21 Gennaio, 2010 15:00 in punto

Adaptive Multi Level Monte Carlo Simulation

Erik Von Schwerin, Applied Mathematics and Computational Sciences, KAUST, Thuwal, Saudi Arabia

Aula Seminari F. Saleri VI Piano MOX- Dipartimento di Matematica, Politecnico di Milano

abstract

abstract

The topic of this talk is a work that generalizes a multilevel Forward Euler Monte Carlo method
introduced in [1] for the approximation of expected values depending on the solution to an Ito stochastic differential equation. The work [1] proposed and analyzed a Forward Euler Multilevel Monte Carlo method based on a hierarchy of uniform time discretizations and control variates to reduce the computational effort required by a standard, single level, Forward Euler Monte Carlo method. The present work introduces and analyzes an adaptive hierarchy of non uniform time discretizations, generated by adaptive algorithms introduced in [3, 2]. These adaptive algorithms apply either deterministic time steps or stochastic time
steps and are based on a posteriori error expansions first developed in [4]. Under sufficient regularity conditions, both our analysis and numerical results, which include one case with singular drift and one with stopped diffusion, exhibit savings in the computational cost to achieve an accuracy of O(TOL), from
O(TOL^-3) to O(TOL^-1*log(TOL))^2.

In the talk I will give a background on the Multi Level Monte Carlo method of uniform time discretizations in [1] and show how to extend this idea to the setting of adaptive time stepping.
This talk presents joint work with H. Hoel, A. Szepessy, and R. Tempone.

Key words: computational finance, Monte Carlo, multi-level, adaptivity, weak approximation, error
control, Euler-Maruyama method, a posteriori error estimates, backward dual functions, adjoints

References
[1] Michael B. Giles. Multilevel Monte Carlo path simulation. Oper. Res., 56(3):607-617, 2008.
[2] Kyoung-Sook Moon, Anders Szepessy, Raul Tempone, and Georgios E. Zouraris. Convergence rates for adaptive weak approximation of stochastic dierential equations. Stoch. Anal. Appl., 23(3):511-558, 2005.
[3] Kyoung-Sook Moon, Erik von Schwerin, Anders Szepessy, and Raul Tempone. An adaptive algorithm
for ordinary, stochastic and partial dierential equations. In Recent advances in adaptive computation, volume 383 of Contemp. Math., pages 325-343. Amer. Math. Soc., Providence, RI, 2005.
[4] Anders Szepessy, Raul Tempone, and Georgios E. Zouraris. Adaptive weak approximation of stochastic dierential equations. Comm. Pure Appl. Math., 54(10):1169-1214, 2001.

20 Gennaio, 2010 16:15

Sezione di Analisi

Problemi matematici nella condensazione di Bose-Einstein

Susanna Terracini, Università di Milano-Bicocca

Aula Seminari MOX - VI piano

The topic of this talk is a work that generalizes a multilevel Forward Euler Monte Carlo method
introduced in [1] for the approximation of expected values depending on the solution to an Ito stochastic differential equation. The work [1] proposed and analyzed a Forward Euler Multilevel Monte Carlo method based on a hierarchy of uniform time discretizations and control variates to reduce the computational effort required by a standard, single level, Forward Euler Monte Carlo method. The present work introduces and analyzes an adaptive hierarchy of non uniform time discretizations, generated by adaptive algorithms introduced in [3, 2]. These adaptive algorithms apply either deterministic time steps or stochastic time
steps and are based on a posteriori error expansions first developed in [4]. Under sufficient regularity conditions, both our analysis and numerical results, which include one case with singular drift and one with stopped diffusion, exhibit savings in the computational cost to achieve an accuracy of O(TOL), from
O(TOL^-3) to O(TOL^-1*log(TOL))^2.

In the talk I will give a background on the Multi Level Monte Carlo method of uniform time discretizations in [1] and show how to extend this idea to the setting of adaptive time stepping.
This talk presents joint work with H. Hoel, A. Szepessy, and R. Tempone.

