital instruments which are converted into shares, or may suffer a principal write-down, if certain trigger event occurs. In this talk we discuss some approaches to the problem of pricing CoCos when its conversion and the other relevant credit events are triggered by the issuer's share price. We introduce a new model of partial information which aims at enhancing the market trigger approach while remaining analytically tractable. We address also CoCos having the additional feature of being callable by the issuer at a series of pre-defined dates. These callable CoCos are thus exposed to a new source of risk – referred to as extension risk – since they have no fixed maturity, and the repayment of the principal may take place at the issuer's convenience.
The discovery that VaR is elicitable while CVaR is not has provoked considerable debate in risk-management circles, particularly in view of the Basel Committee's recommendation that the former be discarded in favour of the latter. It has however not been clear just how to apply the elicitability concept in a risk management environment where we observe successive returns that are not necessarily independent and all have different conditional distributions. This problem is best addressed from the point of view of Dawid's theory of 'prequential statistics' [Dawid, JRSS(A)1984]: we examine the long-run 'consistency' of a sequence of predictions and observed outcomes. To do so requires a 'calibration function' and it turns out that this is closely related to the gradient of the score function of elicitability theory.
We consider the dynamics no-good-deal bounds introduced in [1] and we discuss the representation of continuous time convex prices satisfying such bounds. This is based on extension theorems for convex operator in $L^p$ setting ($p \in [1, \infty]$), which are here presented and discussed in comparison with other results in the same area, see e.g. [2]. Restricting to the $L^2$ case, we study the relationship between convex no-good-deal price operators and risk-indifferent pricing. In particular we address the problem of how to related the risk measure of the risk-indifferent pricing with the no-good-deal prices. The relationship of the resulting convex prices with the non-arbitrage bounds is also addressed.
| [1] | J. Bion Nadal and G. Di Nunno (2013): Dynamic no-good-deal pricing measures and extensions theorems for linear operators on $L^\infty$. Finance and Stochastics, 17, 3, pp. 587-613. |
| [2] | J. Bion-Nadal and G. Di Nunno (2014): Representation of convex operators and their static and dynamic sandwich extensions. ArXiv 1412.2030v1. |
We introduce a multivariate estimator of financial volatility that is based on the theory of Markov chains. The Markov chain framework takes advantage of the discreteness of high-frequency returns. We study the finite sample properties of the estimation in a simulation study and apply it to high-frequency commodity prices.
We consider a continuous Brownian semimartingale which is observed along a, possibly random, discrete sampling scheme, and contaminated by some noise. This noise is additive, but not necessarily white. Our aim is twofold: first we show how one can estimate the autocovariance function of the noise and in particular check whether it is white or not. Second, we propose a way of adapting the pre-averaging method to estimate the integrated volatility, still with the rate $n^{1/4}$ when $n$ is the number of observations, to the case when the noise is not a white noise.
The aim of this talk is to give a review of Lévy based spatio-temporal modelling in stochastic geometry. Lévy bases are independently scattered infinitely divisible random measures that obey a Lévy-Khintchine representation. Using a key relation for the characteristic function of an integral with respect to a Lévy basis, it is possible to develop Lévy based modelling into a flexible, yet tractable modelling tool in stochastic geometry. We will exemplify the use of Lévy based modelling in stochastic geometry by considering Lévy based growth modelling, Lévy random fields, stereology of Lévy particles and Lévy driven Cox point processes.
Huber (1964) defined an M-estimator as the minimizer of the objective function $\sum_{i=1}^{n}\rho (y_{i}-\beta x_{i})$. Particular cases are Least Squares, $\rho (u)=u^{2}$, Median Regression, $\rho (u)=|u|,$ and the Huber-skip, $\rho (u)=\min (u^{2},c^{2})$.
We contribute to the asymptotic theory of M-estimators for multiple linear time series regression by analysing objective functions that need not have a continuous derivative, like the Huber-skip, and by allowing for quite general time series regressors. We prove tightness, consistency, and find a stochastic expansion of the M-estimator, from which one can derive limit distributions under weak assumptions on the objective function and regressors. The results are obtained using some recent results on the supremum of a class of martingales, see Johansen and Nielsen (2014).
