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  • 1.
    Björk, Tomas
    et al.
    Stockholm School of Economics.
    Hult, Henrik
    Dept. of Appl. Math. and Statistics, Universitetsparken 5, 2100 Copenhagen, Denmark.
    A note on Wick products and the fractional Black-Scholes model2005In: Finance and Stochastics, ISSN 0949-2984, E-ISSN 1432-1122, Vol. 9, no 2, p. 197-209Article in journal (Refereed)
    Abstract [en]

    In some recent papers (Elliott and van der Hoek 2003; Hu and Oksendal 2003) a fractional Black-Scholes model has been proposed as an improvement of the classical Black-Scholes model (see also Benth 2003; Biagini et al. 2002; Biagini and Oksendal 2004). Common to these fractional Black-Scholes models is that the driving Brownian motion is replaced by a fractional Brownian motion and that the Ito integral is replaced by the Wick integral, and proofs have been presented that these fractional Black-Scholes models are free of arbitrage. These results on absence of arbitrage complelety contradict a number of earlier results in the literature which prove that the fractional Black-Scholes model (and related models) will in fact admit arbitrage. The objective of the present paper is to resolve this contradiction by pointing out that the definition of the self-financing trading strategies and/or the definition of the value of a portfolio used in the above papers does not have a reasonable economic interpretation, and thus that the results in these papers are not economically meaningful. In particular we show that in the framework of Elliott and van der Hoek (2003), a naive buy-and-hold strategy does not in general qualify as "self-financing". We also show that in Hu and Oksendal (2003), a portfolio consisting of a positive number of shares of a stock with a positive price may, with positive probability, have a negative "value".

  • 2.
    Blanchet, Jose
    et al.
    Columbia University.
    Hult, Henrik
    KTH, School of Engineering Sciences (SCI), Mathematics (Dept.), Mathematical Statistics.
    Leder, Kevin
    University of Minnesota.
    Importance sampling for stochastic recurrence equations with heavy-tailed increments2011In: Proceedings of the 2011 Winter Simulation Conference, 2011, p. 3824-3831Conference paper (Other academic)
    Abstract [en]

    Importance sampling in the setting of heavy tailed random variables has generally focused on models withadditive noise terms. In this work we extend this concept by considering importance sampling for theestimation of rare events in Markov chains of the formXn+1 = An+1Xn+Bn+1; X0 = 0;where the Bn’s and An’s are independent sequences of independent and identically distributed (i.i.d.) randomvariables and the Bn’s are regularly varying and the An’s are suitably light tailed relative to Bn. We focuson efficient estimation of the rare event probability P(Xn > b) as b%¥. In particular we present a stronglyefficient importance sampling algorithm for estimating these probabilities, and present a numerical exampleshowcasing the strong efficiency.

  • 3.
    Blanchet, Jose
    et al.
    Columbia University.
    Hult, Henrik
    KTH, School of Engineering Sciences (SCI), Mathematics (Dept.), Mathematical Statistics.
    Leder, Kevin
    University of Minnesota.
    Rare-Event Simulation for Stochastic Recurrence Equations with Heavy-Tailed Innovations2013In: ACM Transactions on Modeling and Computer Simulation, ISSN 1049-3301, E-ISSN 1558-1195, Vol. 23, no 4, p. 22-Article in journal (Refereed)
    Abstract [en]

    In this article, rare-event simulation for stochastic recurrence equations of the form Xn+1 = A(n+1)X(n) + Bn+1, X-0 = 0 is studied, where {A(n);n >= 1} and {B-n;n >= 1} are independent sequences consisting of independent and identically distributed real-valued random variables. It is assumed that the tail of the distribution of B-1 is regularly varying, whereas the distribution of A(1) has a suitably light tail. The problem of efficient estimation, via simulation, of quantities such as P{X-n > b} and P{sup(k <= n) X-k > b} for large b and n is studied. Importance sampling strategies are investigated that provide unbiased estimators with bounded relative error as b and n tend to infinity.

  • 4. Caravaca, A. S.
    et al.
    Gallina, A. L.
    Tarnawski, L.
    Shavva, V. S.
    Colas, R. A.
    Dalli, J.
    Malin, S. G.
    Hult, Henrik
    KTH, School of Engineering Sciences (SCI), Mathematics (Dept.).
    Arnardottir, H.
    Olofsson, P. S.
    Vagus nerve stimulation promotes resolution of inflammation by a mechanism that involves Alox15 and requires the α7nAChR subunit2022In: Proceedings of the National Academy of Sciences of the United States of America, ISSN 0027-8424, E-ISSN 1091-6490, Vol. 119, no 22, article id e2023285119Article in journal (Refereed)
    Abstract [en]

    Nonresolving inflammation underlies a range of chronic inflammatory diseases, and therapeutic acceleration of resolution of inflammation may improve outcomes. Neural reflexes regulate the intensity of inflammation (for example, through signals in the vagus nerve), but whether activation of the vagus nerve promotes the resolution of inflammation in vivo has been unknown. To investigate this, mice were subjected to electrical vagus nerve stimulation (VNS) or sham surgery at the cervical level followed by zymosan-induced peritonitis. The duration of inflammation resolution was significantly reduced and efferocytosis was significantly increased in mice treated with VNS as compared with sham. Lipid mediator (LM) metabololipidomics revealed that mice treated with VNS had higher levels of specialized proresolving mediators (SPMs), particularly from the omega-3 docosahexaenoic (DHA) and docosapentaenoic (n-3 DPA) metabolomes, in peritoneal exudates. VNS also shifted the ratio between proinflammatory and proresolving LMs toward a proresolving profile, but this effect by VNS was inverted in mice deficient in 12/15-lipoxgenase (Alox15), a key enzyme in this SPM biosynthesis. The significant VNS-mediated reduction of neutrophil numbers in peritoneal exudates was absent in mice deficient in the cholinergic α7-nicotinic acetylcholine receptor subunit (α7nAChR), an essential component of the inflammatory reflex. Thus, VNS increased local levels of SPM and accelerated resolution of inflammation in zymosan-induced peritonitis by a mechanism that involves Alox15 and requires the α7nAChR. 

  • 5.
    Djehiche, Boualem
    et al.
    KTH, School of Engineering Sciences (SCI), Mathematics (Dept.), Mathematical Statistics.
    Hult, Henrik
    KTH, School of Engineering Sciences (SCI), Mathematics (Dept.), Mathematical Statistics.
    Nyquist, Pierre
    KTH, School of Engineering Sciences (SCI), Mathematics (Dept.), Mathematical Statistics.
    Importance sampling for a Markovian intensity model with applications to credit riskManuscript (preprint) (Other academic)
    Abstract [en]

    This paper considers importance sampling for estimation of rare-event probabilities in a Markovian intensity model for credit risk. The main contribution is the design of efficient importance sampling algorithms using subsolutions of a certain Hamilton-Jacobi equation. For certain instances of the credit risk model the proposed algorithm is proved to be asymptotically optimal. The computational gain compared to standard Monte Carlo is illustrated by numerical experiments.