Key words: computational finance, Monte Carlo, multi-level, adaptivity, weak approximation, error
control, Euler-Maruyama method, a posteriori error estimates, backward dual functions, adjoints

References
[1] Michael B. Giles. Multilevel Monte Carlo path simulation. Oper. Res., 56(3):607-617, 2008.
[2] Kyoung-Sook Moon, Anders Szepessy, Raul Tempone, and Georgios E. Zouraris. Convergence rates for adaptive weak approximation of stochastic dierential equations. Stoch. Anal. Appl., 23(3):511-558, 2005.
[3] Kyoung-Sook Moon, Erik von Schwerin, Anders Szepessy, and Raul Tempone. An adaptive algorithm
for ordinary, stochastic and partial dierential equations. In Recent advances in adaptive computation, volume 383 of Contemp. Math., pages 325-343. Amer. Math. Soc., Providence, RI, 2005.
[4] Anders Szepessy, Raul Tempone, and Georgios E. Zouraris. Adaptive weak approximation of stochastic dierential equations. Comm. Pure Appl. Math., 54(10):1169-1214, 2001.

20 Gennaio, 2010 15:15

Seminari FDS

Tra matematica e arte: un’esperienza didattica

Adriana Contarini, Docente Scuole Superiori

Aula Consiglio - VII piano

The topic of this talk is a work that generalizes a multilevel Forward Euler Monte Carlo method
introduced in [1] for the approximation of expected values depending on the solution to an Ito stochastic differential equation. The work [1] proposed and analyzed a Forward Euler Multilevel Monte Carlo method based on a hierarchy of uniform time discretizations and control variates to reduce the computational effort required by a standard, single level, Forward Euler Monte Carlo method. The present work introduces and analyzes an adaptive hierarchy of non uniform time discretizations, generated by adaptive algorithms introduced in [3, 2]. These adaptive algorithms apply either deterministic time steps or stochastic time
steps and are based on a posteriori error expansions first developed in [4]. Under sufficient regularity conditions, both our analysis and numerical results, which include one case with singular drift and one with stopped diffusion, exhibit savings in the computational cost to achieve an accuracy of O(TOL), from
O(TOL^-3) to O(TOL^-1*log(TOL))^2.

In the talk I will give a background on the Multi Level Monte Carlo method of uniform time discretizations in [1] and show how to extend this idea to the setting of adaptive time stepping.
This talk presents joint work with H. Hoel, A. Szepessy, and R. Tempone.

Key words: computational finance, Monte Carlo, multi-level, adaptivity, weak approximation, error
control, Euler-Maruyama method, a posteriori error estimates, backward dual functions, adjoints

References
[1] Michael B. Giles. Multilevel Monte Carlo path simulation. Oper. Res., 56(3):607-617, 2008.
[2] Kyoung-Sook Moon, Anders Szepessy, Raul Tempone, and Georgios E. Zouraris. Convergence rates for adaptive weak approximation of stochastic dierential equations. Stoch. Anal. Appl., 23(3):511-558, 2005.
[3] Kyoung-Sook Moon, Erik von Schwerin, Anders Szepessy, and Raul Tempone. An adaptive algorithm
for ordinary, stochastic and partial dierential equations. In Recent advances in adaptive computation, volume 383 of Contemp. Math., pages 325-343. Amer. Math. Soc., Providence, RI, 2005.
[4] Anders Szepessy, Raul Tempone, and Georgios E. Zouraris. Adaptive weak approximation of stochastic dierential equations. Comm. Pure Appl. Math., 54(10):1169-1214, 2001.

19 Gennaio, 2010 12:00 in punto

Seminari FDS

Nonparametric Leverage Effects

Roberto Renò, Università degli Studi di Siena

Aula seminari, VI piano

abstract

abstract

Vast empirical evidence points to the existence of a negative correlation, named leverage effect, between shocks in volatility and shocks in returns. We provide a nonparametric theory of leverage estimation in the context of a continuous-time stochastic volatility model with jumps in returns, jumps in volatility, or both. Leverage is defined as a flexible function of the state of the firm, as summarized by the spot volatility level. We show that its point-wise functional estimates have asymptotic properties (in terms of rates of convergence, limiting biases, and limiting variances) which crucially depend on the likelihood of the individual jumps and co-jumps as well as on the features of the jump size distributions. Empirically, we find economically important time-variation in leverage with more negative values associated with higher volatility levels.

14 Gennaio, 2010 11:00 in punto

Sezione di Geometria, Algebra e loro applicazioni

La geometria della sicurezza informatica e problemi connessi.

