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Hilbert, Astrid
Publications (10 of 24) Show all publications
Erraoui, M., Hilbert, A. & Louriki, M. (2019). Bridges with Random Length: Gamma Case. Journal of theoretical probability
Open this publication in new window or tab >>Bridges with Random Length: Gamma Case
2019 (English)In: Journal of theoretical probability, ISSN 0894-9840, E-ISSN 1572-9230Article in journal (Refereed) Epub ahead of print
Abstract [en]

In this paper, we generalize the concept of gamma bridge in the sense that the length will be random, that is, the time to reach the given level is random. The main objective of this paper is to show that certain basic properties of gamma bridges with deterministic length stay true also for gamma bridges with random length. We show that the gamma bridge with random length is a pure jump process and that its jumping times are countable and dense in the random interval bounded by 0 and the random length. Moreover, we prove that this process is a Markov process with respect to its completed natural filtration as well as with respect to the usual augmentation of this filtration, which leads us to conclude that its completed natural filtration is right continuous. Finally, we give its canonical decomposition with respect to the usual augmentation of its natural filtration.

Place, publisher, year, edition, pages
Springer, 2019
Keywords
Levy processes, Gamma processes, Gamma bridges, Markov process, Bayes theorem
National Category
Mathematics
Research subject
Natural Science, Mathematics
Identifiers
urn:nbn:se:lnu:diva-90079 (URN)10.1007/s10959-019-00955-4 (DOI)000492647800001 ()
Available from: 2019-11-19 Created: 2019-11-19 Last updated: 2020-03-23
Berrhazi, B.-e., El Fatini, M., Hilbert, A., Mrhardy, N. & Pettersson, R. (2019). Reflected backward doubly stochastic differential equations with discontinuous barrier. Stochastics: An International Journal of Probablitiy and Stochastic Processes
Open this publication in new window or tab >>Reflected backward doubly stochastic differential equations with discontinuous barrier
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2019 (English)In: Stochastics: An International Journal of Probablitiy and Stochastic Processes, ISSN 1744-2508, E-ISSN 1744-2516Article in journal (Refereed) Epub ahead of print
Abstract [en]

In this paper, we investigate reflected backward doubly stochastic differential equations (RBDSDEs) with a lower not necessarily right-continuous obstacle. First, we establish the existence and uniqueness of a solution to RBDSDEs with Lipschitz drivers. In the second part, we present a comparison theorem and we prove the existence of a minimal solution to the RBDSDE with the continuous driver.

Place, publisher, year, edition, pages
Taylor & Francis Group, 2019
Keywords
Backward doubly stochastic differential equations, reflected backward doubly stochastic differential equations, Mertens decomposition, strong optional supermartingale
National Category
Mathematics
Research subject
Natural Science, Mathematics
Identifiers
urn:nbn:se:lnu:diva-90317 (URN)10.1080/17442508.2019.1691207 (DOI)000496847400001 ()
Available from: 2019-11-29 Created: 2019-11-29 Last updated: 2020-03-23
Agram, N., Hilbert, A. & Øksendal, B. (2019). Singular control of SPDEs with space-mean dynamics. Mathematical Control & Related Fields
Open this publication in new window or tab >>Singular control of SPDEs with space-mean dynamics
2019 (English)In: Mathematical Control & Related Fields, ISSN 2156-8472Article in journal (Refereed) Epub ahead of print
Abstract [en]

We consider the problem of optimal singular control of a stochastic partial differential equation (SPDE) with space-mean dependence. Such systems are proposed as models for population growth in a random environment. We obtain sufficient and necessary maximum principles for such control problems. The corresponding adjoint equation is a reflected backward stochastic partial differential equation (BSPDE) with space-mean dependence. We prove existence and uniqueness results for such equations. As an application we study optimal harvesting from a population modelled as an SPDE with space-mean dependence.

