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  • 51.
    Braojos Peláes, Marta Rita
    Uppsala University, Disciplinary Domain of Science and Technology, Mathematics and Computer Science, Department of Mathematics, Analysis and Probability Theory.
    Optimal exercise of an American Option under drift uncertainty2016Independent thesis Advanced level (degree of Master (Two Years)), 20 credits / 30 HE creditsStudent thesis
  • 52.
    Brightwell, Graham
    et al.
    London Sch Econ, Dept Math, London WC2A 2AE, England..
    Janson, Svante
    Uppsala University, Disciplinary Domain of Science and Technology, Mathematics and Computer Science, Department of Mathematics, Analysis and Probability Theory.
    Luczak, Malwina
    Queen Mary Univ London, Sch Math, London, England..
    The Greedy Independent Set in a Random Graph with Given Degrees2017In: Random structures & algorithms (Print), ISSN 1042-9832, E-ISSN 1098-2418, Vol. 51, no 4, p. 565-586Article in journal (Refereed)
    Abstract [en]

    We analyse the size of an independent set in a random graph on n vertices with specified vertex degrees, constructed via a simple greedy algorithm: order the vertices arbitrarily, and, for each vertex in turn, place it in the independent set unless it is adjacent to some vertex already chosen. We find the limit of the expected proportion of vertices in the greedy independent set as n (the jamming constant), expressed as an integral whose upper limit is defined implicitly, valid whenever the second moment of a random vertex degree is uniformly bounded. We further show that the random proportion of vertices in the independent set converges in probability to the jamming constant as n. The results hold under weaker assumptions in a random multigraph with given degrees constructed via the configuration model.

  • 53.
    Brill, Markus
    et al.
    Univ Oxford, Oxford, England.
    Freeman, Rupert
    Duke Univ, Durham, NC 27706 USA.
    Janson, Svante
    Uppsala University, Disciplinary Domain of Science and Technology, Mathematics and Computer Science, Department of Mathematics, Analysis and Probability Theory.
    Lackner, Martin
    Univ Oxford, Oxford, England.
    Phragmen's Voting Methods and Justified Representation2017In: Thirty-First AAAI Conference On Artificial Intelligence, Assoc Advancement Artificial Intelligence , 2017, p. 406-413Conference paper (Refereed)
    Abstract [en]

    In the late 19th century, Lars Edvard Phragmen proposed a load-balancing approach for selecting committees based on approval ballots. We consider three committee voting rules resulting from this approach: two optimization variants-one minimizing the maximal load and one minimizing the variance of loads-and a sequential variant. We study Phragmen's methods from an axiomatic point of view, focussing on justified representation and related properties that have recently been introduced by Aziz et al. (2015a) and Sanchez-Fernandez et al. (2017). We show that the sequential variant satisfies proportional justified representation, making it the first known polynomial-time computable method with this property. Moreover, we show that the optimization variants satisfy perfect representation. We also analyze the computational complexity of Phragmen's methods and provide mixed- integer programming based algorithms for computing them.

  • 54.
    Broman, Erik I.
    Uppsala University, Disciplinary Domain of Science and Technology, Mathematics and Computer Science, Department of Mathematics, Analysis and Probability Theory.
    Stochastic Ordering of Infinite Geometric Galton-Watson Trees2016In: Journal of theoretical probability, ISSN 0894-9840, E-ISSN 1572-9230, Vol. 29, no 3, p. 1069-1082Article in journal (Refereed)
    Abstract [en]

    We consider Galton-Watson trees with Geom(p) offspring distribution. We let T-infinity (p) denote such a tree conditioned on being infinite. We prove that for any 1/2 <= p(1) <= p2 <= 1, there exists a coupling between T-infinity (p(1)) and T-infinity (p(2)) such that P(T-infinity(p(1)) subset of T-infinity(p(2))) = 1.

  • 55.
    Broman, Erik I.
    et al.
    Uppsala University, Disciplinary Domain of Science and Technology, Mathematics and Computer Science, Department of Mathematics.
    Elias, Olof
    Mussini, Filipe
    Uppsala University, Disciplinary Domain of Science and Technology, Mathematics and Computer Science, Department of Mathematics, Analysis and Probability Theory.
    Tykesson, Johan
    The fractal cylinder process: existence and connectivity phase transitionIn: The Annals of Applied Probability, ISSN 1050-5164, E-ISSN 2168-8737Article in journal (Refereed)
    Abstract [en]

    We consider a semi-scale invariant version of the Poisson cylinder model which in a natural way induces a random fractal set. We show that this random fractal exhibits an existence phase transition for any dimension d ≥ 2, and a connectivityphase transition whenever d ≥ 4. We determine the exact value of the critical point of the existence phase transition, and we show that the fractal set is almost surely empty at this critical point.

    A key ingredient when analysing the connectivity phase transition is to consider a restriction of the full process onto a subspace. We show that this restriction results in a fractal ellipsoid model which we describe in detail, as it is key to obtaining our main results.

    In addition we also determine the almost sure Hausdorff dimension of the fractal set.

  • 56.
    Broman, Erik I.
    et al.
    Chalmers Univ Technol, MV Huset, Chalmers Tvargata 3, Gothenburg, Sweden; Gothenburg Univ, MV Huset, Chalmers Tvargata 3, Gothenburg, Sweden.
    Mussini, Filipe
    Uppsala University, Disciplinary Domain of Science and Technology, Mathematics and Computer Science, Department of Mathematics, Analysis and Probability Theory.
    Random cover times using the Poisson cylinder process.2019In: Latin American Journal of Probability and Mathematical Statistics, ISSN 1980-0436, E-ISSN 1980-0436, Vol. 16, no 2, p. 1165-1199Article in journal (Refereed)
    Abstract [en]

    In this paper we deal with the classical problem of random cover times. We investigate the distribution of the time it takes for a Poisson process of cylinders to cover a subset A of the d-dimensional Euclidean space. This Poisson process of cylinders is invariant under rotations, reflections and translations, and in addition we add a time component so that cylinders are “raining from the sky” at unit rate. Our main results concerns the asymptotic of this cover time as the set A grows. If the set A is discrete and well separated, we show convergence of the cover time to a Gumbel distribution. If instead A has positive box dimension (and satisfies a weak additional assumption), we find the correct rate of convergence.

  • 57.
    Broman, Erik I.
    et al.
    Uppsala University, Disciplinary Domain of Science and Technology, Mathematics and Computer Science, Department of Mathematics, Analysis and Probability Theory.
    Tykesson, Johan
    Chalmers, Dept Math, Gothenburg, Sweden.;Gothenburg Univ, S-41124 Gothenburg, Sweden..
    Connectedness of Poisson cylinders in Euclidean space2016In: Annales de l'I.H.P. Probabilites et statistiques, ISSN 0246-0203, E-ISSN 1778-7017, Vol. 52, no 1, p. 102-126Article in journal (Refereed)
    Abstract [en]

    We consider the Poisson cylinder model in R-d, d >= 3. We show that given any two cylinders c(1) and c(2) in the process, there is a sequence of at most d - 2 other cylinders creating a connection between c(1) and c(2). In particular, this shows that the union of the cylinders is a connected set, answering a question appearing in (Probab. Theory Related Fields 154 (2012) 165-191). We also show that there are cylinders in the process that are not connected by a sequence of at most d - 3 other cylinders. Thus, the diameter of the cluster of cylinders equals d - 2.

