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  • JOURNAL ARTICLE
    Crisan D, McMurray E, 2018,

    Smoothing properties of McKean–Vlasov SDEs

    , Probability Theory and Related Fields, Vol: 171, Pages: 97-148, ISSN: 0178-8051

    © 2017, The Author(s). In this article, we develop integration by parts formulae on Wiener space for solutions of SDEs with general McKean–Vlasov interaction and uniformly elliptic coefficients. These integration by parts formulae hold both for derivatives with respect to a real variable and derivatives with respect to a measure understood in the sense of Lions. They allows us to prove the existence of a classical solution to a related PDE with irregular terminal condition. We also develop bounds for the derivatives of the density of the solutions of McKean–Vlasov SDEs.

  • JOURNAL ARTICLE
    Crisan D, Míguez J, 2018,

    Nested particle filters for online parameter estimation in discrete-time state-space Markov models

    , Bernoulli, Vol: 24, Pages: 2429-2460, ISSN: 1350-7265

    © 2018 ISI/BS. We address the problem of approximating the posterior probability distribution of the fixed parameters of a state-space dynamical system using a sequential Monte Carlo method. The proposed approach relies on a nested structure that employs two layers of particle filters to approximate the posterior probability measure of the static parameters and the dynamic state variables of the system of interest, in a vein similar to the recent “sequential Monte Carlo square” (SMC 2 ) algorithm. However, unlike the SMC 2 scheme, the proposed technique operates in a purely recursive manner. In particular, the computational complexity of the recursive steps of the method introduced herein is constant over time. We analyse the approximation of integrals of real bounded functions with respect to the posterior distribution of the system parameters computed via the proposed scheme. As a result, we prove, under regularity assumptions, that the approximation errors vanish asymptotically in L p (p ≥ 1) with convergence rate proportional to 1 N + 1 M , where N is the number of Monte Carlo samples in the parameter space and N × M is the number of samples in the state space. This result also holds for the approximation of the joint posterior distribution of the parameters and the state variables. We discuss the relationship between the SMC 2 algorithm and the new recursive method and present a simple example in order to illustrate some of the theoretical findings with computer simulations.

  • JOURNAL ARTICLE
    Davis M, Obłój J, Siorpaes P, 2018,

    Pathwise stochastic calculus with local times

    , Annales de l'Institut Henri Poincaré (B) Probabilités et Statistiques, Vol: 54, Pages: 1-21, ISSN: 0246-0203

    We study a notion of local time for a continuous path, defined as a limit ofsuitable discrete quantities along a general sequence of partitions of the timeinterval. Our approach subsumes other existing definitions and agrees with theusual (stochastic) local times a.s. for paths of a continuous semimartingale.We establish pathwise version of the It\^o-Tanaka, change of variables andchange of time formulae. We provide equivalent conditions for existence ofpathwise local time. Finally, we study in detail how the limiting objects, thequadratic variation and the local time, depend on the choice of partitions. Inparticular, we show that an arbitrary given non-decreasing process can beachieved a.s. by the pathwise quadratic variation of a standard Brownian motionfor a suitable sequence of (random) partitions; however, such degeneratebehavior is excluded when the partitions are constructed from stopping times.

  • JOURNAL ARTICLE
    Gulisashvili A, Horvath B, Jacquier A, 2018,

    Mass at zero in the uncorrelated SABR model and implied volatility asymptotics

    , Quantitative Finance, Pages: 1-13, ISSN: 1469-7688

    © 2018 Informa UK Limited, trading as Taylor & Francis Group We study the mass at the origin in the uncorrelated stochastic alpha, beta, rho stochastic volatility model and derive several tractable expressions, in particular when time becomes small or large. As an application—in fact the original motivation for this paper—we derive small-strike expansions for the implied volatility when the maturity becomes short or large. These formulae, by definition arbitrage free, allow us to quantify the impact of the mass at zero on existing implied volatility approximations, and in particular how correct/erroneous these approximations become.