  • 6.
    Djehiche, Boualem
    et al.
    KTH, School of Engineering Sciences (SCI), Mathematics (Dept.), Mathematical Statistics.
    Hult, Henrik
    KTH, School of Engineering Sciences (SCI), Mathematics (Dept.), Mathematical Statistics.
    Nyquist, Pierre
    KTH, School of Engineering Sciences (SCI), Mathematics (Dept.), Mathematical Statistics.
    Importance Sampling for a Simple Markovian Intensity Model Using Subsolutions2022In: ACM Transactions on Modeling and Computer Simulation, ISSN 1049-3301, E-ISSN 1558-1195, Vol. 32, no 2, p. 1-25, article id 14Article in journal (Refereed)
    Abstract [en]

    This article considers importance sampling for estimation of rare-event probabilities in a specific collection of Markovian jump processes used for, e.g., modeling of credit risk. Previous attempts at designing importance sampling algorithms have resulted in poor performance and the main contribution of the article is the design of efficient importance sampling algorithms using subsolutions. The dynamics of the jump processes cause the corresponding Hamilton-Jacobi equations to have an intricate state-dependence, which makes the design of efficient algorithms difficult. We provide theoretical results that quantify the performance of importance sampling algorithms in general and construct asymptotically optimal algorithms for some examples. The computational gain compared to standard Monte Carlo is illustrated by numerical examples.

  • 7.
    Djehiche, Boualem
    et al.
    KTH, School of Engineering Sciences (SCI), Mathematics (Dept.), Mathematical Statistics.
    Hult, Henrik
    KTH, School of Engineering Sciences (SCI), Mathematics (Dept.), Mathematical Statistics.
    Nyquist, Pierre
    KTH, School of Engineering Sciences (SCI), Mathematics (Dept.), Mathematical Statistics.
    Min-max representations of viscosity solutions of Hamilton-Jacobi equations and applications in rare-event simulationManuscript (preprint) (Other academic)
    Abstract [en]

    In this paper a duality relation between the Mañé potential and Mather's action functional is derived in the context of convex and state-dependent Hamiltonians. The duality relation is used to obtain min-max representations of viscosity solutions of first order Hamilton-Jacobi equations. These min-max representations naturally suggest classes of subsolutions of Hamilton-Jacobi equations that arise in the theory of large deviations. The subsolutions, in turn, are good candidates for designing efficient rare-event simulation algorithms.

  • 8.
    Donahue, Mary J.
    et al.
    Linköping Univ, Lab Organ Elect, Campus Norrköping, SE-60174 Norrköping, Sweden..
    Ejneby, Malin Silvera
    Linköping Univ, Lab Organ Elect, Campus Norrköping, SE-60174 Norrköping, Sweden.;Linköping Univ, Wallenberg Ctr Mol Med, SE-58185 Linköping, Sweden..
    Jakesova, Marie
    Brno Univ Technol, Cent European Inst Technol, Bioelect Mat & Devices Lab, Purkynova 123, Brno 61200, Czech Republic..
    Caravaca, April S.
    Karolinska Inst, Ctr Mol Med, Dept Med, Ctr Bioelect Med,Lab Immunobiol, Stockholm, Sweden.;Karolinska Univ Hosp, MedTechLabs, Stockholm Ctr Bioelect Med, Solna, Sweden..
    Andersson, Gabriel
    KTH, School of Engineering Sciences (SCI), Mathematics (Dept.).
    Sahalianov, Ihor
    Brno Univ Technol, Cent European Inst Technol, Bioelect Mat & Devices Lab, Purkynova 123, Brno 61200, Czech Republic..
    Derek, Vedran
    Univ Zagreb, Fac Sci, Dept Phys, Bijenicka C 32, Zagreb 10000, Croatia..
    Hult, Henrik
    KTH, School of Engineering Sciences (SCI), Mathematics (Dept.). Karolinska Univ Hosp, MedTechLabs, Stockholm Ctr Bioelect Med, Solna, Sweden..
    Olofsson, Peder S.
    Karolinska Inst, Ctr Mol Med, Dept Med, Ctr Bioelect Med,Lab Immunobiol, Stockholm, Sweden.;Karolinska Univ Hosp, MedTechLabs, Stockholm Ctr Bioelect Med, Solna, Sweden.;Feinstein Inst Med Res, Inst Bioelect Med, Manhasset, NY USA..
    Glowacki, Eric Daniel
    Linköping Univ, Lab Organ Elect, Campus Norrköping, SE-60174 Norrköping, Sweden.;Brno Univ Technol, Cent European Inst Technol, Bioelect Mat & Devices Lab, Purkynova 123, Brno 61200, Czech Republic..
    Wireless optoelectronic devices for vagus nerve stimulation in mice2022In: Journal of Neural Engineering, ISSN 1741-2560, E-ISSN 1741-2552, Vol. 19, no 6, p. 066031-, article id 066031Article in journal (Refereed)
    Abstract [en]

    Objective. Vagus nerve stimulation (VNS) is a promising approach for the treatment of a wide variety of debilitating conditions, including autoimmune diseases and intractable epilepsy. Much remains to be learned about the molecular mechanisms involved in vagus nerve regulation of organ function. Despite an abundance of well-characterized rodent models of common chronic diseases, currently available technologies are rarely suitable for the required long-term experiments in freely moving animals, particularly experimental mice. Due to challenging anatomical limitations, many relevant experiments require miniaturized, less invasive, and wireless devices for precise stimulation of the vagus nerve and other peripheral nerves of interest. Our objective is to outline possible solutions to this problem by using nongenetic light-based stimulation. Approach. We describe how to design and benchmark new microstimulation devices that are based on transcutaneous photovoltaic stimulation. The approach is to use wired multielectrode cuffs to test different stimulation patterns, and then build photovoltaic stimulators to generate the most optimal patterns. We validate stimulation through heart rate analysis. Main results. A range of different stimulation geometries are explored with large differences in performance. Two types of photovoltaic devices are fabricated to deliver stimulation: photocapacitors and photovoltaic flags. The former is simple and more compact, but has limited efficiency. The photovoltaic flag approach is more elaborate, but highly efficient. Both can be used for wireless actuation of the vagus nerve using light impulses. Significance. These approaches can enable studies in small animals that were previously challenging, such as long-term in vivo studies for mapping functional vagus nerve innervation. This new knowledge may have potential to support clinical translation of VNS for treatment of select inflammatory and neurologic diseases.