This paper studies the variance functions of the natural exponential families (NEF) on the real line of the form $(Am^4+Bm^2+C)^{1/2}.$ Surprisingly enough, most of them are discrete families concentrated on $\lambda\mathbb{Z}$ for some constant $\lambda$ and the Laplace transform of their elements are expressed by elliptic functions. The concept of association of two NEF is an auxilliary tool for their study: two families $F$ and $G$ are associated if they are generated by symmetric probabilities and if the analytic continuations of their variance functions satisfy $V_F(m)=V_G(m\sqrt{-1})$. We give some properties of the association before its application to these elliptic NEF. The paper is completed by the study of NEF with variance functions $m(Cm^4+Bm^2+A)^{1/2}.$ They are easier to study and they are concentrated on $\lambda \mathbb{N}.$
We characterize exchangeability of infinitely divisible distributions in terms of the characteristic triplet. This is applied to stable distributions and self-decomposable distributions, and a connection to Lévy copulas is made. We further study general mappings between classes of measures that preserve exchangeability and give various examples which arise from discrete time settings, such as stationary distributions of AR(1) processes, or from continuous time settings, such as Ornstein–Uhlenbeck processes or Upsilon-transforms.
The chapter introduces the generalised partial autocorrelation (GPAC) coefficients of a stationary stochastic process. The latter are related to the generalised autocovariances, the inverse Fourier transform coefficients of a power transformation of the spectral density function. By interpreting the generalised partial autocorrelations as the partial autocorrelation coefficients of an auxiliary process, we derive their properties and relate them to essential features of the original process. \newline\indent Based on a parameterisation suggested by Barndorff-Nielsen and Schou (1973) and on Whittle likelihood, we develop an estimation strategy for the GPAC coefficients. We further prove the GPAC coefficients can be used to estimate the mutual information between the past and the future of a time series.
Recurrence and transience problem has been studied for many stochastic processes. Among additive processes on $\mathbb R ^d$, the dichotomy of recurrence and transience of Lévy processes and selfsimilar additive processes are known, and also that of semi-selfsimilar additive processes on $\mathbb R$ is known, but the dichotomy is not necessarily true for general additive processes.
Suppose $\{ X_t, t\ge 0\}$ is an additive process with a property that for some $p>0$, \begin{equation*} X_{t+p}- X_{s+p} \overset{\scriptscriptstyle\mathrm d}{=} X_t-X_s \qquad \text{for any}\,\, s,t\ge 0. \end{equation*} This process is called a semi-Lévy process with period $p$, which is a generalization of Lévy processes.
Note that, in the examples above, Lévy processes and selfsimilar additive processes are semimartingale, but semi-selfsimilar additive processes and semi-Lévy processes are not necessarily semimartingale.
In this talk, we show the dichotomy of recurrence and transience of semi-Lévy processes. For the proof, the concept of semi-random walks is introduced, which is a discrete version of semi-Lévy process. An example of semi-Lévy processes on $\mathbb R$ constructed from two independent Lévy processes on $\mathbb R$ is precisely investigated.
| [1] | M. Maejima, T. Takamune and Y. Ueda, The dichotomy of recurrence and transience of semi-Lévy processes, J. Theor. Probab. 27 (2014), 982-996. |
In quantum theory a wave function or a density matrix is commonly referred to as "the state of the system", and it provides probabilistic information about the outcome of any experiment that one may perform on the system. If a quantum system is monitored continuously in time, its wave function or density matrix evolves by a combination of unitary and stochastic changes.
In the presentation, I shall review some elementary aspects of quantum theory and quantum measurements, and I shall discuss recent work where we show that monitoring does not only update the current state of a quantum system, but it also provides information about the state of the system at earlier times. While this idea is indeed well understood in the context of classical stochastic processes, it challenges the concept of a physical state in quantum theory. I shall introduce the formalism needed to account for quantum states, which are conditioned on both past and future probing results, and I shall show applications to recent experiment.
