Sara Mazzonetto

Bio: Sara Mazzonetto is an academic researcher from University of Lorraine. The author has contributed to research in topics: Stochastic differential equation & Brownian motion. The author has an hindex of 5, co-authored 13 publications receiving 63 citations. Previous affiliations of Sara Mazzonetto include University of Potsdam & University of Duisburg-Essen.

An explicit representation of the transition densities of the skew Brownian motion with drift and two semipermeable barriers

Abstract: In this paper we obtain an explicit representation of the transition density of the one-dimensional skew Brownian motion with (a constant drift and) two semipermeable barriers. Moreover we propose a rejection method to simulate this density in an exact way.

Exact simulation of Brownian diffusions with drift admitting jumps

Abstract: In this paper, using an algorithm based on the retrospective rejection sampling scheme introduced in [A. Beskos, O. Papaspiliopoulos, and G. O. Roberts, Methodol. Comput. Appl. Probab., 10 (2008), pp. 85--104] and [P. Etore and M. Martinez, ESAIM Probab. Stat., 18 (2014), pp. 686--702], we propose an exact simulation of a Brownian diffusion whose drift admits several jumps. We treat explicitly and extensively the case of two jumps, providing numerical simulations. Our main contribution is to manage the technical difficulty due to the presence of two jumps thanks to a new explicit expression of the transition density of the skew Brownian motion with two semipermeable barriers and a constant drift.

A stochastic Gronwall inequality and applications to moments, strong completeness, strong local Lipschitz continuity, and perturbations

Abstract: There are numerous applications of the classical (deterministic) Gronwall inequality. Recently, Michael Scheutzow discovered a stochastic Gronwall inequality which provides upper bounds for p-th moments, p∈(0,1), of the supremum of nonnegative scalar continuous processes which satisfy a linear integral inequality. In this article we complement this with upper bounds for p-th moments, p∈[2,∞), of the supremum of general Ito processes which satisfy a suitable one-sided affine-linear growth condition. As example applications, we improve known results on strong local Lipschitz continuity in the starting point of solutions of stochastic differential equations (SDEs), on (exponential) moment estimates for SDEs, on strong completeness of SDEs, and on perturbation estimates for SDEs.

A stochastic Gronwall inequality and applications to moments, strong completeness, strong local Lipschitz continuity, and perturbations.

Abstract: There are numerous applications of the classical (deterministic) Gronwall inequality. Recently, Michael Scheutzow discovered a stochastic Gronwall inequality which provides upper bounds for $p$-th moments, $p\in(0,1)$, of the supremum of nonnegative scalar continuous processes which satisfy a linear integral inequality. In this article we complement this with upper bounds for $p$-th moments, $p\in[2,\infty)$, of the supremum of general Ito processes which satisfy a suitable one-sided affine-linear growth condition. As example applications, we improve known results on strong local Lipschitz continuity in the starting point of solutions of stochastic differential equations (SDEs), on (exponential) moment estimates for SDEs, on strong completeness of SDEs, and on perturbation estimates for SDEs.

Existence and uniqueness properties for solutions of a class of Banach space valued evolution equations

Abstract: In this note we provide a self-contained proof of an existence and uniqueness result for a class of Banach space valued evolution equations with an additive forcing term. The framework of our abstract result includes, for example, finite dimensional ordinary differential equations (ODEs), semilinear deterministic partial differential equations (PDEs), as well as certain additive noise driven stochastic partial differential equations (SPDEs) as special cases. The framework of our general result assumes somehow mild regularity conditions on the involved semigroup and also allows the involved semigroup operators to be nonlinear. The techniques used in the proofs of our results are essentially well-known in the relevant literature. The contribution of this note is to provide a rather general existence and uniqueness result which covers several situations as special cases and also to provide a self-contained proof for this existence and uniqueness result.

A First Course in the Numerical Analysis of Differential Equations: Stiff equations

Abstract: Cambridge University Press. Paperback. Book Condition: New. Paperback. 480 pages. Numerical analysis presents different faces to the world. For mathematicians it is a bona fide mathematical theory with an applicable flavour. For scientists and engineers it is a practical, applied subject, part of the standard repertoire of modelling techniques. For computer scientists it is a theory on the interplay of computer architecture and algorithms for real-number calculations. The tension between these standpoints is the driving force of this book, which presents a rigorous account of the fundamentals of numerical analysis of both ordinary and partial differential equations. The exposition maintains a balance between theoretical, algorithmic and applied aspects. This new edition has been extensively updated, and includes new chapters on emerging subject areas: geometric numerical integration, spectral methods and conjugate gradients. Other topics covered include multistep and Runge-Kutta methods; finite difference and finite elements techniques for the Poisson equation; and a variety of algorithms to solve large, sparse algebraic systems. This item ships from multiple locations. Your book may arrive from Roseburg,OR, La Vergne,TN. Paperback.

Generalised multilevel Picard approximations

Abstract: It is one of the most challenging problems in applied mathematics to approximatively solve high-dimensional partial differential equations (PDEs). In particular, most of the numerical approximation schemes studied in the scientific literature suffer under the curse of dimensionality in the sense that the number of computational operations needed to compute an approximation with an error of size at most $ \varepsilon > 0 $ grows at least exponentially in the PDE dimension $ d \in \mathbb{N} $ or in the reciprocal of $ \varepsilon $. Recently, so-called full-history recursive multilevel Picard (MLP) approximation methods have been introduced to tackle the problem of approximately solving high-dimensional PDEs. MLP approximation methods currently are, to the best of our knowledge, the only methods for parabolic semi-linear PDEs with general time horizons and general initial conditions for which there is a rigorous proof that they are indeed able to beat the curse of dimensionality. The main purpose of this work is to investigate MLP approximation methods in more depth, to reveal more clearly how these methods can overcome the curse of dimensionality, and to propose a generalised class of MLP approximation schemes, which covers previously analysed MLP approximation schemes as special cases. In particular, we develop an abstract framework in which this class of generalised MLP approximations can be formulated and analysed and, thereafter, apply this abstract framework to derive a computational complexity result for suitable MLP approximations for semi-linear heat equations. These resulting MLP approximations for semi-linear heat equations essentially are generalisations of previously introduced MLP approximations for semi-linear heat equations.

Extreme at-the-money skew in a local volatility model

Abstract: We consider a local volatility model, with volatility taking two possible values, depending on the value of the underlying with respect to a fixed threshold. When the threshold is taken at the money, we establish exact pricing formulas for European call options and compute short-time asymptotics of the implied volatility surface. We derive an exact formula for the at-the-money implied volatility skew which explodes as $T^{-1/2}$ , reproducing the empirical steep short end of the smile. This behaviour is a consequence of the singularity of the local volatility at the money. Finally, we look at continuous, non-differentiable versions of such a model. We still find, in simulations, exploding implied skews.