  • 9.
    Favero, Martina
    et al.
    KTH, School of Engineering Sciences (SCI), Mathematics (Dept.), Mathematical Statistics.
    Hult, Henrik
    KTH, School of Engineering Sciences (SCI), Mathematics (Dept.), Mathematical Statistics.
    Asymptotic analysis of backward sampling algorithms in Kingman's coalescentManuscript (preprint) (Other academic)
  • 10.
    Favero, Martina
    et al.
    KTH, School of Engineering Sciences (SCI), Mathematics (Dept.), Mathematical Statistics.
    Hult, Henrik
    KTH, School of Engineering Sciences (SCI), Mathematics (Dept.), Mathematical Statistics.
    Asymptotic behaviour of sampling and backward transition probabilities of the coalescent with parent dependent mutationsManuscript (preprint) (Other academic)
  • 11.
    Favero, Martina
    et al.
    Stockholm Univ, Stockholm, Sweden..
    Hult, Henrik
    KTH, School of Engineering Sciences (SCI), Mathematics (Dept.), Mathematical Statistics.
    Asymptotic behaviour of sampling and transition probabilities in coalescent models under selection and parent dependent mutations2022In: Electronic Communications in Probability, E-ISSN 1083-589X, Vol. 27, no none, article id 32Article in journal (Refereed)
    Abstract [en]

    The results in this paper provide new information on asymptotic properties of classical models: the neutral Kingman coalescent under a general finite-alleles, parent-dependent mutation mechanism, and its generalisation, the ancestral selection graph. Several relevant quantities related to these fundamental models are not explicitly known when mutations are parent dependent. Examples include the probability that a sample taken from a population has a certain type configuration, and the transition probabilities of their block counting jump chains. In this paper, asymptotic results are derived for these quantities, as the sample size goes to infinity. It is shown that the sampling probabilities decay polynomially in the sample size with multiplying constant depending on the stationary density of the Wright-Fisher diffusion and that the transition probabilities converge to the limit of frequencies of types in the sample.

  • 12.
    Favero, Martina
    et al.
    KTH, School of Engineering Sciences (SCI), Mathematics (Dept.), Mathematical Statistics.
    Hult, Henrik
    KTH, School of Engineering Sciences (SCI), Mathematics (Dept.), Mathematical Statistics.
    Weak convergence of the scaled jump chain and number of mutations of the Kingman coalescentManuscript (preprint) (Other academic)
  • 13.
    Favero, Martina
    et al.
    KTH, School of Engineering Sciences (SCI), Mathematics (Dept.), Mathematical Statistics.
    Hult, Henrik
    KTH, School of Engineering Sciences (SCI), Mathematics (Dept.), Mathematical Statistics.
    Koski, Timo
    KTH, School of Engineering Sciences (SCI), Mathematics (Dept.), Mathematical Statistics.
    A dual process for the coupled Wright-Fisher diffusion2021In: Journal of Mathematical Biology, ISSN 0303-6812, E-ISSN 1432-1416, Vol. 82, no 1-2, article id 6Article in journal (Refereed)
    Abstract [en]

    The coupled Wright-Fisher diffusion is a multi-dimensional Wright-Fisher diffusion for multi-locus and multi-allelic genetic frequencies, expressed as the strong solution to a system of stochastic differential equations that are coupled in the drift, where the pairwise interaction among loci is modelled by an inter-locus selection. In this paper, an ancestral process, which is dual to the coupled Wright-Fisher diffusion, is derived. The dual process corresponds to the block counting process of coupled ancestral selection graphs, one for each locus. Jumps of the dual process arise from coalescence, mutation, single-branching, which occur at one locus at the time, and double-branching, which occur simultaneously at two loci. The coalescence and mutation rates have the typical structure of the transition rates of the Kingman coalescent process. The single-branching rate not only contains the one-locus selection parameters in a form that generalises the rates of an ancestral selection graph, but it also contains the two-locus selection parameters to include the effect of the pairwise interaction on the single loci. The double-branching rate reflects the particular structure of pairwise selection interactions of the coupled Wright-Fisher diffusion. Moreover, in the special case of two loci, two alleles, with selection and parent independent mutation, the stationary density for the coupled Wright-Fisher diffusion and the transition rates of the dual process are obtained in an explicit form.

  • 14.
    Favero, Martina
    et al.
    KTH, School of Engineering Sciences (SCI), Mathematics (Dept.), Mathematical Statistics.
    Koski, Timo
    KTH, School of Engineering Sciences (SCI), Mathematics (Dept.), Mathematical Statistics.
    Hult, Henrik
    KTH, School of Engineering Sciences (SCI), Mathematics (Dept.), Mathematical Statistics.
    A dual process for the coupled Wright-Fisher diffusionIn: Journal of Mathematical Biology, ISSN 0303-6812, E-ISSN 1432-1416Article in journal (Refereed)
    Abstract [en]

    The coupled Wright-Fisher diffusion is a multi-dimensional Wright-Fisher diffusion for multi-locus and multi-allelic genetic frequencies, expressed as the strong solution to a system of stochastic differential equations that are coupled in the drift, where the pairwise interaction among loci is modelled by an inter-locus selection. In this paper, an ancestral process, which is dual to the coupled Wright-Fisher diffusion, is derived. The dual process corresponds to the block counting process of coupled ancestral selection graphs, one for each locus. Jumps of the dual process arise from coalescence, mutation, single-branching, which occur at one locus at the time, and double-branching, which occur simultaneously at two loci. The coalescence and mutation rates have the typical structure of the transition rates of the Kingman coalescent process. The single-branching rate not only contains the one-locus selection parameters in a form that generalises the rates of an ancestral selection graph, but it also contains the two-locus selection parameters to include the effect of the pairwise interaction on the single loci. The double-branching rate reflects the particular structure of pairwise selection interactions of the coupled Wright-Fisher diffusion. Moreover, in the special case of two loci, two alleles, with selection and parent independent mutation, the stationary density for the coupled Wright-Fisher diffusion and the transition rates of the dual process are obtained in an explicit form.