High frequency inference has generated a wave of research interest among econometricians and practitioners, as indicated from the increasing number of estimators based on intra-day data. However, we also witness a scarcity of methodology to assess the uncertainty – standard error – of the estimator. The root of the problem is that whether with or without the presence of microstructure, standard errors rely on estimating the asymptotic variance (AVAR), and often this asymptotic variance involves substantially more complex quantities than the original parameter to be estimated.
Standard errors are important: they are used both to assess the precision of estimators in the form of confidence intervals, to create "feasible statistics" for testing, and also when building forecasting models based on, say, daily estimates.
The contribution of this paper is to provide an alternative and general solution to this problem, which we call Observed Asymptotic Variance. It is a general nonparametric method for assessing asymptotic variance (AVAR), and it provides consistent estimators of AVAR for a broad class of parameters that are the integral of a spot parameter process. The spot process can be volatility, covariance, leverage effect, high frequency betas, and more generally any semimartingale, with continuous and jump components. The construction and the analysis of estimators work well in the presence of microstructure noise, and when the observation times are irregular or asynchronous in the multivariate case. The edge effect – phasing in and phasing out the information on the boundary of the data interval – of any relevant estimator is also analyzed and treated rigorously.
As part of the theoretical development, the paper shows how to feasibly disentangle the effect from estimation error and the variation in the spot parameter process alone. For the latter, we obtain a consistent estimator of the quadratic variation (QV) of the parameter to be estimated, for example, the QV of the leverage effect.
A multivariate Lévy-driven moving average is a stochastic process constructed by taking a convolution of a deterministic, matrix-valued kernel function and a multivariate Lévy process. We study the infinite-dimensional conditional distributions of a multivariate Lévy-driven moving average, under the assumption that the process has continuous sample paths. More specifically, we present sufficient conditions, under which such a process has the conditional full support (CFS) property, ensuring that any conditional distribution of the process has the largest possible support in the relevant space of continuous functions. The CFS property is important in connection to some recent no-arbitrage and super-replication results under transaction costs. These sufficient conditions require that the kernel function and the driving Lévy process are non-degenerate in some sense, and we discuss how they can be verified in practise.
The Dyson Brownian motion is the eigenvalue process of an Hermitian Brownian process and it is described as a system of stochastic differential equations with non smooth drift, the repulsion force between eingenvalues. In this talk the dynamics of the eigenvalue (semimartingale) process of a Hermitian Lévy process will be discussed. Similar to the Dyson Brownian motion, the repulsion force among the eigenvalues appears in the bounded variation part of its semimartingale decomposition only when there is a Gaussian component. The eigenvalue process of an Hermitian Lévy process with jumps of rank one will be also considered. Conditions for the simultaneous jumps of the corresponding eigenvalue process will be discussed. This is joint work with Alfonso Rocha-Arteaga.
In a model for the limit order book with arrivals and cancellations, we derive an SPDE with one heating source and two cooling elements on a finite rod for the order volume which we solve in terms of local time. Moreover, via Brownian excursion theory, we provide a hyperbolic function table for the Laplace transforms of various times of trade. A bivariate Laplace-Mellin transform is introduced for the joint excursion height and length and expressed in terms of the Riemann Xi function. Finally, we show that two diferent disintegrations of the Ito measure are equivalent to Jacobi's Theta transformation formula.