  • 58.
    Broman, Erik
    et al.
    Uppsala University, Disciplinary Domain of Science and Technology, Mathematics and Computer Science, Department of Mathematics, Analysis and Probability Theory.
    Tykesson, Johan
    Poisson cylinders in hyperbolic space2015In: Electronic Journal of Probability, ISSN 1083-6489, E-ISSN 1083-6489, Vol. 20, p. 1-25Article in journal (Refereed)
    Abstract [en]

    We consider the Poisson cylinder model in d-dimensional hyperbolic space. We show that in contrast to the Euclidean case, there is a phase transition in the connectivity of the collection of cylinders as the intensity parameter varies. We also show that for any non-trivial intensity, the diameter of the collection of cylinders is infinite.

  • 59.
    Cai, Xing Shi
    et al.
    Uppsala University, Disciplinary Domain of Science and Technology, Mathematics and Computer Science, Department of Mathematics, Analysis and Probability Theory.
    Holmgren, Cecilia
    Uppsala University, Disciplinary Domain of Science and Technology, Mathematics and Computer Science, Department of Mathematics, Analysis and Probability Theory.
    Cutting resilient networks - complete binary trees2019In: The Electronic Journal of Combinatorics, ISSN 1097-1440, E-ISSN 1077-8926, Vol. 26, no 4, article id P4.43Article in journal (Refereed)
    Abstract [en]

    In our previous work [2, 3], we introduced the random k-cut number for rooted graphs. In this paper, we show that the distribution of the k-cut number in complete binary trees of size n, after rescaling, is asymptotically a periodic function of lg n - lg lg n. Thus there are different limit distributions for different subsequences, where these limits are similar to weakly 1-stable distributions. This generalizes the result for the case k = 1, i.e., the traditional cutting model, by Janson [12].

  • 60.
    Cai, Xing Shi
    et al.
    Uppsala University, Disciplinary Domain of Science and Technology, Mathematics and Computer Science, Department of Mathematics, Analysis and Probability Theory.
    Holmgren, Cecilia
    Uppsala University, Disciplinary Domain of Science and Technology, Mathematics and Computer Science, Department of Mathematics, Analysis and Probability Theory.
    Devroye, Luc
    McGill Univ, Montreal, PQ, Canada.
    Skerman, Fiona
    Uppsala University, Disciplinary Domain of Science and Technology, Mathematics and Computer Science, Department of Mathematics.
    kappa-cut on paths and some trees2019In: Electronic Journal of Probability, ISSN 1083-6489, E-ISSN 1083-6489, Vol. 24, no 53Article in journal (Refereed)
    Abstract [en]

    We define the (random) kappa-cut number of a rooted graph to model the difficulty of the destruction of a resilient network. The process is as the cut model of Meir and Moon [21] except now a node must be cut kappa times before it is destroyed. The first order terms of the expectation and variance of chi(n), the kappa-cut number of a path of length n, are proved. We also show that chi(n), after rescaling, converges in distribution to a limit B-kappa, which has a complicated representation. The paper then briefly discusses the kappa-cut number of some trees and general graphs. We conclude by some analytic results which may be of interest.

  • 61.
    Cai, Xing Shi
    et al.
    Uppsala University, Disciplinary Domain of Science and Technology, Mathematics and Computer Science, Department of Mathematics.
    Janson, Svante
    Uppsala University, Disciplinary Domain of Science and Technology, Mathematics and Computer Science, Department of Mathematics, Analysis and Probability Theory.
    Non-fringe subtrees in conditioned Galton-Watson trees2018In: The Electronic Journal of Combinatorics, ISSN 1097-1440, E-ISSN 1077-8926, Vol. 25, no 3, article id P3.40Article in journal (Refereed)
    Abstract [en]

    We study S(T-n), the number of subtrees in a conditioned Galton-Watson tree of size n. With two very different methods, we show that log(S(T-n)) has a Central Limit Law and that the moments of S(T-n) are of exponential scale.

  • 62.
    Cai, Xing Shi
    et al.
    Uppsala University, Disciplinary Domain of Science and Technology, Mathematics and Computer Science, Department of Mathematics, Analysis and Probability Theory.
    Lopez, Jose L.
    Univ Publ Navarra, Dept Estadist Matemat & Informat, Pamplona, Spain;Univ Publ Navarra, INAMAT, Pamplona, Spain.
    A note on the asymptotic expansion of the Lerch's transcendent2019In: Integral transforms and special functions, ISSN 1065-2469, E-ISSN 1476-8291, Vol. 30, no 10, p. 844-855Article in journal (Refereed)
    Abstract [en]

    In Ferreira and Lopez [Asymptotic expansions of the Hurwitz-Lerch zeta function. J Math Anal Appl. 2004;298(1):210-224], the authors derived an asymptotic expansion of the Lerch's transcendent Phi(z, s, a) for large vertical bar a vertical bar, valid for Ra > 0, Rs > 0 and z is an element of C \ [1, infinity). In this paper, we study the special case z >= 1 not covered in Ferreira and Lopez [Asymptotic expansions of the Hurwitz-Lerch zeta function. J Math Anal Appl. 2004; 298(1): 210-224], deriving a complete asymptotic expansion of the Lerch's transcendent Phi(z, s, a) for z > 1 and Rs > 0 as Ra goes to infinity. We also show that when a is a positive integer, this expansion is convergent for Rz >= 1. As a corollary, we get a full asymptotic expansion for the sum Sigma(m)(n=1) z(n)/n(s) for fixed z > 1 as m -> infinity. Some numerical results show the accuracy of the approximation.

  • 63.
    Casanova, Adrian Gonzalez
    et al.
    Tech Univ Berlin, Inst Math, Str 17 Juni 136, D-10623 Berlin, Germany..
    Kurt, Noemi
    Tech Univ Berlin, Inst Math, Str 17 Juni 136, D-10623 Berlin, Germany..
    Wakolbinger, Anton
    Goethe Univ Frankfurt, Math Inst, Box 111932, D-60054 Frankfurt, Germany..
    Yuan, Linglong
    Uppsala University, Disciplinary Domain of Science and Technology, Mathematics and Computer Science, Department of Mathematics, Analysis and Probability Theory.
    An individual-based model for the Lenski experiment, and the deceleration of the relative fitness2016In: Stochastic Processes and their Applications, ISSN 0304-4149, E-ISSN 1879-209X, Vol. 126, no 8, p. 2211-2252Article in journal (Refereed)
    Abstract [en]

    The Lenski experiment investigates the long-term evolution of bacterial populations. In this paper we present an individual-based probabilistic model that captures essential features of the experimental design, and whose mechanism does not include epistasis in the continuous-time (intraday) part of the model, but leads to an epistatic effect in the discrete-time (interday) part. We prove that under some assumptions excluding clonal interference, the rescaled relative fitness process converges in the large population limit to a power law function, similar to the one obtained by Wiser et al. (2013), there attributed to effects of clonal interference and epistasis.