  • 15.
    Garcia-Pareja, Celia
    et al.
    KTH.
    Hult, Henrik
    KTH, School of Engineering Sciences (SCI), Mathematics (Dept.).
    Koski, Timo
    KTH, School of Engineering Sciences (SCI), Mathematics (Dept.). ;Univ Helsinki, Helsinki Inst Informat Technol, Pietari Kalmin Katu 5, Helsinki 00560, Finland..
    EXACT SIMULATION OF COUPLED WRIGHT-FISHER DIFFUSIONS2021In: Advances in Applied Probability, ISSN 0001-8678, E-ISSN 1475-6064, Vol. 53, no 4, p. 923-950Article in journal (Refereed)
    Abstract [en]

    In this paper an exact rejection algorithm for simulating paths of the coupled Wright- Fisher diffusion is introduced. The coupled Wright-Fisher diffusion is a family of multivariate Wright-Fisher diffusions that have drifts depending on each other through a coupling term and that find applications in the study of networks of interacting genes. The proposed rejection algorithm uses independent neutral Wright-Fisher diffusions as candidate proposals, which are only needed at a finite number of points. Once a candidate is accepted, the remainder of the path can be recovered by sampling from neutral multivariate Wright-Fisher bridges, for which an exact sampling strategy is also provided. Finally, the algorithm's complexity is derived and its performance demonstrated in a simulation study.

  • 16.
    Gudmundsson, Thorbjörn
    et al.
    KTH, School of Engineering Sciences (SCI), Mathematics (Dept.), Mathematical Statistics.
    Hult, Henrik
    KTH, School of Engineering Sciences (SCI), Mathematics (Dept.), Mathematical Statistics.
    Markov chain monte carlo for computing rare-event probabilities for a heavy-tailed random walk2014In: Journal of Applied Probability, ISSN 0021-9002, E-ISSN 1475-6072, Vol. 51, no 2, p. 359-376Article in journal (Refereed)
    Abstract [en]

    In this paper a method based on a Markov chain Monte Carlo (MCMC) algorithm is proposed to compute the probability of a rare event. The conditional distribution of the underlying process given that the rare event occurs has the probability of the rare event as its normalizing constant. Using the MCMC methodology, a Markov chain is simulated, with the aforementioned conditional distribution as its invariant distribution, and information about the normalizing constant is extracted from its trajectory. The algorithm is described in full generality and applied to the problem of computing the probability that a heavy-tailed random walk exceeds a high threshold. An unbiased estimator of the reciprocal probability is constructed whose normalized variance vanishes asymptotically. The algorithm is extended to random sums and its performance is illustrated numerically and compared to existing importance sampling algorithms.

  • 17.
    Gudmundsson, Thorbjörn
    et al.
    KTH, School of Engineering Sciences (SCI), Mathematics (Dept.), Mathematical Statistics.
    Hult, Henrik
    KTH, School of Engineering Sciences (SCI), Mathematics (Dept.), Mathematical Statistics.
    Markov chain Monte Carlo for rare-event simulation for light-tailed random walkManuscript (preprint) (Other academic)
  • 18.
    Gudmundsson, Thorbjörn
    et al.
    KTH, School of Engineering Sciences (SCI), Mathematics (Dept.), Mathematical Statistics.
    Hult, Henrik
    KTH, School of Engineering Sciences (SCI), Mathematics (Dept.), Mathematical Statistics.
    Markov chain Monte Carlo for rare-event simulation for Markov chainsManuscript (preprint) (Other academic)
  • 19.
    Gudmundsson, Thorbjörn
    et al.
    KTH, School of Engineering Sciences (SCI), Mathematics (Dept.), Mathematical Statistics.
    Hult, Henrik
    KTH, School of Engineering Sciences (SCI), Mathematics (Dept.), Mathematical Statistics.
    Markov chain Monte Carlo for rare-event simulation for stochastic recurrence equations with heavy-tailed innovationsManuscript (preprint) (Other academic)
  • 20.
    Hargreaves, Brandon
    et al.
    Brown University.
    Hult, Henrik
    Reda, Sherief
    Brown University.
    Within-die process variations: How accurately can they be statistically modeled?2008Conference paper (Refereed)
    Abstract [en]

    Within-die process variations arise during integrated circuit (IC) fabrication in the sub-100nm regime. These variations are of paramount concern as they deviate the performance of ICs from their designers’ original intent. These deviations reduce the parametric yield and revenues from integrated circuit fabrication. In this paper we provide a complete treatment to the subject of within-die variations. We propose a scan-chain based system, vMeter, to extract within-die variations in an automated fashion. We implement our system in a sample of 90nm chips, and collect the within-die variations data. Then we propose a number of novel statistical analysis techniques that accurately model the within-die variation trends and capture the spatial correlations. We propose the use of maximum-likelihood techniques to find the required parameters to fit the model to the data. The accuracy of our models is statistically verified through residua lanalysis and variograms. Using our successful modeling technique, we propose a procedure to generate synthetic within-dievariation patterns that mimic, or imitate, real silicon data.

  • 21.
    Hult, Henrik
    KTH, School of Engineering Sciences (SCI), Mathematics (Dept.).
    Approximating some Volterra type stochastic integrals with applications to parameter estimation2003In: Stochastic Processes and their Applications, ISSN 0304-4149, E-ISSN 1879-209X, Vol. 105, no 1, p. 1-32Article in journal (Refereed)
  • 22.
    Hult, Henrik
    et al.
    KTH, School of Engineering Sciences (SCI), Mathematics (Dept.), Mathematical Statistics.
    Favero, Martina
    KTH, School of Engineering Sciences (SCI), Mathematics (Dept.), Mathematical Statistics.
    Estimates of the proportino of SARS-CoV-2 infected individuals in SwedenManuscript (preprint) (Other academic)
    Abstract [en]

    In this paper a Bayesian SEIR model is studied to estimate the

    proportion of the population infected with SARS-CoV-2, the virus responsi-

    ble for COVID-19. To capture heterogeneity in the population and the eect

    of interventions to reduce the rate of epidemic spread, the model uses a time-

    varying contact rate, whose logarithm has a Gaussian process prior. A Poisson

    point process is used to model the occurrence of deaths due to COVID-19 and

    the model is calibrated using data of daily death counts in combination with

    a snapshot of the the proportion of individuals with an active infection, per-

    formed in Stockholm in late March. The methodology is applied to regions in

    Sweden. The results show that the estimated proportion of the population who

    has been infected is around 13:5% in Stockholm, by 2020-05-15, and ranges be-

    tween 2.5%-15.6% in the other investigated regions. In Stockholm where the

    peak of daily death counts is likely behind us, parameter uncertainty does not

    heavily inuence the expected daily number of deaths, nor the expected cumu-

    lative number of deaths. It does, however, impact the estimated cumulative

    number of infected individuals. In the other regions, where random sampling

    of the number of active infections is not available, parameter sharing is used

    to improve estimates, but the parameter uncertainty remains substantial.

