Luciano Tubaro

Bio: Luciano Tubaro is an academic researcher from University of Trento. The author has contributed to research in topics: Stochastic differential equation & Hilbert space. The author has an hindex of 18, co-authored 60 publications receiving 1531 citations.

Expansion of the global error for numerical schemes solving stochastic differential equations

Abstract: Given the solution (Xt ) of a Stochastic Differential System, two situat,ions are considered: computat,ion of Ef(Xt ) by a Monte–Carlo method and, in the ergodic case, integration of a function f w.r.t. the invariant probability law of (Xt ) by simulating a simple t,rajectory. For each case it is proved the expansion of the global approximat,ion error—for a class of discret,isat,ion schemes and of funct,ions f—in powers of the discretisation step size, extending in the fist case a result of Gragg for deterministic O.D.E. Some nn~nerical examples are shown to illust,rate the applicat,ion of extrapolation methods, justified by the foregoing expansion, in order to improve the approximation accuracy

Fully nonlinear stochastic partial differential equations

Abstract: The authors study a class of fully nonlinear stochastic partial differential equations by the reduction to a family of deterministic fully nonlinear equations using the stochastic characteristic method.

An estimate of Burkholder type for stochastic processes defined by the stochastic integral

Abstract: We develop a method to prove an estimate of Burkholder type for a class of processes defined by a stochastic convolution with a semigroup for p ≥ 2; this result, for p = 2, has been obtained also by Kotelenez [6] using a martingale inequality due to him[5]

Stochastic Partial Differential Equations and Applications

Abstract: The Semi-Martingale Property of the Square of White Noise Integrators Luigi Accardi and Andreas Boukas SPDEs Leading to Local, Relativistic Quantum Vector Fields with Indefinite Metric and Nontrivial S-Matrix Sergio Albeverio, Hanno Gottschalk, and Jiang-Lun Wu Considerations on the Controllability of Stochastic Linear Heat Equations Viorel Barbu and Gianmario Tessitore Stochastic Differential Equations for Trace-Class Operators and Quantum Continual Measurements Alberto Barchielli and Anna Maria Paganoni Invariant Measures of Diffusion Processes: Regularity, Existence, and Uniqueness Problems Vladimir I. Bogachev and Michael Rockner On the Theory of Random Attractors and Some Open Problems Tomas Caraballo and Jose Antonio Langa Invariant Densities for Stochastic Semilinear Evolution Equations and Related Properties of Transition Semigroups Anna Chojnowska-Michalik On Some Generalized Solutions of Stochastic PDEs Pao-Liu Chow Riemannian Geometry on the Path Space B. Cruzeiro and P. Malliavin A Note on Regularizing Properties of Ornstein-Uhlenbeck Semigroups in Infinite Dimensions Giuseppe Da Prato, Marco Fuhrman, and Jerzy Zabczyk White Noise Approach to Stochastic Partial Differential Equations T. Deck, S. Kruse, J. Potthoff, and H. Watanabe Some Results on Invariant States for Quantum Markov Semigroups Franco Fagnola and Rolando Rebolledo Stochastic Problems in Fluid Dynamics Franco Flandoli Limit Theorems for Random Interface Models of Ginzburg-Landau "j Type Giambattista Giacomin Second Order Hamilton-Jacobi Equations in Hilbert Spaces and Stochastic Optimal Control Fausto Gozzi Approximations of Stochastic Partial Differential Equations Istvan Gyongy Regularity and Continuity of Solutions to Stochastic Evolution Equations Anna Karczewska Some New Results in the Theory of SPDEs in Sobolev Spaces N. V. Krylov Lyapunov Function Approaches and Asymptotic Stability of Stochastic Evolution Equations in Hilbert Spaces-A Survey of Recent Developments Kai Liu and Aubrey Truman Strong Feller Infinite-Dimensional Diffusions Bohdan Maslowski and Jan Seidler Optimal Stopping Time and Impulse Control Problems for the Stochastic Navier-Stokes Equations J. L. Menaldi and S. S. Sritharan On Martingale Problem Solutions for Stochastic Navier-Stokes Equation R. Mikulevicius and B. Rozovskii SPDEs Driven by a Homogeneous Wiener Process Szymon Peszat Applications of Malliavin Calculus to SPDEs Marta Sanz-Sole Stochastic Curvature Driven Flows Nung Kwan Yip