National Category
Computational Mathematics
Research subject
Mathematics, Mathematics
Identifiers
urn:nbn:se:lnu:diva-82304 (URN)10.3934/mcrf.2020004 (DOI)
Available from: 2019-04-26 Created: 2019-04-26 Last updated: 2020-03-09
Assing, S. & Hilbert, A. (2018). On the collapse of trial solutions for a damped-driven nonlinear Schrödinger equation. Nonlinearity, 31(11), 4955-4978
Open this publication in new window or tab >>On the collapse of trial solutions for a damped-driven nonlinear Schrödinger equation
2018 (English)In: Nonlinearity, ISSN 0951-7715, E-ISSN 1361-6544, Vol. 31, no 11, p. 4955-4978Article in journal (Refereed) Published
Abstract [en]

We consider the focusing 2D nonlinear Schrodinger equation, perturbed by a damping term, and driven by multiplicative noise. We show that a physically motivated trial solution does not collapse for any admissible initial condition although the exponent of the nonlinearity is critical. Our method is based on the construction of a global solution to a singular stochastic Hamiltonian system used to connect trial solution and Schrodinger equation.

Place, publisher, year, edition, pages
Institute of Physics Publishing (IOPP), 2018
Keywords
Schrödinger equation, singular Hamiltonian system, stochastic differential equations
National Category
Mathematics
Research subject
Natural Science, Mathematics
Identifiers
urn:nbn:se:lnu:diva-78607 (URN)10.1088/1361-6544/aad64a (DOI)000446784000003 ()2-s2.0-85055341017 (Scopus ID)
Available from: 2018-11-01 Created: 2018-11-01 Last updated: 2019-08-29Bibliographically approved
Basna, R., Hilbert, A. & Kolokoltsov, V. (2017). An Approximate Nash Equilibrium for Pure Jump Markov Games of Mean-field-type on Continuous State Space. Stochastics: An International Journal of Probablitiy and Stochastic Processes, 89(6-7), 967-993
Open this publication in new window or tab >>An Approximate Nash Equilibrium for Pure Jump Markov Games of Mean-field-type on Continuous State Space
2017 (English)In: Stochastics: An International Journal of Probablitiy and Stochastic Processes, ISSN 1744-2508, E-ISSN 1744-2516, Vol. 89, no 6-7, p. 967-993Article in journal (Refereed) Published
Abstract [en]

We investigate mean-field games from the point of view of a large number of indistinguishable players, which eventually converges to infinity. The players are weakly coupled via their empirical measure. The dynamics of the states of the individual players is governed by a non-autonomous pure jump type semi group in a Euclidean space, which is not necessarily smoothing. Investigations are conducted in the framework of non-linear Markovian semi groups. We show that the individual optimal strategy results from a consistent coupling of an optimal control problem with a forward non-autonomous dynamics. In the limit as the number N of players goes to infinity this leads to a jump-type analog of the well-known non-linear McKean–Vlasov dynamics. The case where one player has an individual preference different from the ones of the remaining players is also covered. The two results combined reveal an epsilon-NashEquilibrium for the N-player games.

Place, publisher, year, edition, pages
London: Taylor & Francis, 2017
Keywords
Mean-field games-Non-linear Markov Processes, Optimal Control
National Category
Probability Theory and Statistics
Research subject
Mathematics, Mathematics
Identifiers
urn:nbn:se:lnu:diva-55851 (URN)10.1080/17442508.2017.1297812 (DOI)000415805700010 ()2-s2.0-85015257280 (Scopus ID)
Projects
Sweden
Available from: 2016-08-31 Created: 2016-08-31 Last updated: 2019-08-29Bibliographically approved
Doerr, B., Fischer, P., Hilbert, A. & Witt, C. (2017). Detecting Structural Breaks in Time Series via Genetic Algorithms. Soft Computing - A Fusion of Foundations, Methodologies and Applications, 21(16), 4707-4720
Open this publication in new window or tab >>Detecting Structural Breaks in Time Series via Genetic Algorithms
2017 (English)In: Soft Computing - A Fusion of Foundations, Methodologies and Applications, ISSN 1432-7643, E-ISSN 1433-7479, Vol. 21, no 16, p. 4707-4720Article in journal (Refereed) Published
Abstract [en]