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  • 23.
    Hult, Henrik
    et al.
    KTH, School of Engineering Sciences (SCI), Mathematics (Dept.), Mathematical Statistics.
    Kiessling, Jonas
    KTH, School of Engineering Sciences (SCI), Mathematics (Dept.), Mathematics (Div.).
    Algorithmic trading with Markov chainsManuscript (preprint) (Other academic)
    Abstract [en]

    An order book consists of a list of all buy and sell offers, represented by price and quantity, available to a market agent. The order book changes rapidly, within fractions of a second, due to new orders being entered into the book. The volume at a certain price level may increase due to limitorders, i.e. orders to buy or sell placed at the end of the queue, or decrease because of market orders or cancellations.

    In this paper a high-dimensional Markov chain is used to represent the state and evolution of the entire order book. The design and evaluation of optimal algorithmic strategies for buying and selling is studied within the theory of Markov decision processes. General conditions are provided that guarantee the existence of optimal strategies. Moreover, a value-iteration algorithm is presented that enables finding optimal strategies numerically.

    As an illustration a simple version of the Markov chain model is calibrated to high-frequency observations of the order book in a foreign exchange market. In this model, using an optimally designed strategy for buying one unit provides a significant improvement, in terms of the expected buy price, over a naive buy-one-unit strategy.

  • 24.
    Hult, Henrik
    et al.
    KTH, School of Engineering Sciences (SCI), Mathematics (Dept.), Mathematical Statistics.
    Lindhe, Adam
    KTH, School of Engineering Sciences (SCI), Mathematics (Dept.), Mathematical Statistics.
    Nyquist, Pierre
    KTH, School of Engineering Sciences (SCI), Mathematics (Dept.), Mathematical Statistics.
    On the projected Aubry set of the rate function associated with large deviations for stochastic approximationsManuscript (preprint) (Other academic)
    Abstract [en]

    In this article, we look at the problem of minimizing an action potential that arises from large deviation theory for stochastic approximations. The solutions to the minimising problem satisfy, in the sense of a viscosity solution, a Hamilton-Jacobi equation. From weak KAM theory, we know that these viscosity solutions are characterised by the projected Aubryset. The main result of this paper is that, for a specific rate function corresponding to the astochastic approximation algorithm, we prove that the projected Aubry set is equal to the forward limit set to the limit ODE.

    Download full text (pdf)
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  • 25.
    Hult, Henrik
    et al.
    KTH, School of Engineering Sciences (SCI), Mathematics (Dept.), Mathematical Statistics.
    Lindhe, Adam
    KTH, School of Engineering Sciences (SCI), Mathematics (Dept.), Mathematical Statistics.
    Nyquist, Pierre
    KTH, School of Engineering Sciences (SCI), Mathematics (Dept.), Mathematical Statistics.
    Wu, Guo-Jhen
    KTH, School of Engineering Sciences (SCI), Mathematics (Dept.).
    A weak convergence approach to large deviations for stochastic approximationsManuscript (preprint) (Other academic)
    Abstract [en]

    Large deviations for stochastic approximations is a well-studied field that yields convergence properties for many useful algorithms in statistics, machine learning and statistical physics. In this article, we prove, under certain assumptions, a large deviation principle for a stochastic approximation with state-dependent Markovian noise and with decreasing step size. Common algorithms that satisfy these conditions include stochastic gradient descent, persistent contrastive divergence and the Wang-Landau algorithm. The proof is based don't he weak convergence approach to the theory of large deviations and uses a representation formula to rewrite the problem into a stochastic control problem. The resulting rate function is an action potential over a local rate function that is the Fenchel-Legendre transform of a limiting Hamiltonian.

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  • 26.
    Hult, Henrik
    et al.
    KTH, School of Engineering Sciences (SCI), Mathematics (Dept.), Mathematical Statistics.
    Lindskog, F.
    Hammarlid, O.
    Rehn, C. J.
    Utility-Based Investment Principles2012In: Risk and Portfolio Analysis, Springer Nature , 2012, p. 127-157Chapter in book (Refereed)
    Abstract [en]

    In the previous chapter we measured the quality of an investment in terms of the expected value E[V1] and the variance Var(V1) of the future portfolio value V1 and determined portfolio weights (subject to constraints) that maximize a suitable trade-off$$\mathrm{E}[{V }_{1}] - c\mathrm{Var}({V }_{1})/(2{V }_{0})$$ between a large expected value and a small variance. Attractive features of this approach are that the probability distribution of V1 does not have to be specified in detail and that explicit expressions for the optimal portfolio weights are found that have intuitive interpretations. We saw that this approach makes perfect sense if we consider portfolio values V1 that can be expressed as linear combinations of asset returns whose joint distribution is a multivariate normal distribution. However, unless there are good reasons to assume a multivariate normal distribution (or, more generally, as will be made clear in Chap. 9, an elliptical distribution), solutions provided by the quadratic investment principles can be rather misleading. Here we want to allow for a probability distribution of any kind, and this calls for more general investment principles that are not only based on the variance and expected value of V1. 

  • 27.
    Hult, Henrik
    et al.
    KTH, School of Engineering Sciences (SCI), Mathematics (Dept.), Mathematical Statistics.
    Lindskog, Filip
    KTH, School of Engineering Sciences (SCI), Mathematics (Dept.), Mathematical Statistics.
    Extremal behavior of regularly varying stochastic processes2005In: Stochastic Processes and their Applications, ISSN 0304-4149, E-ISSN 1879-209X, Vol. 115, no 2, p. 249-274Article in journal (Refereed)
    Abstract [en]

    We study a formulation of regular variation for multivariate stochastic processes on the unit interval with sample paths that are almost surely right-continuous with left limits and we provide necessary and sufficient conditions for such stochastic processes to be regularly varying. A version of the Continuous Mapping Theorem is proved that enables the derivation of the tail behavior of rather general mappings of the regularly varying stochastic process. For a wide class of Markov processes with increments satisfying a condition of weak dependence in the tails we obtain simplified sufficient conditions for regular variation. For such processes we show that the possible regular variation limit measures concentrate on step functions with one step, from which we conclude that the extremal behavior of such processes is due to one big jump or an extreme starting point. By combining this result with the Continuous Mapping Theorem, we are able to give explicit results on the tail behavior of various vectors of functionals acting on such processes. Finally, using the Continuous Mapping Theorem we derive the tail behavior of filtered regularly varying Levy processes.