  • 64.
    Cassar, Peter
    Uppsala University, Disciplinary Domain of Science and Technology, Mathematics and Computer Science, Department of Mathematics, Analysis and Probability Theory.
    Forecasting of a Loan Book Using Monte Carlo Methods2016Independent thesis Advanced level (degree of Master (Two Years)), 20 credits / 30 HE creditsStudent thesis
  • 65.
    Castro, Alejandro J.
    et al.
    Uppsala University, Disciplinary Domain of Science and Technology, Mathematics and Computer Science, Department of Mathematics, Analysis and Probability Theory.
    Hytonen, Tuomas P.
    Univ Helsinki, Gustaf Hallstromin Katu 2b, FIN-00014 Helsinki, Finland..
    Bounds for partial derivatives: necessity of UMD and sharp constants2016In: Mathematische Zeitschrift, ISSN 0025-5874, E-ISSN 1432-1823, Vol. 282, no 3-4, p. 635-650Article in journal (Refereed)
    Abstract [en]

    We prove the necessity of the UMD condition, with a quantitative estimate of the UMD constant, for any inequality in a family of bounds between different partial derivatives of . In particular, we show that the estimate characterizes the UMD property, and the best constant K is equal to one half of the UMD constant. This precise value of K seems to be new even for scalar-valued functions.

  • 66.
    Castro, Alejandro J.
    et al.
    Uppsala University, Disciplinary Domain of Science and Technology, Mathematics and Computer Science, Department of Mathematics, Analysis and Probability Theory.
    Nowak, Adam
    Polish Acad Sci, Inst Math, Sniadeckich 8, PL-00656 Warsaw, Poland..
    Szarek, Tomasz Z.
    Polish Acad Sci, Inst Math, Sniadeckich 8, PL-00656 Warsaw, Poland..
    Riesz-Jacobi Transforms as Principal Value Integrals2016In: Journal of Fourier Analysis and Applications, ISSN 1069-5869, E-ISSN 1531-5851, Vol. 22, no 3, p. 493-541Article in journal (Refereed)
    Abstract [en]

    We establish an integral representation for the Riesz transforms naturally associated with classical Jacobi expansions. We prove that the Riesz-Jacobi transforms of odd orders express as principal value integrals against kernels having non-integrable singularities on the diagonal. On the other hand, we show that the Riesz-Jacobi transforms of even orders are not singular operators. In fact they are given as usual integrals against integrable kernels plus or minus, depending on the order, the identity operator. Our analysis indicates that similar results, existing in the literature and corresponding to several other settings related to classical discrete and continuous orthogonal expansions, should be reinvestigated so as to be refined and in some cases also corrected.

  • 67.
    Castro, Alejandro J.
    et al.
    Uppsala University, Disciplinary Domain of Science and Technology, Mathematics and Computer Science, Department of Mathematics, Analysis and Probability Theory. Nazarbayev Univ, Dept Math, Astana 010000, Kazakhstan..
    Rodriguez-Lopez, Salvador
    Stockholm Univ, Dept Math, SE-10691 Stockholm, Sweden..
    Staubach, Wolfgang
    Uppsala University, Disciplinary Domain of Science and Technology, Mathematics and Computer Science, Department of Mathematics, Analysis and Probability Theory.
    L2-solvability of the Dirichlet, Neumann and regularity problems for parabolic equations with time-independent Hölder-continuous coefficients2018In: Transactions of the American Mathematical Society, ISSN 0002-9947, E-ISSN 1088-6850, Vol. 370, no 1, p. 265-319Article in journal (Refereed)
    Abstract [en]

    We establish the L-2-solvability of Dirichlet, Neumann and regularity problems for divergence-form heat (or diffusion) equations with timeindependent Holder-continuous diffusion coefficients on bounded Lipschitz domains in R-n. This is achieved through the demonstration of invertibility of the relevant layer potentials, which is in turn based on Fredholm theory and a systematic transference scheme which yields suitable parabolic Rellich-type estimates.

  • 68.
    Castro, Alejandro J.
    et al.
    Nazarbayev Univ, Dept Math, Astana 010000, Kazakhstan.
    Rodriguez-Lopez, Salvador
    Stockholm Univ, Dept Math, SE-10691 Stockholm, Sweden.
    Staubach, Wolfgang
    Uppsala University, Disciplinary Domain of Science and Technology, Mathematics and Computer Science, Department of Mathematics, Analysis and Probability Theory.
    Transference of local to global L-2 maximal estimates for dispersive partial differential equations2019In: Journal of Mathematical Analysis and Applications, ISSN 0022-247X, E-ISSN 1096-0813, Vol. 471, no 1-2, p. 411-422Article in journal (Refereed)
    Abstract [en]

    In this paper we give an elementary proof for transference of local to global maximal estimates for dispersive PDEs. This is done by transferring local L-2 estimates for certain oscillatory integrals with rough phase functions, to the corresponding global estimates. The elementary feature of our approach is that it entirely avoids the use of the wave packet techniques which are quite common in this context, and instead is based on scalings and classical oscillatory integral estimates.

  • 69.
    Castro, Alejandro, J.
    et al.
    Uppsala University, Disciplinary Domain of Science and Technology, Mathematics and Computer Science, Department of Mathematics, Analysis and Probability Theory.
    Rodríguez-López, Salvador
    Staubach, Wolfgang
    L2 Solvability of the Dirichlet, Neumann and the regularity problems for parabolic equations with time-independent Hölder-continuous coefficientsIn: Transactions of the American Mathematical Society, ISSN 0002-9947, E-ISSN 1088-6850Article in journal (Refereed)
    Abstract [en]

    We establish the $L^2$-solvability of Dirichlet, Neumann and regularity problems for divergence-form heat (or diffusion) equations with time-independent H\"older-continuous diffusion coefficients, on bounded Lipschitz domains in $\mathbb{R}^n$. This is achieved through the demonstration of invertibility of the relevant layer-potentials which is in turn based on Fredholm theory and a systematic transference scheme which yields suitable parabolic Rellich-type estimates.

  • 70.
    Castro, Alejandro J.
    et al.
    Nazarbayev Univ, Dept Math, Astana 010000, Kazakhstan.
    Strömqvist, Martin
    Uppsala University, Disciplinary Domain of Science and Technology, Mathematics and Computer Science, Department of Mathematics, Analysis and Probability Theory.
    Homogenization of a parabolic Dirichlet problem by a method of Dahlberg2018In: Publicacions matemàtiques, ISSN 0214-1493, E-ISSN 2014-4350, Vol. 62, no 2, p. 439-473Article in journal (Refereed)
    Abstract [en]

    Consider the linear parabolic operator in divergence form: Hu := partial derivative(t)u (X, t) - div(A (X) del u (X, t)). We employ a method of Dahlberg to show that the Dirichlet problem for H in the upper half plane is well-posed for boundary data in L-p, for any elliptic matrix of coef-ficients A which is periodic and satisfies a Dini-type condition. This result allows us to treat a homogenization problem for the equation partial derivative(t)u(epsilon) (X, t) - div(A (X/epsilon) del u(epsilon) (X, t)) in Lipschitz domains with L-p-boundary data.