Let $\xi$ be a real valued Lévy process that either has a finite life time or drifts towards $-\infty$, and consider the exponential functional of $\xi$, defined by \begin{equation*} I=\int^{\infty}_{0}\exp\{\xi_{s}\}ds. \end{equation*} This random variable plays a key rol in several areas of probability theory, for instance in fragmentation, coalescence and branching processes, financial and insurance mathematics, Brownian motion in hyperbolic spaces, random processes in random environment, positive self-similar Markov processes, etc. Obtaining distributional properties of these r.v. has been the subject of many research articles. The paper by Bertoin and Yor [1] is a thorough review on the topic. In this talk we will start by recalling some of the main known results. Then we will establish that the density of $I$, say $k$, is the unique probability density function that solves the equation \begin{equation*} \int^{\infty}_{t}k(s)ds=\int_{\mathbb{R}}U(dy)k(te^{-y}),\qquad \text{a.e. $t>0$}, \end{equation*} where $U$ denotes the potential measure of $\xi,$ \begin{equation*} U(dy)=\mathbb{E}\left(\int^{\infty}_{0}1_{\{\xi_{s}\in dy\}}ds\right),\qquad \text{on $\mathbb{R}$}. \end{equation*} We will show how this formula can be used to provide elementary proofs of several known results for $I$, in particular of the striking factorisation obtained in [2]. To finish we will establish that a similar identity holds when we replace $\xi$ by a Markov additive Lévy process, and explain how our methods allow to derive analogous results to the Lévy processes case, in particular of the main result in [2].
| [1] | J. Bertoin and M. Yor. Exponential functionals of Lévy processes. Probab. Surv., 2:191-212, 2005. |
| [2] | P. Patie and M. Savov. Exponential functional of Lévy processes: generalized Weierstrass products and Wiener-Hopf factorization. C. R. Math. Acad. Sci. Paris, 351(9\nobreakdash-10):393-396, 2013. |
Estimating volatility from recent high frequency data, we revisit the question of the smoothness of the volatility process. Our main result is that log-volatility behaves essentially as a fractional Brownian motion with Hurst exponent $H$ of order $0.1$, at any reasonable time scale. This leads us to adopt the fractional stochastic volatility (FSV) model of Comte and Renault. We call our model Rough FSV (RFSV) to underline that, in contrast to FSV, $H<1/2$. We demonstrate that our RFSV model is remarkably consistent with financial time series data; one application is that it enables us to obtain improved forecasts of realized volatility. Furthermore, we find that although volatility is not long memory in the RFSV model, classical statistical procedures aiming at detecting volatility persistence tend to conclude the presence of long memory in data generated from it. This sheds light on why long memory of volatility has been widely accepted as a stylized fact. Finally, we provide a quantitative market microstructure-based foundation for our findings, relating the roughness of volatility to high frequency trading and order splitting.
Dynkin Isomorphism Theorem relates local times of strongly symmetric Markov processes to Gaussian processes. We propose an isomorphism theorem for Poissonian processes, i.e., infinitely divisible processes without Gaussian component. The framework of our result involves the concept of Lévy measures of general infinitely divisible process as well as series and integral representations. We will illustrate this result showing how Dynkin's Isomorphism Theorem follows in our setting. Other applications of the isomorphism theorem will also be discussed.
Brownian semi-stationary processes have been proposed as a class of stochastic models for time series of the turbulent velocity field. We show, by detailed comparison, that these processes are able to reproduce the main characteristics of turbulent data. Furthermore, we present an algorithm that allows to estimate the model parameters from second and third order statistics.
This paper proposes a novel model of financial prices where: (i) prices are discrete; (ii) prices change in continuous time; (iii) a high proportion of price changes are reversed in a fraction of a second. Our model is analytically tractable and directly formulated in terms of the calendar time and price impact curve. The resulting c`adl`ag price process is a piecewise constant semimartingale with finite activity, finite variation and no Brownian motion component. We use moment-based estimations to fit four high frequency futures data sets and demonstrate the descriptive power of our proposed model. This model is able to describe the observed dynamics of price changes over three different orders of magnitude of time intervals
We formulate several problems of optimal stopping for Brownian motion: old Chernoff's problem about a drift of Brownian process,testing of the three statistical hypotheses. We consider also some financial problem of type "When to sell Apple"?
The Sand Gangat the University of Aarhus was a group of researchers in earth science, physics and statistics who, at Ole Barndorff-Nielsen's initiative, collaborated on investigating the physics of wind blown sand. A review will be given of some recent research by members of this group, Keld Rømer Rasmussen and myself.