Stochastic Equations in Infinite Dimensions

Abstract: Part I. Foundations: 1. Random variables 2. Probability measures 3. Stochastic processes 4. The stochastic integral Part II. Existence and Uniqueness: 5. Linear equations with additive noise 6. Linear equations with multiplicative noise 7. Existence and uniqueness for nonlinear equations 8. Martingale solutions Part III. Properties of Solutions: 9. Markov properties and Kolmogorov equations 10. Absolute continuity and Girsanov's theorem 11. Large time behaviour of solutions 12. Small noise asymptotic.

Multilevel Monte Carlo Path Simulation

Abstract: We show that multigrid ideas can be used to reduce the computational complexity of estimating an expected value arising from a stochastic differential equation using Monte Carlo path simulations. In the simplest case of a Lipschitz payoff and a Euler discretisation, the computational cost to achieve an accuracy of O(e) is reduced from O(e-3) to O(e-2 (log e)2). The analysis is supported by numerical results showing significant computational savings.

A theory of regularity structures

Abstract: We introduce a new notion of “regularity structure” that provides an algebraic framework allowing to describe functions and/or distributions via a kind of “jet” or local Taylor expansion around each point. The main novel idea is to replace the classical polynomial model which is suitable for describing smooth functions by arbitrary models that are purpose-built for the problem at hand. In particular, this allows to describe the local behaviour not only of functions but also of large classes of distributions. We then build a calculus allowing to perform the various operations (multiplication, composition with smooth functions, integration against singular kernels) necessary to formulate fixed point equations for a very large class of semilinear PDEs driven by some very singular (typically random) input. This allows, for the first time, to give a mathematically rigorous meaning to many interesting stochastic PDEs arising in physics. The theory comes with convergence results that allow to interpret the solutions obtained in this way as limits of classical solutions to regularised problems, possibly modified by the addition of diverging counterterms. These counterterms arise naturally through the action of a “renormalisation group” which is defined canonically in terms of the regularity structure associated to the given class of PDEs. Our theory also allows to easily recover many existing results on singular stochastic PDEs (KPZ equation, stochastic quantisation equations, Burgers-type equations) and to understand them as particular instances of a unified framework. One surprising insight is that in all of these instances local solutions are actually “smooth” in the sense that they can be approximated locally to arbitrarily high degree as linear combinations of a fixed family of random functions/distributions that play the role of “polynomials” in the theory. As an example of a novel application, we solve the long-standing problem of building a natural Markov process that is symmetric with respect to the (finite volume) measure describing the $$\Phi ^4_3$$ Euclidean quantum field theory. It is natural to conjecture that the Markov process built in this way describes the Glauber dynamic of $$3$$ -dimensional ferromagnets near their critical temperature.

Multidimensional Stochastic Processes as Rough Paths

Abstract: Rough path analysis provides a fresh perspective on Ito's important theory of stochastic differential equations. Key theorems of modern stochastic analysis (existence and limit theorems for stochastic flows, Freidlin-Wentzell theory, the Stroock-Varadhan support description) can be obtained with dramatic simplifications. Classical approximation results and their limitations (Wong-Zakai, McShane's counterexample) receive 'obvious' rough path explanations. Evidence is building that rough paths will play an important role in the future analysis of stochastic partial differential equations and the authors include some first results in this direction. They also emphasize interactions with other parts of mathematics, including Caratheodory geometry, Dirichlet forms and Malliavin calculus. Based on successful courses at the graduate level, this up-to-date introduction presents the theory of rough paths and its applications to stochastic analysis. Examples, explanations and exercises make the book accessible to graduate students and researchers from a variety of fields.