  • 71.
    Castro, Alejandro
    et al.
    Uppsala University, Disciplinary Domain of Science and Technology, Mathematics and Computer Science, Department of Mathematics, Analysis and Probability Theory.
    Nyström, Kaj
    Uppsala University, Disciplinary Domain of Science and Technology, Mathematics and Computer Science, Department of Mathematics, Analysis and Probability Theory.
    Sande, Olow
    Uppsala University, Disciplinary Domain of Science and Technology, Mathematics and Computer Science, Department of Mathematics, Analysis and Probability Theory.
    Boundedness of single layer potentials associated to divergence form parabolic equations with complex coefficients2016In: Calculus of Variations and Partial Differential Equations, ISSN 0944-2669, E-ISSN 1432-0835, Vol. 55, no 5, article id 124Article in journal (Refereed)
    Abstract [en]

    We consider parabolic operators of the form $$\partial_t+\mathcal{L},\ \mathcal{L}:=-\mbox{div}\, A(X,t)\nabla,$$ in$\mathbb R_+^{n+2}:=\{(X,t)=(x,x_{n+1},t)\in \mathbb R^{n}\times \mathbb R\times \mathbb R:\ x_{n+1}>0\}$, $n\geq 1$. We assume that $A$ is an $(n+1)\times (n+1)$-dimensional matrix which is bounded, measurable, uniformly elliptic and complex, and we assume, in addition, that the entries of A are independent of the spatial coordinate $x_{n+1}$ as well as of the time coordinate $t$. We prove that the boundedness of associated single layer potentials, with data in $L^2$, can be reduced to two crucial estimates (Theorem \ref{th0}), one being a square function estimate involving the single layer potential. By establishing a local parabolic Tb-theorem for square functions we are then able to verify the two  crucial estimates in the case of real, symmetric operators (Theorem \ref{th2}). Our results are crucial when addressing the solvability of the classical Dirichlet, Neumann and Regularity problems for the operator $\partial_t+\mathcal{L}$ in $\mathbb R_+^{n+2}$, with $L^2$-data on $\mathbb R^{n+1}=\partial\mathbb R_+^{n+2}$, and by way of layer potentials.

  • 72.
    Colesanti, Andrea
    et al.
    U.Dini, Firenze, Italy.
    Nyström, Kaj
    Uppsala University, Disciplinary Domain of Science and Technology, Mathematics and Computer Science, Department of Mathematics, Analysis and Probability Theory.
    Salani, Paolo
    U.Dini, Firenze, Italy.
    Xiao, Jie
    Memorial University of Newfoundland, St. John's, Canada.
    Yang, Deane
    Polytechnic Institute of New York University, New York, USA.
    Zhang, Gaoyong
    Polytechnic Institute of New York University, New York, USA.
    The Hadamard variational formula and the Minkowski problem for p-Capacity2015In: Advances in Mathematics, ISSN 0001-8708, E-ISSN 1090-2082, Vol. 285, p. 1511-1585Article in journal (Refereed)
    Abstract [en]

    A Hadamard variational formula for p-capacity of convex bodies in R-n is established when 1 < p < n. The formula is applied to solve the Minkowski problem for p-capacity which involves a degenerate Monge Ampere type equation. zkiniqueness for the Minkowski problem for p-capacity is established when 1 < p < n and existence and regularity when 1 < p < 2. These results are (non-linear) extensions of the now classical solution of Jerison of the Minkowski problem for electrostatic capacity (p = 2).

  • 73.
    Coletti, Cristian F.
    et al.
    Univ Fed ABC, Sao Paulo, Brazil..
    Miranda, Daniel
    Univ Fed ABC, Sao Paulo, Brazil..
    Mussini, Filipe
    Uppsala University, Disciplinary Domain of Science and Technology, Mathematics and Computer Science, Department of Mathematics, Analysis and Probability Theory.
    Invariance Under Quasi-isometries of Subcritical and Supercritical Behavior in the Boolean Model of Percolation2016In: Journal of statistical physics, ISSN 0022-4715, E-ISSN 1572-9613, Vol. 162, no 3, p. 685-700Article in journal (Refereed)
    Abstract [en]

    In this work we study the Poisson Boolean model of percolation in locally compact Polish metric spaces and we prove the invariance of subcritical and supercritical phases under mm-quasi-isometries. More precisely, we prove that if a metric space M is mm-quasi-isometric to another metric space N and the Poisson Boolean model in M exhibits any of the following: (a) a subcritical phase; (b) a supercritical phase; or (c) a phase transition, then respectively so does the Poisson Boolean model of percolation in N. Then we use these results in order to understand the phase transition phenomenon in a large family of metric spaces. Indeed, we study the Poisson Boolean model of percolation in the context of Riemannian manifolds, in a large family of nilpotent Lie groups and in Cayley graphs. Also, we prove the existence of a subcritical phase in Gromov spaces with bounded growth at some scale.

  • 74.
    Cooper, Colin
    et al.
    Univ London, Kings Coll, Dept Comp Sci, London WC2R 2LS, England..
    Frieze, Alan
    Carnegie Mellon Univ, Dept Math Sci, Pittsburgh, PA 15217 USA..
    Ince, Nate
    Carnegie Mellon Univ, Dept Math Sci, Pittsburgh, PA 15217 USA..
    Janson, Svante
    Uppsala University, Disciplinary Domain of Science and Technology, Mathematics and Computer Science, Department of Mathematics, Analysis and Probability Theory.
    Spencer, Joel
    NYU, Courant Inst, New York, NY 10012 USA..
    On the Length of a Random Minimum Spanning Tree2016In: Combinatorics, probability & computing, ISSN 0963-5483, E-ISSN 1469-2163, Vol. 25, no 1, p. 89-107Article in journal (Refereed)
    Abstract [en]

    We study the expected value of the length L-n of the minimum spanning tree of the complete graph K-n when each edge e is given an independent uniform [0, 1] edge weight. We sharpen the result of Frieze [6] that lim(n ->infinity) E(L-n) = zeta(3) and show that E(L-n) = zeta(3) + c(1)/n + c(2) + o(1)/n4/3, where c(1), c(2) are explicitly defined constants.

  • 75.
    Costa, David G.
    et al.
    Univ Nevada, Dept Math Sci, Las Vegas, NV 89154 USA..
    do O, Joao Marcos
    Univ Fed Paraiba, Dept Math, BR-58051900 Joao Pessoa, Paraiba, Brazil..
    Tintarev, Kyril
    Uppsala University, Disciplinary Domain of Science and Technology, Mathematics and Computer Science, Department of Mathematics, Analysis and Probability Theory.
    Compactness Properties Of Critical Nonlinearities And Nonlinear Schrödinger Equations2013In: Proceedings of the Edinburgh Mathematical Society, ISSN 0013-0915, E-ISSN 1464-3839, Vol. 56, no 2, p. 427-441Article in journal (Refereed)
    Abstract [en]

    We prove the compactness of critical Sobolev embeddings with applications to nonlinear singular Schrodinger equations and provide a unified treatment in dimensions N > 2 and N = 2, based on a somewhat unexpectedly broad array of parallel properties between spaces D-1,D-2(R-N) and H-0(1) of the unit disc. These properties include Leray inequality for N = 2 as a counterpart of Hardy inequality for N > 2, pointwise estimates by ground states r((2-N)/2) and root log(1/r) of the respective Hardy-type inequalities, as well as compactness of the limiting Sobolev embeddings once the Sobolev norm is appended by a potential term whose growth at singularities exceeds that of the corresponding Hardy-type potential.

