The Experts below are selected from a list of 16368 Experts worldwide ranked by ideXlab platform
Ismaël Castillo - One of the best experts on this subject based on the ideXlab platform.
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On Bayesian Supremum norm contraction rates
The Annals of Statistics, 2014Co-Authors: Ismaël CastilloAbstract:Building on ideas from Castillo and Nickl (6), a method is pro- vided to study nonparametric Bayesian posterior convergence rates when 'strong' measures of distances, such as the sup-norm, are considered. In particular, we show that likelihood methods can achieve optimal minimax sup-norm rates in density estimation on the unit interval. The introduced methodology is used to prove that commonly used families of prior distri- butions on densities, namely log-density priors and dyadic random density histograms, can indeed achieve optimal sup-norm rates of convergence. New results are also derived in the Gaussian white noise model as a further il- lustration of the presented techniques. AMS 2000 subject classications: Primary 62G20; secondary 62G05, 62G07.
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on bayesian Supremum norm contraction rates
arXiv: Statistics Theory, 2013Co-Authors: Ismaël CastilloAbstract:Building on ideas from Castillo and Nickl [Ann. Statist. 41 (2013) 1999-2028], a method is provided to study nonparametric Bayesian posterior convergence rates when "strong" measures of distances, such as the sup-norm, are considered. In particular, we show that likelihood methods can achieve optimal minimax sup-norm rates in density estimation on the unit interval. The introduced methodology is used to prove that commonly used families of prior distributions on densities, namely log-density priors and dyadic random density histograms, can indeed achieve optimal sup-norm rates of convergence. New results are also derived in the Gaussian white noise model as a further illustration of the presented techniques.
Kees Van Schaik - One of the best experts on this subject based on the ideXlab platform.
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optimally stopping at a given distance from the ultimate Supremum of a spectrally negative levy process
Advances in Applied Probability, 2021Co-Authors: Monica Carvajal B Pinto, Kees Van SchaikAbstract:We consider the optimal prediction problem of stopping a spectrally negative Levy process as close as possible to a given distance from its ultimate Supremum, under a squared-error penalty function. Under some mild conditions, the solution is fully and explicitly characterised in terms of scale functions. We find that the solution has an interesting non-trivial structure: if b is larger than a certain threshold then it is optimal to stop as soon as the difference between the running Supremum and the position of the process exceeds a certain level (less than b), while if b is smaller than this threshold then it is optimal to stop immediately (independent of the running Supremum and position of the process). We also present some examples.
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predicting the time at which a levy process attains its ultimate Supremum
Acta Applicandae Mathematicae, 2014Co-Authors: Erik J Baurdoux, Kees Van SchaikAbstract:We consider the problem of finding a stopping time that minimises the L 1-distance to ?, the time at which a Levy process attains its ultimate Supremum. This problem was studied in Du Toit and Peskir (Proc. Math. Control Theory Finance, pp. 95---112, 2008) for a Brownian motion with drift and a finite time horizon. We consider a general Levy process and an infinite time horizon (only compound Poisson processes are excluded. Furthermore due to the infinite horizon the problem is interesting only when the Levy process drifts to ??). Existing results allow us to rewrite the problem as a classic optimal stopping problem, i.e. with an adapted payoff process. We show the following. If ? has infinite mean there exists no stopping time with a finite L 1-distance to ?, whereas if ? has finite mean it is either optimal to stop immediately or to stop when the process reflected in its Supremum exceeds a positive level, depending on whether the median of the law of the ultimate Supremum equals zero or is positive. Furthermore, pasting properties are derived. Finally, the result is made more explicit in terms of scale functions in the case when the Levy process has no positive jumps.
Mateusz Kwaśnicki - One of the best experts on this subject based on the ideXlab platform.
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fluctuation theory for levy processes with completely monotone jumps
Electronic Journal of Probability, 2019Co-Authors: Mateusz KwaśnickiAbstract:We study the Wiener–Hopf factorization for Levy processes $X_{t}$ with completely monotone jumps. Extending previous results of L.C.G. Rogers, we prove that the space-time Wiener–Hopf factors are complete Bernstein functions of both the spatial and the temporal variable. As a corollary, we prove complete monotonicity of: (a) the tail of the distribution function of the Supremum of $X_{t}$ up to an independent exponential time; (b) the Laplace transform of the Supremum of $X_{t}$ up to a fixed time $T$, as a function of $T$. The proof involves a detailed analysis of the holomorphic extension of the characteristic exponent $f(\xi )$ of $X_{t}$, including a peculiar structure of the curve along which $f(\xi )$ takes real values.
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fluctuation theory for l evy processes with completely monotone jumps
arXiv: Probability, 2018Co-Authors: Mateusz KwaśnickiAbstract:We study the Wiener-Hopf factorization for Levy processes $X_t$ with completely monotone jumps. Extending previous results of L.C.G. Rogers, we prove that the space-time Wiener-Hopf factors are complete Bernstein functions of both the spatial and the temporal variable. As a corollary, we prove complete monotonicity of: (a) the tail of the distribution function of the Supremum of $X_t$ up to an independent exponential time; (b) the Laplace transform of the Supremum of $X_t$ up to a fixed time $T$, as a function of $T$. The proof involves a detailed analysis of the holomorphic extension of the characteristic exponent $f(\xi)$ of $X_t$, including a peculiar structure of the curve along which $f(\xi)$ takes real values.