The starting point for the Sand Gang was the hyperbolic distribution and related distributions, such as the NIG-distribution, that were developed as models of the log size distribution of sand deposits, but have also turned out to give a good description of the distribution of turbulent velocity fluctuations and of returns from financial assets. A simple model is presented for which the distribution of the logarithm of the grain size in an aeolian sand deposit is a NIG-distribution, or possibly another normal variance-mean mixture.
The transport process itself is a central problem, which can only be investigated and described in a satisfactory way by means of probabilistic models: grains of varying sizes and irregular shapes move through a randomly fluctuating wind and repeatedly hit and rebound from a randomly arranged sand surface. A fundamental difficulty in empirical studies is that the high concentration of moving grains in the so-called saltation layer close to the surface makes direct observation of grain behaviour at this height impossible. R.A. Bagnold (1941) and P.R. Owen (1964) proposed different hypotheses about the saltation layer, based on which formulae for the transport rate can be developed; see Sørensen (1991, 2004) and Durán and Herrmann (2006). This theory is briefly reviewed. Parameters in a model of the transport process are estimated using wind tunnel observations by Keld Rømer Rasmussen of grain speeds at nine heights at four wind speeds. In this way interesting light is cast on the hypotheses by Bagnold and Owen.
Financial markets based on Lévy processes are typically incomplete and option prices depend on risk preferences of individual agents. In this context, the notion of utility indifference price has gained popularity in the academic circles. Although theoretically very appealing, this pricing method remains difficult to apply in practice, due to the high computational cost of solving the nonlinear partial integro-differential equation associated to the indifference price.
In this work, we develop closed form approximations to exponential utility indifference prices in exponential Lévy models. To this end, we first establish a new non-asymptotic approximation of the indifference price which extends earlier results on small risk aversion asymptotics of this quantity. Next, we use this formula to derive a closed-form approximation of the indifference price by treating the Lévy model as a perturbation of the Black-Scholes model. This extends the methodology introduced in a recent paper for smooth linear functionals of Lévy processes (A. Cerny, S. Denkl and J. Kallsen, arXiv:1309.7833) to nonlinear and non-smooth functionals. Our closed formula represents the indifference price as the linear combination of the Black-Scholes price and correction terms which depend on the variance, skewness and kurtosis of the underlying Lévy process, and the derivatives of the Black-Scholes price. As a by-product, we obtain a simple explicit formula for the spread between the buyer's and the seller's indifference price. This formula allows to quantify, in a model-independent fashion, how sensitive a given product is to jump risk in the limit of small jump size.
Quoting from Ole Barndorff-Nielsen and Albert Shiryaev's Change of time and change of measure, (2010), World Scientific: "Random change of time is key to understanding the nature of various stochastic processes and gives rise to interesting mathematical results and insights of importance for the modelling and interpretation of empirically observed dynamic processes." This is a theme that underlies the analysis to be presented here. Continuity of local time for Brownian motion ranks among the most notable mathematical results in the theory of stochastic processes. An extension of previous results on an explicit role of continuity of (natural) local time is obtained for applications to recent classes of problems in physics, biology and finance involving discontinuities in a dispersion coefficient. The main theorem and its corollary provide physical principles that relate macroscale continuity of deterministic quantities to microscale continuity of the (stochastic) local time.
We consider stochastic control problems where the controller has access to some information about the future value of one or several of the stochastic parameters of the system. For example, in a financial market it could be information about the value of a stock at some future time. Such controllers/traders are called insiders. The handling of such insider control problems represents a mathematical challenge, because we are no longer in a classical semimartingale context, at least not a priori.
We approach this problem by setting up a general anticipative calculus machinery, involving forward integrals, white noise theory, Donsker-delta function and associated Hida-Malliavin calculus. We prove a sufficient and a necessary maximum principle for general insider control. Then we apply these results to solve some optimal portfolio problems for an insider in a financial market modeled by Itô-Lévy processes.