  • 76.
    Cribäck, Kevin
    Uppsala University, Disciplinary Domain of Science and Technology, Mathematics and Computer Science, Department of Mathematics, Analysis and Probability Theory.
    Micro payments: Viable technical platforms and models for a bankto provide payments on micro amounts2018Independent thesis Advanced level (degree of Master (Two Years)), 20 credits / 30 HE creditsStudent thesis
  • 77. Crisan, D.
    et al.
    Li, Kai
    Uppsala University, Disciplinary Domain of Science and Technology, Mathematics and Computer Science, Department of Mathematics, Analysis and Probability Theory.
    Generalised particle filters with Gaussian mixtures2015In: Stochastic Processes and their Applications, ISSN 0304-4149, E-ISSN 1879-209X, Vol. 125, no 7, p. 2643-2673Article in journal (Refereed)
    Abstract [en]

    Stochastic filtering is defined as the estimation of a partially observed dynamical system. Approximating the solution of the filtering problem with Gaussian mixtures has been a very popular method since the 1970s. Despite nearly fifty years of development, the existing work is based on the success of the numerical implementation and is not theoretically justified. This paper fills this gap and contains a rigorous analysis of a new Gaussian mixture approximation to the solution of the filtering problem. We deduce the L-2-convergence rate for the approximating system and show some numerical examples to test the new algorithm.

  • 78.
    Curien, Nicolas
    et al.
    Ecole Normale Super, Dept Math & Applicat, F-75230 Paris 05, France..
    Konstantopoulos, Takis
    Uppsala University, Disciplinary Domain of Science and Technology, Mathematics and Computer Science, Department of Mathematics, Analysis and Probability Theory. Uppsala Univ, Dept Math, S-75106 Uppsala, Sweden..
    Iterating Brownian Motions, Ad Libitum2014In: Journal of theoretical probability, ISSN 0894-9840, E-ISSN 1572-9230, Vol. 27, no 2, p. 433-448Article in journal (Refereed)
    Abstract [en]

    Let B-1, B-2, aEuro broken vertical bar be independent one-dimensional Brownian motions parameterized by the whole real line such that B (i) (0)=0 for every ia parts per thousand yen1. We consider the nth iterated Brownian motion W (n) (t)=B (n) (B (n-1)(a <-(B (2)(B (1)(t)))a <-)). Although the sequence of processes (W (n) ) (na parts per thousand yen1) does not converge in a functional sense, we prove that the finite-dimensional marginals converge. As a consequence, we deduce that the random occupation measures of W (n) converge to a random probability measure mu (a). We then prove that mu (a) almost surely has a continuous density which should be thought of as the local time process of the infinite iteration W (a) of independent Brownian motions. We also prove that the collection of random variables (W (a)(t),taa"ea-{0}) is exchangeable with directing measure mu(infinity).

  • 79.
    Cwikel, Michael
    et al.
    Technion Israel Inst Technol, Dept Math, IL-32000 Haifa, Israel..
    Tintarev, Kyril
    Uppsala University, Disciplinary Domain of Science and Technology, Mathematics and Computer Science, Department of Mathematics, Analysis and Probability Theory.
    On interpolation of cocompact imbeddings2013In: REVISTA MATEMATICA COMPLUTENSE, ISSN 1139-1138, Vol. 26, no 1, p. 33-55Article in journal (Refereed)
    Abstract [en]

    Cocompactness is a useful weaker counterpart of compactness in the study of imbeddings between function spaces. In this paper we prove that, under quite general conditions, cocompactness of imbeddings of Banach spaces persists under both real and complex interpolation. As an application, we obtain that subcritical continuous imbeddings of fractional Sobolev spaces and Besov spaces are cocompact relative to lattice shifts. We deduce this by interpolating the known cocompact imbeddings for classical Sobolev spaces ("vanishing" lemmas of Lieb and Lions). We also apply cocompactness to prove compactness of imbeddings of some radial subspaces and to show the existence of minimizers in some isoperimetric problems. Our research complements a range of previous results, and recalls that there is a natural conceptual framework for unifying them.

  • 80.
    Dareiotis, Konstantinos
    Uppsala University, Disciplinary Domain of Science and Technology, Mathematics and Computer Science, Department of Mathematics, Analysis and Probability Theory.
    Symmetrization of exterior parabolic problems and probabilistic interpretation2017In: STOCHASTICS AND PARTIAL DIFFERENTIAL EQUATIONS-ANALYSIS AND COMPUTATIONS, ISSN 2194-0401, Vol. 5, no 1, p. 38-52Article in journal (Refereed)
    Abstract [en]

    We prove a comparison theorem for the spatial mass of the solutions of two exterior parabolic problems, one of them having symmetrized geometry, using approximation of the Schwarz symmetrization by polarizations, as it was introduced in Brock and Solynin (TransAmMath Soc 352(4): 1759-1796, 2000). This comparison provides an alternative proof, based on PDEs, of the isoperimetric inequality for the Wiener sausage, which was proved in Peres and Sousi (Geom Funct Anal 22(4): 10001014, 2012).

  • 81.
    Dareiotis, Konstantinos
    et al.
    Max Planck Inst Math Sci, Inselstr 22, D-04103 Leipzig, Germany.
    Ekström, Erik
    Uppsala University, Disciplinary Domain of Science and Technology, Mathematics and Computer Science, Department of Mathematics, Applied Mathematics and Statistics. Uppsala University, Disciplinary Domain of Science and Technology, Mathematics and Computer Science, Department of Mathematics, Analysis and Probability Theory.
    Density symmetries for a class of 2-D diffusions with applications to finance2019In: Stochastic Processes and their Applications, ISSN 0304-4149, E-ISSN 1879-209X, Vol. 129, no 2, p. 452-472Article in journal (Refereed)
    Abstract [en]

    We study densities of two-dimensional diffusion processes with one non-negative component. For such diffusions, the density may explode at the boundary, thus making a precise specification of the boundary condition in the corresponding forward Kolmogorov equation problematic. We overcome this by extending a classical symmetry result for densities of one-dimensional diffusions to our case, thereby reducing the study of forward equations with exploding boundary data to the study of a related backward equation with non-exploding boundary data. We also discuss applications of this symmetry for option pricing in stochastic volatility models and in stochastic short rate models. (C) 2018 Elsevier B.V. All rights reserved.

  • 82.
    Dareiotis, Konstantinos
    et al.
    Uppsala University, Disciplinary Domain of Science and Technology, Mathematics and Computer Science, Department of Mathematics, Analysis and Probability Theory.
    Gerencser, Mate
    Univ Edinburgh, Edinburgh EH8 9YL, Midlothian, Scotland..
    Local L-infinity-estimates, weak Harnack inequality, and stochastic continuity of solutions of SPDEs2017In: Journal of Differential Equations, ISSN 0022-0396, E-ISSN 1090-2732, Vol. 262, no 1, p. 615-632Article in journal (Refereed)
    Abstract [en]

    We consider stochastic partial differential equations under minimal assumptions: the coefficients are merely bounded and measurable and satisfy the stochastic parabolicity condition. In particular, the diffusion term is allowed to be scaling-critical. We derive local supremum estimates with a stochastic adaptation of De Giorgi's iteration and establish a weak Harnack inequality for the solutions. The latter is then used to obtain pointwise almost sure continuity.