Angelo Sonnino, Università della Basilicata

Aula seminari del terzo piano

abstract

abstract

L implementazione di un criptosistema in una rete di comunicazione pone alcuni specifici problemi di natura matematica. In questa trattazione si discutono alcuni di questi problemi quali, nel caso di un criptosistema ellittico, l immersione dello spazio dei messaggi in una curva ellittica ed il calcolo del prodotto fra elementi di un campo di Galois di ordine elevato, con particolare riguardo all ottimizzazione per sistemi con limitata disponibilità di tempo per il trattamento dei dati.

7 Gennaio, 2010 17:00

Sezione di Analisi

Problemi matematici nella condensazione di Bose-Einstein

Susanna Terracini, Università di Milano-Bicocca

Sala Consiglio - VII piano

L implementazione di un criptosistema in una rete di comunicazione pone alcuni specifici problemi di natura matematica. In questa trattazione si discutono alcuni di questi problemi quali, nel caso di un criptosistema ellittico, l immersione dello spazio dei messaggi in una curva ellittica ed il calcolo del prodotto fra elementi di un campo di Galois di ordine elevato, con particolare riguardo all ottimizzazione per sistemi con limitata disponibilità di tempo per il trattamento dei dati.

22 Dicembre, 2009 11:15

Sezione di Analisi

Mortar Finite Element approaches for eddy current computations in moving conductors

Francesca Rapetti, Université de Nice Sophia-Antipolis, Laboratoire de Mathématiques J.A. Dieudonné

Aula Seminari F. Saleri VI Piano MOX- Dipartimento di Matematica, Politecnico di Milano

abstract

abstract

In this talk, we consider the particular problem of computing eddy currents in moving conductors. We present the problem from two points of view, the Eulerian and the Lagrangian one, and an overview on the most used numerical techniques to tackle with it. We then describe two Lagrangian non-conforming finite element approaches based on the mortar element method, namely the sliding mesh and overlapping mesh techiques. We will illustrate the application of these approaches to realistic cases.

18 Dicembre, 2009 14:00

Sezione di Geometria, Algebra e loro applicazioni

$A^{\omega}=B^{\omega}$ for $A$ and $B$ finite sets.

Boris Melnikov, State University of Togliattigrad

auletta seminari III piano

In this talk, we consider the particular problem of computing eddy currents in moving conductors. We present the problem from two points of view, the Eulerian and the Lagrangian one, and an overview on the most used numerical techniques to tackle with it. We then describe two Lagrangian non-conforming finite element approaches based on the mortar element method, namely the sliding mesh and overlapping mesh techiques. We will illustrate the application of these approaches to realistic cases.

17 Dicembre, 2009 15:00 in punto

Sezione di Fisica Matematica

Distorsion, Dislocation and Disclination or Why Liquid Crystals are disinclined to be disinclined

Tim Sluckin, University of Southampton

Aula Seminari, VI piano

abstract

abstract

The original observation of liquid crystals in 1888 emphasised that the materials in question underwent a double melting phenomenon. A new cloudy fluid phase interposed itself between solid and liquid phases. Under a polarising microscope, however, the cloudy phase turned out to exhibit quite beautiful patterns, which became known as textures. The interpretation of these textures in terms of defects in a director field of an anisotropic fluid was one of the triumphs of twentieth century physics. More recently it has also provided a paradigm for the application of methods of topology to condensed matter physics. In this talk I shall begin by giving surveying the history of liquid crystals, and of the idea of a defect, concentrating on issues of interest to a mathematical audience. I shall conclude by discussing some of my own recent work in this area.

16 Dicembre, 2009 16:00 in punto

Sezione di Probabilità Quantistica

Quantum Stochastic Calculus and Continual Measurement with Unbounded Coefficients

Ricardo Castro Santis, Centro de Análisis Estocástico, Pontificia Universidad Católica de Chile

Aula seminari III piano

The original observation of liquid crystals in 1888 emphasised that the materials in question underwent a double melting phenomenon. A new cloudy fluid phase interposed itself between solid and liquid phases. Under a polarising microscope, however, the cloudy phase turned out to exhibit quite beautiful patterns, which became known as textures. The interpretation of these textures in terms of defects in a director field of an anisotropic fluid was one of the triumphs of twentieth century physics. More recently it has also provided a paradigm for the application of methods of topology to condensed matter physics. In this talk I shall begin by giving surveying the history of liquid crystals, and of the idea of a defect, concentrating on issues of interest to a mathematical audience. I shall conclude by discussing some of my own recent work in this area.