Martijn Pistorius - One of the best experts on this subject based on the ideXlab platform.
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on the drawdown of completely asymmetric levy processes
Stochastic Processes and their Applications, 2012Co-Authors: Aleksandar Mijatovic, Martijn PistoriusAbstract:Abstract The drawdown process Y of a completely asymmetric Levy process X is equal to X reflected at its running Supremum X ¯ : Y = X ¯ − X . In this paper we explicitly express in terms of the scale function and the Levy measure of X the law of the sextuple of the first-passage time of Y over the level a > 0 , the time G ¯ τ a of the last Supremum of X prior to τ a , the infimum X ¯ τ a and Supremum X ¯ τ a of X at τ a and the undershoot a − Y τ a − and overshoot Y τ a − a of Y at τ a . As application we obtain explicit expressions for the laws of a number of functionals of drawdowns and rallies in a completely asymmetric exponential Levy model.
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on the drawdown of completely asymmetric levy processes
2012Co-Authors: Aleksandar Mijatovic, Martijn PistoriusAbstract:The {\em drawdown} process $Y$ of a completely asymmetric L\'{e}vy process $X$ is equal to $X$ reflected at its running Supremum $\bar{X}$: $Y = \bar{X} - X$. In this paper we explicitly express in terms of the scale function and the L\'{e}vy measure of $X$ the law of the sextuple of the first-passage time of $Y$ over the level $a>0$, the time $\bar{G}_{\tau_a}$ of the last Supremum of $X$ prior to $\tau_a$, the infimum $\unl X_{\tau_a}$ and Supremum $\ovl X_{\tau_a}$ of $X$ at $\tau_a$ and the undershoot $a - Y_{\tau_a-}$ and overshoot $Y_{\tau_a}-a$ of $Y$ at $\tau_a$. As application we obtain explicit expressions for the laws of a number of functionals of drawdowns and rallies in a completely asymmetric exponential L\'{e}vy model.
Alexey Kuznetsov - One of the best experts on this subject based on the ideXlab platform.
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on the density of the Supremum of a stable process
Stochastic Processes and their Applications, 2013Co-Authors: Alexey KuznetsovAbstract:Abstract We study the density of the Supremum of a strictly stable Levy process. Our first goal is to investigate convergence properties of the series representation for this density, which was established recently by Hubalek and Kuznetsov (2011) [24] . Our second goal is to investigate in more detail the important case when α is rational: we derive an explicit formula for the Mellin transform of the Supremum. We perform several numerical experiments and discuss their implications. Finally, we state some interesting connections that this problem has to other areas of Mathematics and Mathematical Physics and we also suggest several open problems.
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On the density of the Supremum of a stable process
arXiv: Probability, 2011Co-Authors: Alexey KuznetsovAbstract:We study the density of the Supremum of a strictly stable L\'evy process. As was proved recently in F. Hubalek and A. Kuznetsov "A convergent series representation for the density of the Supremum of a stable process" (Elect. Comm. in Probab., 16, 84-95, 2011), for almost all irrational values of the stability parameter $\alpha$ this density can be represented by an absolutely convergent series. We show that this result is not valid for all irrational values of $\alpha$: we construct a dense uncountable set of irrational numbers $\alpha$ for which the series does not converge absolutely. Our second goal is to investigate in more detail the important case when $\alpha$ is rational. We derive an explicit formula for the Mellin transform of the Supremum, which is given in terms of Gamma function and dilogarithm. In order to illustrate the usefulness of these results we perform several numerical experiments and discuss their implications. Finally, we state some interesting connections that this problem has to other areas of Mathematics and Mathematical Physics, such as q-series, Diophantine approximations and quantum dilogarithms, and we also suggest several open problems.
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a convergent series representation for the density of the Supremum of a stable process
Electronic Communications in Probability, 2011Co-Authors: Friedrich Hubalek, Alexey KuznetsovAbstract:We study the density of the Supremum of a strictly stable Levy process. We prove that for almost all values of the index $\alpha$ - except for a dense set of Lebesgue measure zero - the asymptotic series which were obtained in Kuznetsov (2010) "On extrema of stable processes" are in fact absolutely convergent series representations for the density of the Supremum.
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wiener hopf factorization and distribution of extrema for a family of levy processes
Annals of Applied Probability, 2010Co-Authors: Alexey KuznetsovAbstract:In this paper we introduce a ten-parameter family of Levy processes for which we obtain Wiener–Hopf factors and distribution of the Supremum process in semi-explicit form. This family allows an arbitrary behavior of small jumps and includes processes similar to the generalized tempered stable, KoBoL and CGMY processes. Analytically it is characterized by the property that the characteristic exponent is a meromorphic function, expressed in terms of beta and digamma functions. We prove that the Wiener–Hopf factors can be expressed as infinite products over roots of a certain transcendental equation, and the density of the Supremum process can be computed as an exponentially converging infinite series. In several special cases when the roots can be found analytically, we are able to identify the Wiener–Hopf factors and distribution of the Supremum in closed form. In the general case we prove that all the roots are real and simple, and we provide localization results and asymptotic formulas which allow an efficient numerical evaluation. We also derive a convergence acceleration algorithm for infinite products and a simple and efficient procedure to compute the Wiener–Hopf factors for complex values of parameters. As a numerical example we discuss computation of the density of the Supremum process.