  • 83.
    Dareiotis, Konstantinos
    et al.
    Uppsala University, Disciplinary Domain of Science and Technology, Mathematics and Computer Science, Department of Mathematics, Analysis and Probability Theory.
    Kumar, Chaman
    Univ Edinburgh, Sch Math, Edinburgh EH9 3JZ, Midlothian, Scotland..
    Sabanis, Sotirios
    Univ Edinburgh, Sch Math, Edinburgh EH9 3JZ, Midlothian, Scotland..
    On tamed euler approximations of sdes driven by levy noise with applications to delay equations2016In: SIAM Journal on Numerical Analysis, ISSN 0036-1429, E-ISSN 1095-7170, Vol. 54, no 3, p. 1840-1872Article in journal (Refereed)
    Abstract [en]

    We extend the taming techniques for explicit Euler approximations of stochastic differential equations driven by Levy noise with superlinearly growing drift coefficients. Strong convergence results are presented for the case of locally Lipschitz coefficients. Moreover, rate of convergence results are obtained in agreement with classical literature when the local Lipschitz continuity assumptions are replaced by global assumptions and, in addition, the drift coefficients satisfy polynomial Lipschitz continuity. Finally, we further extend these techniques to the case of delay equations.

  • 84.
    Dareiotis, Konstantinos
    et al.
    Uppsala University, Disciplinary Domain of Science and Technology, Mathematics and Computer Science, Department of Mathematics, Analysis and Probability Theory.
    Leahy, James-Michael
    Univ Southern Calif, Los Angeles, CA 90089 USA..
    Finite difference schemes for linear stochastic integro-differential equations2016In: Stochastic Processes and their Applications, ISSN 0304-4149, E-ISSN 1879-209X, Vol. 126, no 10, p. 3202-3234Article in journal (Refereed)
    Abstract [en]

    We study the rate of convergence of an explicit and an implicit explicit finite difference scheme for linear stochastic integro-differential equations of parabolic type arising in non-linear filtering of jump diffusion processes. We show that the rate is of order one in space and order one-half in time.

  • 85.
    De Angelis, Tiziano
    et al.
    Univ Leeds, Sch Math, Leeds LS2 9JT, W Yorkshire, England..
    Ekström, Erik
    Uppsala University, Disciplinary Domain of Science and Technology, Mathematics and Computer Science, Department of Mathematics, Analysis and Probability Theory.
    The dividend problem with a finite horizon2017In: The Annals of Applied Probability, ISSN 1050-5164, E-ISSN 2168-8737, Vol. 27, no 6, p. 3525-3546Article in journal (Refereed)
    Abstract [en]

    We characterise the value function of the optimal dividend problem with a finite time horizon as the unique classical solution of a suitable Hamilton-Jacobi-Bellman equation. The optimal dividend strategy is realised by a Skorokhod reflection of the fund's value at a time-dependent optimal boundary. Our results are obtained by establishing for the first time a new connection between singular control problems with an absorbing boundary and optimal stopping problems on a diffusion reflected at 0 and created at a rate proportional to its local time.

  • 86.
    Denisov, D.
    et al.
    Cardiff Univ, Sch Math, Cardiff CF24 4AG, S Glam, Wales..
    Foss, S.
    Heriot Watt Univ, Sch Math & Comp Sci, Edinburgh EH14 4AS, Midlothian, Scotland..
    Konstantopoulos, Takis
    Uppsala University, Disciplinary Domain of Science and Technology, Mathematics and Computer Science, Department of Mathematics, Analysis and Probability Theory.
    Limit Theorems For A Random Directed Slab Graph2012In: The Annals of Applied Probability, ISSN 1050-5164, E-ISSN 2168-8737, Vol. 22, no 2, p. 702-733Article in journal (Refereed)
    Abstract [en]

    We consider a stochastic directed graph on the integers whereby a directed edge between i and a larger integer j exists with probability p(j-i) depending solely on the distance between the two integers. Under broad conditions, we identify a regenerative structure that enables us to prove limit theorems for the maximal path length in a long chunk of the graph. The model is an extension of a special case of graphs studied in [Markov Process. Related Fields 9 (2003) 413-468]. We then consider a similar type of graph but on the "slab" Z x I, where I is a finite partially ordered set. We extend the techniques introduced in the first part of the paper to obtain a central limit theorem for the longest path. When I is linearly ordered, the limiting distribution can be seen to be that of the largest eigenvalue of a |I| x |I| random matrix in the Gaussian unitary ensemble (GUE).

  • 87.
    Dettmann, Carl P.
    et al.
    Univ Bristol, Sch Math, Bristol BS8 1TW, Avon, England..
    Marklof, Jens
    Univ Bristol, Sch Math, Bristol BS8 1TW, Avon, England..
    Strömbergsson, Andreas
    Uppsala University, Disciplinary Domain of Science and Technology, Mathematics and Computer Science, Department of Mathematics, Analysis and Probability Theory.
    Universal Hitting Time Statistics for Integrable Flows2017In: Journal of statistical physics, ISSN 0022-4715, E-ISSN 1572-9613, Vol. 166, no 3-4, p. 714-749Article in journal (Refereed)
    Abstract [en]

    The perceived randomness in the time evolution of "chaotic" dynamical systems can be characterized by universal probabilistic limit laws, which do not depend on the fine features of the individual system. One important example is the Poisson law for the times at which a particle with random initial data hits a small set. This was proved in various settings for dynamical systems with strong mixing properties. The key result of the present study is that, despite the absence of mixing, the hitting times of integrable flows also satisfy universal limit laws which are, however, not Poisson. We describe the limit distributions for "generic" integrable flows and a natural class of target sets, and illustrate our findings with two examples: the dynamics in central force fields and ellipse billiards. The convergence of the hitting time process follows from a new equidistribution theorem in the space of lattices, which is of independent interest. Its proof exploits Ratner's measure classification theorem for unipotent flows, and extends earlier work of Elkies and McMullen.

  • 88.
    Djindja, Domingos
    Uppsala University, Disciplinary Domain of Science and Technology, Mathematics and Computer Science, Department of Mathematics, Analysis and Probability Theory.
    Valuation of American put options with exercise restrictions2014Independent thesis Advanced level (degree of Master (Two Years)), 20 credits / 30 HE creditsStudent thesis
  • 89. Dudek, Andrzej
    et al.
    Frieze, Alan
    Rucinski, Andrzej
    Sileikis, Matas
    Uppsala University, Disciplinary Domain of Science and Technology, Mathematics and Computer Science, Department of Mathematics, Analysis and Probability Theory.
    Loose Hamilton Cycles in Regular Hypergraphs2015In: Combinatorics, probability & computing, ISSN 0963-5483, E-ISSN 1469-2163, Vol. 24, no 1, p. 179-194Article in journal (Refereed)
    Abstract [en]

    We establish a relation between two uniform models of random k-graphs (for constant k >= 3) on n labelled vertices: H-(k)(n, m), the random k-graph with exactly m edges, and H-(k)(n, d), the random d-regular k-graph. By extending the switching technique of McKay and Wormald to k-graphs, we show that, for some range of d = d(n) and a constant c > 0, if m similar to cnd, then one can couple H-(k)(n, m) and H-(k)(n, d) so that the latter contains the former with probability tending to one as n -> infinity. In view of known results on the existence of a loose Hamilton cycle in H-(k)(n, m), we conclude that H-(k)(n, d) contains a loose Hamilton cycle when d >> log n (or just d >= C log n, if k = 3) and d = o(n(1/ 2)).