Detecting structural breaks is an essential task for the statistical analysis of time series, for example, for fitting parametric models to it. In short, structural breaks are points in time at which the behaviour of the time series substantially changes. Typically, no solid background knowledge of the time series under consideration is available. Therefore, a black-box optimization approach is our method of choice for detecting structural breaks. We describe a genetic algorithm framework which easily adapts to a large number of statistical settings. To evaluate the usefulness of different crossover and mutation operations for this problem, we conduct extensive experiments to determine good choices for the parameters and operators of the genetic algorithm. One surprising observation is that use of uniform and one-point crossover together gave significantly better results than using either crossover operator alone. Moreover, we present a specific fitness function which exploits the sparse structure of the break points and which can be evaluated particularly efficiently. The experiments on artificial and real-world time series show that the resulting algorithm detects break points with high precision and is computationally very efficient. A reference implementation with the data used in this paper is available as an applet at the following address: http://​www.​imm.​dtu.​dk/​~pafi/​TSX/​. It has also been implemented as package SBRect for the statistics language R.

Place, publisher, year, edition, pages
Springer, 2017
National Category
Probability Theory and Statistics
Research subject
Natural Science, Mathematics
Identifiers
urn:nbn:se:lnu:diva-51861 (URN)10.1007/s00500-016-2079-0 (DOI)000407133600017 ()2-s2.0-84959103818 (Scopus ID)
Available from: 2016-04-01 Created: 2016-04-01 Last updated: 2019-08-29Bibliographically approved
Djechiche, B., Hilbert, A. & Nassar, H. (2016). On the Functional Hodrick-Prescott Filter with Non-compact Operators. Random Operators and Stochastic Equations, 24(1), 33-42
Open this publication in new window or tab >>On the Functional Hodrick-Prescott Filter with Non-compact Operators
2016 (English)In: Random Operators and Stochastic Equations, ISSN 0926-6364, E-ISSN 1569-397X, Vol. 24, no 1, p. 33-42Article in journal (Refereed) Published
Abstract [en]

We study a version of the functional Hodrick-Prescott filter where the associated operator is not necessarily compact, but merely closed and densely defined with closed range. We show that the associate doptimal smoothing operator preserves the structure obtained in the compact case, when the underlying distribution of the data is Gaussian.

Place, publisher, year, edition, pages
Walter de Gruyter, 2016
Keywords
Inverse problems, adaptive estimation, Hodrick–Prescott filter, smoothing, trend extraction, Gaussian measures on a Hilbert space
National Category
Other Mathematics
Research subject
Natural Science, Mathematics
Identifiers
urn:nbn:se:lnu:diva-40160 (URN)10.1515/rose-2016-0003 (DOI)000410277200003 ()2-s2.0-84960539968 (Scopus ID)
Available from: 2015-02-15 Created: 2015-02-15 Last updated: 2018-04-17Bibliographically approved
Basna, R., Hilbert, A. & Kolokoltsov, V. (2014). An Epsilon Nash Equilibrium For Non-Linear Markov Games of Mean-Field-Type on Finite Spaces. Communications on Stochastic Analysis, 8(4), 449-468
Open this publication in new window or tab >>An Epsilon Nash Equilibrium For Non-Linear Markov Games of Mean-Field-Type on Finite Spaces
2014 (English)In: Communications on Stochastic Analysis, ISSN 0973-9599, Vol. 8, no 4, p. 449-468Article in journal (Refereed) Published
Abstract [en]

We investigate mean field games from the point of view of a large number of indistinguishable players which eventually converges to in- finity. The players are weakly coupled via their empirical measure. The dynamics of the individual players is governed by pure jump type propagators over a finite space. Investigations are conducted in the framework of non-linear Markov processes. We show that the individual optimal strategy results from a consistent coupling of an optimal control problem with a forward non-autonomous dynamics. In the limit as the number N of players goes to infinity this leads to a jump-type analog of the well-known non-linear McKean-Vlasov dynamics. The case where one player has an individual preference different from the ones of the remaining players is also covered. The two results combined reveal a 1 N -Nash Equilibrium for the approximating system of N players.