  • 28.
    Hult, Henrik
    et al.
    KTH, School of Engineering Sciences (SCI), Mathematics (Dept.).
    Lindskog, Filip
    KTH, School of Engineering Sciences (SCI), Mathematics (Dept.), Mathematical Statistics.
    Heavy-tailed insurance portfolios: buffer capital and ruin probabilities2006Report (Other academic)
  • 29.
    Hult, Henrik
    et al.
    KTH, School of Engineering Sciences (SCI), Mathematics (Dept.).
    Lindskog, Filip
    KTH, School of Engineering Sciences (SCI), Mathematics (Dept.).
    Multivariate extremes, aggregation and dependence in elliptical distributions2002In: Advances in Applied Probability, ISSN 0001-8678, E-ISSN 1475-6064, Vol. 32, no 3, p. 587-608Article in journal (Refereed)
  • 30.
    Hult, Henrik
    et al.
    KTH, School of Engineering Sciences (SCI), Mathematics (Dept.), Mathematical Statistics.
    Lindskog, Filip
    KTH, School of Engineering Sciences (SCI), Mathematics (Dept.), Mathematical Statistics.
    On Kesten's counterexample to the Cramer-Wold device for regular variation2006In: Bernoulli, ISSN 1350-7265, E-ISSN 1573-9759, Vol. 12, no 1, p. 133-142Article in journal (Refereed)
    Abstract [en]

    In 2002 Basrak, Davis and Mikosch showed that an analogue of the Cramer-Wold device holds for regular variation of random vectors if the index of regular variation is not an integer. This characterization is of importance when studying stationary solutions to stochastic recurrence equations. In this paper we construct counterexamples showing that for integer-valued indices, regular variation of all linear combinations does not imply that the vector is regularly varying. The construction is based on unpublished notes by Harry Kesten.

  • 31. Hult, Henrik
    et al.
    Lindskog, Filip
    KTH, School of Engineering Sciences (SCI), Mathematics (Dept.), Mathematical Statistics.
    On regular variation for infinitely divisible random vectors and additive processes2006In: Advances in Applied Probability, ISSN 0001-8678, E-ISSN 1475-6064, Vol. 38, no 1, p. 134-148Article in journal (Refereed)
    Abstract [en]

    We study the tail behavior of regularly varying infinitely divisible random vectors and additive processes, i.e. stochastic processes with independent but not necessarily stationary increments. We show that the distribution of an infinitely divisible random vector is tail equivalent to its Levy measure and we study the asymptotic decay of the probability for an additive process to hit sets far away from the origin. The results are extensions of known univariate results to the multivariate setting; we exemplify some of the difficulties that arise in the multivariate case.

  • 32.
    Hult, Henrik
    et al.
    KTH, School of Engineering Sciences (SCI), Mathematics (Dept.), Mathematical Statistics.
    Lindskog, Filip
    KTH, School of Engineering Sciences (SCI), Mathematics (Dept.), Mathematical Statistics.
    Ruin probabilities under general investments and heavy-tailed claims2011In: Finance and Stochastics, ISSN 0949-2984, E-ISSN 1432-1122, Vol. 15, no 2, p. 243-265Article in journal (Refereed)
    Abstract [en]

    In this paper, the asymptotic decay of finite time ruin probabilities is studied. An insurance company is considered that faces heavy-tailed claims and makes investments in risky assets whose prices evolve according to quite general semimartingales. In this setting, the ruin problem corresponds to determining hitting probabilities for the solution to a randomly perturbed stochastic integral equation. A large deviation result for the hitting probabilities is derived that holds uniformly over a family of semimartingales. This result gives the asymptotic decay of finite time ruin probabilities under sufficiently conservative investment strategies, including ruin-minimizing strategies. In particular, as long as the insurance company invests sufficiently conservatively, the investment strategy has only a moderate impact on the asymptotics of the ruin probability.

  • 33.
    Hult, Henrik
    et al.
    KTH, School of Engineering Sciences (SCI), Mathematics (Dept.), Mathematical Statistics.
    Lindskog, Filip
    KTH, School of Engineering Sciences (SCI), Mathematics (Dept.).
    Hammarlid, O.
    Rehn, C. J.
    Convex Optimization2012In: Springer Series in Operations Research and Financial Engineering, Springer Nature , 2012, p. 33-38Chapter in book (Refereed)
    Abstract [en]

    Many of the investment and hedging problems we will encounter can be formulated as a minimization of a function over a set determined by the investor’s risk and budget constraints and other restrictions on the type of positions that the investor can take. Such problems become particularly tractable if both the function to be minimized and the set over which the minimization is done are convex. The minimization problem is in this case called a convex optimization problem. This chapter presents basic results for solving convex optimization problems that will be applied in subsequent chapters.

  • 34.
    Hult, Henrik
    et al.
    KTH, School of Engineering Sciences (SCI), Mathematics (Dept.), Mathematical Statistics.
    Lindskog, Filip
    KTH, School of Engineering Sciences (SCI), Mathematics (Dept.), Mathematics (Div.).
    Hammarlid, O.
    Rehn, C. J.
    Empirical Methods2012In: Risk and Portfolio Analysis, Springer Nature , 2012, p. 197-229Chapter in book (Refereed)
    Abstract [en]

    In this chapter we consider a modeling approach that uses a set of historical data, such as bond prices, share prices, claim sizes, or exchange rates, to model the value at a future time T > 0 of portfolios whose values depend on a given set of assets and possibly also liabilities. Here we want the data to speak for themselves in the sense that the model for the future values should only be based on information available in the given historical data samples. The assumption we make is therefore that the information in the samples is representative of future values and that no additional probability beliefs of the modeler are relevant.

  • 35.
    Hult, Henrik
    et al.
    KTH, School of Engineering Sciences (SCI), Mathematics (Dept.), Mathematical Statistics.
    Lindskog, Filip
    KTH, School of Engineering Sciences (SCI), Mathematics (Dept.).
    Hammarlid, O.
    Rehn, C. J.
    Multivariate Models2012In: Risk and Portfolio Analysis, Springer Nature , 2012, p. 273-330Chapter in book (Refereed)
    Abstract [en]

    In this chapter, we consider multivariate models for the joint distribution of several risk factors such as returns or log returns for different assets, zero rate changes for different maturity times, changes in implied volatility, and losses due to defaults on risky loans. Our aim is to specify a good model for the future value g(X) of a portfolio, where the function g is known and its argument X is a random vector of, for instance, log returns and zero rate changes over a given future time period. Since the function g is known, what remains is to make a good choice of probability distribution for random vector X. 