  • 90.
    Duits, Maurice
    et al.
    Royal Inst Technol KTH, Dept Math, Stockholm Lindstedsvagen 25, SE-10044 Stockholm, Sweden.
    Kozhan, Rostyslav
    Uppsala University, Disciplinary Domain of Science and Technology, Mathematics and Computer Science, Department of Mathematics, Analysis and Probability Theory.
    Relative Szego Asymptotics for Toeplitz Determinants2019In: International mathematics research notices, ISSN 1073-7928, E-ISSN 1687-0247, Vol. 2019, no 17, p. 5441-5496Article in journal (Refereed)
    Abstract [en]

    We study the asymptotic behaviour, as n -> infinity, of ratios of Toeplitz determinants D-n(e(h)d mu)/D-n(d mu) defined by a measure mu on the unit circle and a sufficiently smooth function h. The approach we follow is based on the theory of orthogonal polynomials. We prove that the second order asymptotics depends on h and only a few Verblunsky coefficients associated to mu. As a result, we establish a relative version of the Strong Szego Limit Theorem for a wide class of measures mu with essential support on a single arc. In particular, this allows the measure to have a singular component within or outside of the arc.

  • 91.
    Duse, Erik
    et al.
    KTH Royal Inst Technol, Stockholm, Sweden.
    Johansson, Kurt
    KTH Royal Inst Technol, Stockholm, Sweden.
    Metcalfe, Anthony
    Uppsala University, Disciplinary Domain of Science and Technology, Mathematics and Computer Science, Department of Mathematics, Analysis and Probability Theory.
    The Cusp-Airy process2016In: Electronic Journal of Probability, ISSN 1083-6489, E-ISSN 1083-6489, Vol. 21, p. 1-50, article id 57Article in journal (Refereed)
    Abstract [en]

    At a typical cusp point of the disordered region in a random tiling model we expect to see a determinantal process called the Pearcey process in the appropriate scaling limit. However, in certain situations another limiting point process appears that we call the Cusp-Airy process, which is a kind of two sided extension of the Airy kernel point process. We will study this problem in a class of random lozenge tiling models coming from interlacing particle systems. The situation was briefly studied previously by Okounkov and Reshetikhin under the name cuspidal turning point but their formula is not completely correct.

  • 92.
    Duse, Erik
    et al.
    Royal Inst Technol KTH, Dept Math, Stockholm, Sweden..
    Metcalfe, Anthony
    Uppsala University, Disciplinary Domain of Science and Technology, Mathematics and Computer Science, Department of Mathematics, Analysis and Probability Theory.
    Asymptotic geometry of discrete interlaced patterns: Part I2015In: International Journal of Mathematics, ISSN 0129-167X, Vol. 26, no 11, article id 1550093Article in journal (Refereed)
    Abstract [en]

    A discrete Gelfand-Tsetlin pattern is a configuration of particles in Z(2). The particles are arranged in a finite number of consecutive rows, numbered from the bottom. There is one particle on the first row, two particles on the second row, three particles on the third row, etc., and particles on adjacent rows satisfy an interlacing constraint. We consider the uniform probability measure on the set of all discrete Gelfand-Tsetlin patterns of a fixed size where the particles on the top row are in deterministic positions. This measure arises naturally as an equivalent description of the uniform probability measure on the set of all tilings of certain polygons with lozenges. We prove a determinantal structure, and calculate the correlation kernel. We consider the asymptotic behavior of the system as the size increases under the assumption that the empirical distribution of the deterministic particles on the top row converges weakly. We consider the asymptotic "shape" of such systems. We provide parameterizations of the asymptotic boundaries and investigate the local geometric properties of the resulting curves. We show that the boundary can be partitioned into natural sections which are determined by the behavior of the roots of a function related to the correlation kernel. This paper should be regarded as a companion piece to the paper [E. Duse and A. Metcalfe, Asymptotic geometry of discrete interlaced patterns: Part II, in preparation], in which we resolve some of the remaining issues. Both of these papers serve as background material for the papers [E. Duse and A. Metcalfe, Universal edge fluctuations of discrete interlaced particle systems, in preparation; E. Duse and K. Johansson and A. Metcalfe, Cusp Airy process of discrete interlaced particle systems, in preparation], in which we examine the edge asymptotic behavior.

  • 93.
    Dyrssen, Hannah
    Uppsala University, Disciplinary Domain of Science and Technology, Mathematics and Computer Science, Department of Mathematics, Analysis and Probability Theory.
    Option Pricing in Jump-to-Default Models2015Licentiate thesis, comprehensive summary (Other academic)
  • 94.
    Dyrssen, Hannah
    Uppsala University, Disciplinary Domain of Science and Technology, Mathematics and Computer Science, Department of Mathematics, Applied Mathematics and Statistics. Uppsala University, Disciplinary Domain of Science and Technology, Mathematics and Computer Science, Department of Mathematics, Analysis and Probability Theory.
    The perpetual American put option in jump-to-default models2017In: Stochastics: An International Journal of Probablitiy and Stochastic Processes, ISSN 1744-2508, E-ISSN 1744-2516, Vol. 89, no 2, p. 510-520Article in journal (Refereed)
    Abstract [en]

    We study the perpetual American put option in a general jump-to-default model, deriving an explicit expression for the price of the option.

    We find that in some cases the optimal stopping boundary vanishes and thus it is not optimal to exercise the option before default occurs. Precise conditions for when this situation arises are given.

    Furthermore we present a necessary and sufficient condition for convexity of the option price, and also show that a nonincreasing intensity is sufficient, but not necessary, to have convexity.

    From this we also get conditions for when option prices are monotone in the model parameters.

  • 95.
    Dyrssen, Hannah
    Uppsala University, Disciplinary Domain of Science and Technology, Mathematics and Computer Science, Department of Mathematics, Applied Mathematics and Statistics. Uppsala University, Disciplinary Domain of Science and Technology, Mathematics and Computer Science, Department of Mathematics, Analysis and Probability Theory.
    Valuation and Optimal Strategies in Markets Experiencing Shocks2017Doctoral thesis, comprehensive summary (Other academic)
    Abstract [en]

    This thesis treats a range of stochastic methods with various applications, most notably in finance. It is comprised of five articles, and a summary of the key concepts and results these are built on.

    The first two papers consider a jump-to-default model, which is a model where some quantity, e.g. the price of a financial asset, is represented by a stochastic process which has continuous sample paths except for the possibility of a sudden drop to zero. In Paper I prices of European-type options in this model are studied together with the partial integro-differential equation that characterizes the price. In Paper II the price of a perpetual American put option in the same model is found in terms of explicit formulas. Both papers also study the parameter monotonicity and convexity properties of the option prices.

    The third and fourth articles both deal with valuation problems in a jump-diffusion model. Paper III concerns the optimal level at which to exercise an American put option with finite time horizon. More specifically, the integral equation that characterizes the optimal boundary is studied. In Paper IV we consider a stochastic game between two players and determine the optimal value and exercise strategy using an iterative technique.

    Paper V employs a similar iterative method to solve the statistical problem of determining the unknown drift of a stochastic process, where not only running time but also each observation of the process is costly.

