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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 Univ, 500 W 120th St, New York, NY 10027 USA..
    Hult, Henrik
    KTH, School of Engineering Sciences (SCI), Mathematics (Dept.), Mathematical Statistics.
    Leder, Kevin
    Univ Minnesota, Minneapolis, MN 55455 USA..
    IMPORTANCE SAMPLING FOR STOCHASTIC RECURRENCE EQUATIONS WITH HEAVY TAILED INCREMENTS2011In: PROCEEDINGS OF THE 2011 WINTER SIMULATION CONFERENCE (WSC) / [ed] Jain, S Creasey, R Himmelspach, J, IEEE , 2011, p. 3824-3831Conference paper (Refereed)
    Abstract [en]

    Importance sampling in the setting of heavy tailed random variables has generally focused on models with additive noise terms. In this work we extend this concept by considering importance sampling for the estimation of rare events in Markov chains of the form Xn+1 = A(n+1)X(n) + Bn+1; X-0 = 0, where the B-n's and A(n)'s are independent sequences of independent and identically distributed ( i.i.d.) random variables and the B-n's are regularly varying and the An's are suitably light tailed relative to B-n. We focus on efficient estimation of the rare event probability P(X-n > b) as b NE arrow infinity. In particular we present a strongly efficient importance sampling algorithm for estimating these probabilities, and present a numerical example showcasing 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.
    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.

  • 4.
    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.

  • 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.
    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.

  • 7.
    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.

  • 8.
    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)
  • 9.
    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)
  • 10.
    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)
  • 11.
    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.

  • 12.
    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)
  • 13.
    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.

  • 14.
    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.

  • 15.
    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)
  • 16.
    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)
  • 17.
    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.

  • 18. 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.

  • 19.
    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.

  • 20.
    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.

  • 21.
    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.

  • 22.
    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)
  • 23.
    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)
  • 24.
    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)
  • 25.
    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.

  • 26.
    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.

  • 27.
    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.

  • 28.
    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.

  • 29.
    Hult, Henrik
    et al.
    KTH, School of Engineering Sciences (SCI), Mathematics (Dept.).
    Svensson, Jens
    KTH, School of Engineering Sciences (SCI), Mathematics (Dept.).
    On Importance Sampling with Mixtures for Random Walks with Heavy Tails2012In: ACM Transactions on Modeling and Computer Simulation, ISSN 1049-3301, E-ISSN 1558-1195, Vol. 22, no 2, p. 8-Article in journal (Refereed)
    Abstract [en]

    State-dependent importance sampling algorithms based on mixtures are considered. The algorithms are designed to compute tail probabilities of a heavy-tailed random walk. The increments of the random walk are assumed to have a regularly varying distribution. Sufficient conditions for obtaining bounded relative error are presented for rather general mixture algorithms. Two new examples, called the generalized Pareto mixture and the scaling mixture, are introduced. Both examples have good asymptotic properties and, in contrast to some of the existing algorithms, they are very easy to implement. Their performance is illustrated by numerical experiments. Finally, it is proved that mixture algorithms of this kind can be designed to have vanishing relative error.

  • 30.
    Lindskog, Filip
    et al.
    KTH, School of Engineering Sciences (SCI), Mathematics (Dept.), Mathematical Statistics.
    Hult, Henrik
    KTH, School of Engineering Sciences (SCI), Mathematics (Dept.), Mathematical Statistics.
    Hammarlid, Ola
    Rehn, Carl-Johan
    Risk and portfolio analysis: principles and methods2012Book (Refereed)
  • 31.
    Nordström, Marcus
    et al.
    KTH, School of Engineering Sciences (SCI), Mathematics (Dept.), Mathematical Statistics.
    Hult, Henrik
    KTH, School of Engineering Sciences (SCI), Mathematics (Dept.), Mathematical Statistics.
    Maki, Atsuto
    KTH, School of Electrical Engineering and Computer Science (EECS), Robotics, perception and learning, RPL.
    Löfman, Fredrik
    Raysearch Labs, Stockholm, Sweden..
    Pareto Dose Prediction Using Fully Convolutional Networks Operating in 3D2018In: Medical physics (Lancaster), ISSN 0094-2405, Vol. 45, no 6, p. E176-E176Article in journal (Other academic)
  • 32.
    Nordström, Marcus
    et al.
    KTH. RaySearch Labs, Stockholm, Sweden..
    Soderberg, J.
    RaySearch Labs, Stockholm, Sweden..
    Shusharina, N.
    Massachusetts Gen Hosp, Boston, MA 02114 USA..
    Edmunds, D.
    Massachusetts Gen Hosp, Boston, MA 02114 USA..
    Lofman, F.
    RaySearch Labs, Stockholm, Sweden..
    Hult, Henrik
    KTH, School of Engineering Sciences (SCI), Mathematics (Dept.), Mathematical Statistics.
    Maki, Atsuto
    KTH, School of Electrical Engineering and Computer Science (EECS), Robotics, Perception and Learning, RPL.
    Bortfeld, T.
    Massachusetts Gen Hosp, Boston, MA 02114 USA..
    Interactive Deep Learning-Based Delineation of Gross Tumor Volume for Postoperative Glioma Patients2019In: Medical physics (Lancaster), ISSN 0094-2405, Vol. 46, no 6, p. E426-E427Article in journal (Other academic)
1 - 32 of 32
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