  • 36.
    Hult, Henrik
    et al.
    KTH, School of Engineering Sciences (SCI), Mathematics (Dept.), Mathematical Statistics.
    Lindskog, Filip
    KTH, School of Engineering Sciences (SCI), Mathematics (Dept.).
    Hammarlid, O.
    Rehn, C. J.
    Parametric Models and Their Tails2012In: Springer Series in Operations Research and Financial Engineering, Springer Nature , 2012, p. 231-271Chapter in book (Refereed)
    Abstract [en]

    In this chapter we consider approaches to selecting a parametric family of distributions for a random variable and approaches to estimating the parameters. We also present techniques for analyzing the tails of the chosen probability distribution and the effect of the tails on the estimation of risk measures. Finally, we consider a semiparametric approach to the estimation of tail probabilities. It provides an alternative to relying on a full parametric model in order to produce estimates of tail probabilities beyond the range of the sample data.

  • 37.
    Hult, Henrik
    et al.
    KTH, School of Engineering Sciences (SCI), Mathematics (Dept.).
    Lindskog, Filip
    KTH, School of Engineering Sciences (SCI), Mathematics (Dept.).
    Hammarlid, O.
    Rehn, C. J.
    Quadratic Hedging Principles2012In: Risk and Portfolio Analysis, Springer Nature , 2012, p. 39-83Chapter in book (Refereed)
    Abstract [en]

    Fix a future time T and let L be the value of a liability at that time. One example of L is the portfolio value of derivative instruments issued by a bank. Another example is the value of future claims from insurance products sold by an insurance company. Typically the holder of the liability does not want to speculate on a favorable outcome of this random variable. The ideal approach to managing the risk of an unfavorable outcome of L would be to purchase a portfolio whose value A (A for assets) at the future time T exactly matches that of the liability. In that case, A = L, and the risk of an unfavorable outcome of L is removed completely by purchasing the asset portfolio. The problem with this approach is that it is not always possible to find a portfolio of assets whose future value corresponds exactly to that of the liability; one example is when the liability is made up of insurance claims. 

  • 38.
    Hult, Henrik
    et al.
    KTH, School of Engineering Sciences (SCI), Mathematics (Dept.), Mathematical Statistics.
    Lindskog, Filip
    KTH, School of Engineering Sciences (SCI), Mathematics (Dept.).
    Hammarlid, O.
    Rehn, C. J.
    Quadratic Investment Principles2012In: Risk and Portfolio Analysis, Springer Nature , 2012, p. 85-126Chapter in book (Refereed)
    Abstract [en]

    In this chapter, we present investment principles solely based on means and variances of asset returns and budget restrictions. To begin with, we only consider risky assets in the sense that the variances of the returns are strictly positive. We will then consider the more interesting situation where we also have the possibility to invest (or deposit) money in a risk-free asset. 

  • 39.
    Hult, Henrik
    et al.
    KTH, School of Engineering Sciences (SCI), Mathematics (Dept.), Mathematical Statistics.
    Lindskog, Filip
    KTH, School of Engineering Sciences (SCI), Mathematics (Dept.).
    Hammarlid, O.
    Rehn, C. J.
    Risk Measurement Principles2012In: Risk and Portfolio Analysis, Springer Nature , 2012, p. 159-194Chapter in book (Refereed)
    Abstract [en]

    In this chapter, we take a close look at the principles of risk measurement. We argue that it is natural to quantify the riskiness of a position in monetary units so that the measurement of the risk of a position can be interpreted as the size of buffer capital that should be added to the position to provide a sufficient protection against undesirable outcomes. In the investment problems in Chap. 4, variance was used to quantify the riskiness of a portfolio. However, variance, being just the expected squared deviation from the mean value, does not differentiate between good positive deviations and bad negative deviations and cannot easily be translated into meaningful monetary values unless the future value we consider is close to normally distributed. The risk premium considered in Chap. 5 is more natural than the variance as a summary of the riskiness and potential reward of a position. However, the risk premium is difficult to use effectively to control the risk taking of a financial institution or to determine whether the aggregate position of a company or business unit is acceptable from a risk perspective. In this chapter, we will present measures of risk, including the widely used value-at-risk and expected shortfall, analyze their properties, and evaluate their performance in a large number of examples. 

  • 40.
    Hult, Henrik
    et al.
    KTH, School of Engineering Sciences (SCI), Mathematics (Dept.), Mathematical Statistics.
    Lindskog, Filip
    KTH, School of Engineering Sciences (SCI), Mathematics (Dept.), Mathematics (Div.).
    Hammarlid, Ola
    Swedbank AB (publ), Stockholm, Sweden.
    Rehn, Carl Johan
    E. Öhman J:or Fondkommission AB, Stockholm, Sweden.
    Interest Rates and Financial Derivatives2012In: Risk and Portfolio Analysis, Springer Nature , 2012, p. 3-31Chapter in book (Refereed)
    Abstract [en]

    In this chapter we present the basic theory of interest rate instruments and the pricing of financial derivatives. The material we have chosen to present here is interesting and relevant in its own right but particularly so as the basis for the principles and methods considered in subsequent chapters.

  • 41.
    Hult, Henrik
    et al.
    KTH, School of Engineering Sciences (SCI), Mathematics (Dept.), Mathematical Statistics.
    Lindskog, Filip
    KTH, School of Engineering Sciences (SCI), Mathematics (Dept.), Mathematical Statistics.
    Hammarlid, Ola
    Swedbank AB, Stockholm, SE-105 34, Sweden.
    Rehn, Carl Johan
    E. Öhman J:or Fondkommission AB, Stockholm, Sweden.
    Preface2012In: Risk and Portfolio Analysis: Principles and Methods, Springer Nature , 2012, p. vii-xChapter in book (Refereed)
  • 42.
    Hult, Henrik
    et al.
    KTH, School of Engineering Sciences (SCI), Mathematics (Dept.), Mathematical Statistics.
    Lindskog, Filip
    KTH, School of Engineering Sciences (SCI), Mathematics (Dept.), Mathematical Statistics.
    Nykvist, Johan
    KTH, School of Engineering Sciences (SCI), Mathematics (Dept.), Mathematical Statistics.
    A simple time-consistent model for the forward density process2013In: International Journal of Theoretical and Applied Finance, ISSN 0219-0249, Vol. 16, no 8, p. 13500489-Article in journal (Refereed)
    Abstract [en]

    In this paper, a simple model for the evolution of the forward density of the future value of an asset is proposed. The model allows for a straightforward initial calibration to option prices and has dynamics that are consistent with empirical findings from option price data. The model is constructed with the aim of being both simple and realistic, and avoid the need for frequent re-calibration. The model prices of n options and a forward contract are expressed as time-varying functions of an (n + 1)-dimensional Brownian motion and it is investigated how the Brownian trajectory can be determined from the trajectories of the price processes. An approach based on particle filtering is presented for determining the location of the driving Brownian motion from option prices observed in discrete time. A simulation study and an empirical study of call options on the S&P 500 index illustrate that the model provides a good fit to option price data.