    List of papers
    1. Pricing equations in jump-to-default models
    Open this publication in new window or tab >>Pricing equations in jump-to-default models
    2014 (English)In: Int. J. Theor. Appl. Finance, ISSN 0219-0249, Vol. 17, no 3Article in journal (Refereed) Published
    National Category
    Probability Theory and Statistics
    Identifiers
    urn:nbn:se:uu:diva-313325 (URN)10.1142/S0219024914500198 (DOI)
    Available from: 2017-01-19 Created: 2017-01-19 Last updated: 2017-03-14
    2. The perpetual American put option in jump-to-default models
    Open this publication in new window or tab >>The perpetual American put option in jump-to-default models
    2017 (English)In: Stochastics: An International Journal of Probablitiy and Stochastic Processes, ISSN 1744-2508, E-ISSN 1744-2516, Vol. 89, no 2, p. 510-520Article in journal (Refereed) Published
    Abstract [en]

    We study the perpetual American put option in a general jump-to-default model, deriving an explicit expression for the price of the option.

    We find that in some cases the optimal stopping boundary vanishes and thus it is not optimal to exercise the option before default occurs. Precise conditions for when this situation arises are given.

    Furthermore we present a necessary and sufficient condition for convexity of the option price, and also show that a nonincreasing intensity is sufficient, but not necessary, to have convexity.

    From this we also get conditions for when option prices are monotone in the model parameters.

    National Category
    Probability Theory and Statistics
    Research subject
    Mathematics with specialization in Applied Mathematics
    Identifiers
    urn:nbn:se:uu:diva-313326 (URN)10.1080/17442508.2016.1267177 (DOI)000392492800004 ()
    Available from: 2017-01-19 Created: 2017-01-19 Last updated: 2017-11-29Bibliographically approved
    3. The integral equation for the American put boundary in models with jumps
    Open this publication in new window or tab >>The integral equation for the American put boundary in models with jumps
    (English)Article in journal (Other academic) Submitted
    Abstract [en]

    The price of the American put option is frequently studied as the solution to an associated free-boundary problem. This free boundary, the optimal exercise boundary, determines the value of the option. In spectrally negative models the early exercise premium representation for the value of the option gives rise to an integral equation for the boundary. We study this integral equation and prove that the optimal exercise boundary is the unique solution and thus that the equation characterizes the free boundary. In a spectrally positive model, this approach does not give an equation for the boundary. We instead find lower and upper bounds for the true boundary which can be found by solving related equations.

    National Category
    Probability Theory and Statistics
    Research subject
    Mathematics with specialization in Applied Mathematics
    Identifiers
    urn:nbn:se:uu:diva-316576 (URN)
    Available from: 2017-03-03 Created: 2017-03-03 Last updated: 2017-03-14
    4. Optimal stopping games for a process with jumps
    Open this publication in new window or tab >>Optimal stopping games for a process with jumps
    (English)Article in journal (Other academic) Submitted
    Abstract [en]

    This paper presents a study of a general two-player optimal stopping game in a jump-diffusion model. An iterative scheme to find the value of this game is derived, specifically the value is shown to be the limit of a sequence of stopping games for a related diffusion but with a running reward. Furthermore the convergence is uniform and exponential. The special case of a cancellable put option is studied.

    National Category
    Probability Theory and Statistics
    Identifiers
    urn:nbn:se:uu:diva-316577 (URN)
    Available from: 2017-03-03 Created: 2017-03-03 Last updated: 2017-03-14
    5. Sequential testing of a Wiener process with costly observations
    Open this publication in new window or tab >>Sequential testing of a Wiener process with costly observations
    (English)Article in journal (Other academic) Submitted
    Abstract [en]

    We consider the sequential testing of two simple hypotheses for the drift of a Brownian motion when each observation of the underlying process is associated with a positive cost. In this setting where continuous monitoring of the underlying process is not feasible, the question is not only whether to stop or to continue at a given observation time, but also, if continuing,how to distribute the next observation time. Adopting a Bayesian methodology, we show that the value function can be characterized as the unique fixed point of an associated operator, and that it can be constructed using an iterative scheme. Moreover, the optimal sequential distribution of observation times can be described in terms of the fixed point.

    National Category
    Probability Theory and Statistics
    Research subject
    Mathematics with specialization in Applied Mathematics
    Identifiers
    urn:nbn:se:uu:diva-316571 (URN)
    Available from: 2017-03-03 Created: 2017-03-03 Last updated: 2017-03-22
  • 96.
    Dyrssen, Hannah
    et al.
    Uppsala University, Disciplinary Domain of Science and Technology, Mathematics and Computer Science, Department of Mathematics.
    Ekström, Erik
    Uppsala University, Disciplinary Domain of Science and Technology, Mathematics and Computer Science, Department of Mathematics, Analysis and Probability Theory.
    Sequential testing of a Wiener process with costly observations2018In: Sequential Analysis, ISSN 0747-4946, E-ISSN 1532-4176, Vol. 37, no 1, p. 47-58Article in journal (Refereed)
    Abstract [en]

    We consider the sequential testing of two simple hypotheses for the drift of a Brownian motion when each observation of the underlying process is associated with a positive cost. In this setting where continuous monitoring of the underlying process is not feasible, the question is not only whether to stop or to continue at a given observation time but also, if continuing, how to distribute the next observation time. Adopting a Bayesian methodology, we show that the value function can be characterized as the unique fixed point of an associated operator and that it can be constructed using an iterative scheme. Moreover, the optimal sequential distribution of observation times can be described in terms of the fixed point.

  • 97.
    Edward, Viktor
    Uppsala University, Disciplinary Domain of Science and Technology, Mathematics and Computer Science, Department of Mathematics, Analysis and Probability Theory.
    Mean-Reverting Stochastic Models for the Electricity Spot Market2014Independent thesis Basic level (degree of Bachelor), 10 credits / 15 HE creditsStudent thesis
  • 98.
    Edwards, Samuel
    Uppsala University, Disciplinary Domain of Science and Technology, Mathematics and Computer Science, Department of Mathematics, Analysis and Probability Theory.
    Effective equidistribution of horospheres in infinite-volume quotients of SO(n, 1) by geometrically finite groupsManuscript (preprint) (Other academic)
    Abstract [en]

    We use the method of Burger to study the rate of equidistribution for translates of pieces of horospheres in Γ\ SO0(n, 1) for geometrically finite discrete subgroups Γ < SO0(n, 1) with infinite covolume.

  • 99.
    Edwards, Samuel
    Uppsala University, Disciplinary Domain of Science and Technology, Mathematics and Computer Science, Department of Mathematics, Analysis and Probability Theory.
    On the equidistribution of translates of orbits of symmetric subgroups in Γ\GManuscript (preprint) (Other academic)
    Abstract [en]

    We use the method of Burger to study the rate of equidistribution for translates of orbits of symmetric subgroups in homogeneous spaces Γ\G for semisimple Lie groups G and lattices Γ.

  • 100.
    Edwards, Samuel
    Uppsala University, Disciplinary Domain of Science and Technology, Mathematics and Computer Science, Department of Mathematics, Analysis and Probability Theory.
    On the rate of equidistribution of expanding translates of horospheres in finite-volume quotients of SL(2,C)2017In: Journal of Modern Dynamics, ISSN 1930-5311, E-ISSN 1930-532X, Vol. 11, p. 155-188Article in journal (Refereed)
    Abstract [en]

    Let Gamma be a lattice in G = SL(2, C). We give an effective equidistribution result with precise error terms for expanding translates of pieces of horospherical orbits in Gamma\G. Our method of proof relies on the theory of unitary representations.

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