National Category
Mathematics Probability Theory and Statistics
Identifiers
urn:nbn:se:lnu:diva-40159 (URN)
Available from: 2015-02-15 Created: 2015-02-15 Last updated: 2017-12-04Bibliographically approved
Fischer, P. & Hilbert, A. (2014). Fast Detection of Structural Breaks. In: Manfred Gilli, Gil Gonzalez-Rodriguez & Alicia Nieto-Reyes (Ed.), Proceedings of COMPSTAT 2014, 21th International Conference on Computational Statistics, Geneva, August 19-22, 2014: . Paper presented at The 21th International Conference on Computational Statistics (COMPSTAT), Geneva, August 19-22, 2014 (pp. 9-16). The International Statistical Institute
Open this publication in new window or tab >>Fast Detection of Structural Breaks
2014 (English)In: Proceedings of COMPSTAT 2014, 21th International Conference on Computational Statistics, Geneva, August 19-22, 2014 / [ed] Manfred Gilli, Gil Gonzalez-Rodriguez & Alicia Nieto-Reyes, The International Statistical Institute, 2014, p. 9-16Conference paper, Published paper (Refereed)
Abstract [en]

A fundamental task in the analysis of time series is to detect structural breaks. A break indicates a significant change in the behaviour of the series. One method to formalise the notion of a break point, is to fit statistical models piecewise to the series. To find break points, the endpoints of the pieces are varied as is their number. A structural break is indicated by a significant change of the model parameters in adjacent pieces. Both, varying the pieces and repeatedly fitting models to them, are usually computationally very expensive. By combining genetic algorithms with a preprocessing of the time series we design a very fast algorithm for structural break detection. It reduces the time for model-fitting from linear to logarithmic in the length of the series. We show how this method can be used to find structural breaks for time series which are piecewise generated by AR(p)-models. Moreover, we introduce a nonparametric model for which the speed-up can also be achieved. Additionally we briefly present simulation results which demonstrate the manifold applications of these methods. A reference implementation is available at http://www2.imm.dtu.dk/~pafi/StructBreak/index.html

Place, publisher, year, edition, pages
The International Statistical Institute, 2014
National Category
Probability Theory and Statistics
Research subject
Natural Science, Mathematics
Identifiers
urn:nbn:se:lnu:diva-34517 (URN)978-2-8399-1347-8 (ISBN)
Conference
The 21th International Conference on Computational Statistics (COMPSTAT), Geneva, August 19-22, 2014
Available from: 2014-06-01 Created: 2014-06-01 Last updated: 2019-06-19Bibliographically approved
Doerr, B., Fischer, P., Hilbert, A. & Witt, C. (2013). Evolutionary Algorithms for the Detection of Structural Breaks in Time Series: extended abstract. In: Proceedings of the 15th annual conference on Genetic and evolutionary computation: . Paper presented at Genetic and evolutionary computation conference (GECCO)Amsterdam, Netherlands — July 06 - 10, 2013 (pp. 119-120). ACM Press
Open this publication in new window or tab >>Evolutionary Algorithms for the Detection of Structural Breaks in Time Series: extended abstract
2013 (English)In: Proceedings of the 15th annual conference on Genetic and evolutionary computation, ACM Press, 2013, p. 119-120Conference paper, Published paper (Refereed)
Abstract [en]

Detecting structural breaks is an essential task for the statistical analysis of time series, for example,  for fitting parametric models to it. In short, structural breaks  are points in time at which the behavior of the time series changes. Typically, no solid background knowledge of the time series under consideration is available. Therefore, a black-box optimization approach is our method of choice for detecting structural breaks. We describe a \ea framework which easily adapts to a large number of statistical settings. The experiments on artificial and real-world time series show that the algorithm detects break points with high precision and is computationally very efficient.

A reference implementation is availble at the following address:

http://www2.imm.dtu.dk/\~\/pafi/SBX/launch.html

Place, publisher, year, edition, pages
ACM Press, 2013
Keywords
Evolutionary Algorithms, Statistics, Break points
National Category
Computational Mathematics
Identifiers
urn:nbn:se:lnu:diva-28152 (URN)10.1145/2464576.2464635 (DOI)2-s2.0-84882359094 (Scopus ID)978-1-4503-1964-5 (ISBN)
Conference
Genetic and evolutionary computation conference (GECCO)Amsterdam, Netherlands — July 06 - 10, 2013
Available from: 2013-08-14 Created: 2013-08-14 Last updated: 2014-04-23Bibliographically approved
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