  • 43.
    Hult, Henrik
    et al.
    KTH, School of Engineering Sciences (SCI), Mathematics (Dept.), Mathematical Statistics.
    Lindskog, Henrik
    KTH, School of Engineering Sciences (SCI), Mathematics (Dept.), Mathematical Statistics.
    Regular variation for measures on metric spaces2006In: Publications de l'Institut Mathématique (Beograd), ISSN 0350-1302, E-ISSN 1820-7405, Vol. 80, no 94, p. 121-140Article in journal (Refereed)
    Abstract [en]

    The foundations of regular variation for Borel measures on a com- plete separable space S, that is closed under multiplication by nonnegative real numbers, is reviewed. For such measures an appropriate notion of convergence is presented and the basic results such as a Portmanteau theorem, a mapping theorem and a characterization of relative compactness are derived. Regu- lar variation is defined in this general setting and several statements that are equivalent to this definition are presented. This extends the notion of regular variation for Borel measures on the Euclidean space Rd to more general metric spaces. Some examples, including regular variation for Borel measures on Rd, the space of continuous functions C and the Skorohod space D, are provided.

  • 44.
    Hult, Henrik
    et al.
    KTH, School of Engineering Sciences (SCI), Mathematics (Dept.), Mathematical Statistics.
    Nykvist, Johan
    KTH, School of Engineering Sciences (SCI), Mathematics (Dept.), Mathematical Statistics.
    A note on efficient importance sampling for one-dimensional diffusionsManuscript (preprint) (Other academic)
  • 45.
    Hult, Henrik
    et al.
    KTH, School of Engineering Sciences (SCI), Mathematics (Dept.), Mathematical Statistics.
    Nykvist, Johan
    KTH, School of Engineering Sciences (SCI), Mathematics (Dept.), Mathematical Statistics.
    Efficient importance sampling to assess the risk of voltage collapse in power systemsManuscript (preprint) (Other academic)
  • 46.
    Hult, Henrik
    et al.
    KTH, School of Engineering Sciences (SCI), Mathematics (Dept.), Mathematical Statistics.
    Nykvist, Johan
    KTH, School of Engineering Sciences (SCI), Mathematics (Dept.), Mathematical Statistics.
    Efficient importance sampling to compute loss probabilities in financial portfoliosManuscript (preprint) (Other academic)
  • 47.
    Hult, Henrik
    et al.
    KTH, School of Engineering Sciences (SCI), Mathematics (Dept.), Mathematical Statistics.
    Nyquist, Pierre
    KTH, School of Engineering Sciences (SCI), Mathematics (Dept.), Mathematical Statistics.
    Large deviations for weighted empirical measures arising in importance sampling2016In: Stochastic Processes and their Applications, ISSN 0304-4149, E-ISSN 1879-209X, Vol. 126, no 1Article in journal (Refereed)
    Abstract [en]

    Importance sampling is a popular method for efficient computation of various properties of a distribution such as probabilities, expectations, quantiles etc. The output of an importance sampling algorithm can be represented as a weighted empirical measure, where the weights are given by the likelihood ratio between the original distribution and the sampling distribution. In this paper the efficiency of an importance sampling algorithm is studied by means of large deviations for the weighted empirical measure. The main result, which is stated as a Laplace principle for the weighted empirical measure arising in importance sampling, can be viewed as a weighted version of Sanov's theorem. The main theorem is applied to quantify the performance of an importance sampling algorithm over a collection of subsets of a given target set as well as quantile estimates. The proof of the main theorem relies on the weak convergence approach to large deviations developed by Dupuis and Ellis.

  • 48.
    Hult, Henrik
    et al.
    KTH, School of Engineering Sciences (SCI), Mathematics (Dept.), Mathematical Statistics.
    Samorodnitsky, Gennady
    Large deviations for point processes based on stationary sequences with heavy tails2010In: Journal of Applied Probability, ISSN 0021-9002, E-ISSN 1475-6072, Vol. 47, no 1, p. 1-40Article in journal (Refereed)
    Abstract [en]

    In this paper we propose a framework that facilitates the study of large deviations for point processes based on stationary sequences with regularly varying tails. This framework allows us to keep track both of the magnitude of the extreme values of a process and the order in which these extreme values appear. Particular emphasis is put on (infinite) linear processes with random coefficients. The proposed framework provides a fairly complete description of the joint asymptotic behavior of the large values of the stationary sequence. We apply the general result on large deviations for point processes to derive the asymptotic decay of certain probabilities related to partial sum processes as well as ruin probabilities.

  • 49.
    Hult, Henrik
    et al.
    KTH, School of Engineering Sciences (SCI), Mathematics (Dept.).
    Samorodnitsky, Gennady
    Cornell University, ORIE.
    Tail probabilities for infinite series of regularly varying random vectors2008In: Bernoulli, ISSN 1350-7265, E-ISSN 1573-9759, Vol. 14, no 3, p. 838-864Article in journal (Refereed)
    Abstract [en]

    A random vector X with representation X = Sigma(j >= 0)A(j)Z(j) is considered. Here, (Z(j)) is a sequence of independent and identically distributed random vectors and (A(j)) is a sequence of random matrices, 'predictable' with respect to the sequence (Z(j)). The distribution of Z(1) is assumed to be multivariate regular varying. Moment conditions on the matrices (A(j)) are determined under which the distribution of X is regularly varying and, in fact, 'inherits' its regular variation from that of the (Z(j))'s. We compute the associated limiting measure. Examples include linear processes, random coefficient linear processes such as stochastic recurrence equations, random sums and stochastic integrals.

  • 50.
    Hult, Henrik
    et al.
    KTH, School of Engineering Sciences (SCI), Mathematics (Dept.).
    Svensson, Jens
    KTH, School of Engineering Sciences (SCI), Mathematics (Dept.).
    Efficient calculation of risk measures by importance sampling -- the heavy tailed caseManuscript (preprint) (Other academic)
    Abstract [en]

    Computation of extreme quantiles and tail-based risk measures using standard Monte Carlo simulation can be inefficient. A method to speed up computations is provided by importance sampling. We show that importance sampling algorithms, designed for efficient tail probability estimation, can significantly improve Monte Carlo estimators of tail-based risk measures. In the heavy-tailed setting, when the random variable of interest has a regularly varying distribution, we provide sufficient conditions for the asymptotic relative error of importance sampling estimators of risk measures, such as Value-at-Risk and expected shortfall, to be small. The results are illustrated by some numerical examples.

12 1 - 50 of 68
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