  • JOURNAL ARTICLE
    Abdulle A, Pavliotis GA, Vaes U, 2017,

    Spectral Methods for Multiscale Stochastic Differential Equations

    , SIAM-ASA JOURNAL ON UNCERTAINTY QUANTIFICATION, Vol: 5, Pages: 720-761, ISSN: 2166-2525
  • CONFERENCE PAPER
    Amiri MM, Gunduz D, 2017,

    Decentralized Caching and Coded Delivery over Gaussian Broadcast Channels

    , IEEE International Symposium on Information Theory (ISIT), Publisher: IEEE, Pages: 2785-2789
  • JOURNAL ARTICLE
    Ananova A, Cont R, 2017,

    Pathwise integration with respect to paths of finite quadratic variation

    , Journal des Mathematiques Pures et Appliquees, Vol: 107, Pages: 737-757, ISSN: 0021-7824

    © 2016 The Author(s) We study a pathwise integral with respect to paths of finite quadratic variation, defined as the limit of non-anticipative Riemann sums for gradient-type integrands. We show that the integral satisfies a pathwise isometry property, analogous to the well-known Ito isometry for stochastic integrals. This property is then used to represent the integral as a continuous map on an appropriately defined vector space of integrands. Finally, we obtain a pathwise ‘signal plus noise’ decomposition for regular functionals of an irregular path with non-vanishing quadratic variation, as a unique sum of a pathwise integral and a component with zero quadratic variation.

  • JOURNAL ARTICLE
    Arulkumaran K, Deisenroth MP, Brundage M, Bharath AAet al., 2017,

    A brief survey of deep reinforcement learning

    , IEEE Signal Processing Magazine, Vol: 34, Pages: 26-38, ISSN: 1053-5888

    Deep reinforcement learning (DRL) is poised to revolutionize the field of artificial intelligence (AI) and represents a step toward building autonomous systems with a higherlevel understanding of the visual world. Currently, deep learning is enabling reinforcement learning (RL) to scale to problems that were previously intractable, such as learning to play video games directly from pixels. DRL algorithms are also applied to robotics, allowing control policies for robots to be learned directly from camera inputs in the real world. In this survey, we begin with an introduction to the general field of RL, then progress to the main streams of value-based and policy-based methods. Our survey will cover central algorithms in deep RL, including the deep Q-network (DQN), trust region policy optimization (TRPO), and asynchronous advantage actor critic. In parallel, we highlight the unique advantages of deep neural networks, focusing on visual understanding via RL. To conclude, we describe several current areas of research within the field.

  • JOURNAL ARTICLE
    Bennedsen M, Lunde A, Pakkanen MS, 2017,

    Hybrid scheme for Brownian semistationary processes

    , Finance and Stochastics, Vol: 21, Pages: 931-965, ISSN: 1432-1122

    We introduce a simulation scheme for Brownian semistationary processes, whichis based on discretizing the stochastic integral representation of the processin the time domain. We assume that the kernel function of the process isregularly varying at zero. The novel feature of the scheme is to approximatethe kernel function by a power function near zero and by a step functionelsewhere. The resulting approximation of the process is a combination ofWiener integrals of the power function and a Riemann sum, which is why we callthis method a hybrid scheme. Our main theoretical result describes theasymptotics of the mean square error of the hybrid scheme and we observe thatthe scheme leads to a substantial improvement of accuracy compared to theordinary forward Riemann-sum scheme, while having the same computationalcomplexity. We exemplify the use of the hybrid scheme by two numericalexperiments, where we examine the finite-sample properties of an estimator ofthe roughness parameter of a Brownian semistationary process and study MonteCarlo option pricing in the rough Bergomi model of Bayer et al. (2015),respectively.

  • JOURNAL ARTICLE
    Beskos A, Crisan D, Jasra A, Kamatani K, Zhou Yet al., 2017,

    A STABLE PARTICLE FILTER FOR A CLASS OF HIGH-DIMENSIONAL STATE-SPACE MODELS

    , ADVANCES IN APPLIED PROBABILITY, Vol: 49, Pages: 24-48, ISSN: 0001-8678

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