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Merener N.,Torcuato Di Tella University | Vicchi L.,Instituto Nacional Of Matematica Pura E Aplicada
Journal of Computational Finance | Year: 2015

We develop an efficient Monte Carlo method for the valuation of financial contracts on discretely realized variance. We work with a general stochastic volatility model that makes realized variance dependent on the full path of the asset price. The variance contract price is a high-dimensional integral over the fundamental sources of randomness. We identify a two-dimensional manifold that drives most of the uncertainty in realized variance, and we compute the contract price by combining precise integration over this manifold, implemented as fine stratification or deterministic sampling with quasirandom numbers, with conditional Monte Carlo on the remaining dimensions. For a subclass of models and a class of nonlinear payoffs, we derive approximate theoretical results that quantify the variance reduction achieved by our method. Numerical tests for the discretized versions of the widely used Hull–White and Heston models show that the algorithm performs significantly better than a standard Monte Carlo, even for fixed computational budgets. © 2015 Incisive Risk Information (IP) Limited.


Goncalves P.,Pontifical Catholic University of Rio de Janeiro | Goncalves P.,University of Minho | Jara M.,Instituto Nacional Of Matematica Pura E Aplicada | Jara M.,University of Paris Dauphine
Archive for Rational Mechanics and Analysis | Year: 2014

We introduce what we call the second-order Boltzmann-Gibbs principle, which allows one to replace local functionals of a conservative, one-dimensional stochastic process by a possibly nonlinear function of the conserved quantity. This replacement opens the way to obtain nonlinear stochastic evolutions as the limit of the fluctuations of the conserved quantity around stationary states. As an application of this second-order Boltzmann-Gibbs principle, we introduce the notion of energy solutions of the KPZ and stochastic Burgers equations. Under minimal assumptions, we prove that the density fluctuations of one-dimensional, stationary, weakly asymmetric, conservative particle systems are sequentially compact and that any limit point is given by energy solutions of the stochastic Burgers equation. We also show that the fluctuations of the height function associated to these models are given by energy solutions of the KPZ equation in this sense. Unfortunately, we lack a uniqueness result for these energy solutions. We conjecture these solutions to be unique, and we show some regularity results for energy solutions of the KPZ/Burgers equation, supporting this conjecture. © 2013 Springer-Verlag Berlin Heidelberg.


Ashoori E.,Technical University of Delft | Marchesin D.,Instituto Nacional Of Matematica Pura E Aplicada | Rossen W.R.,Technical University of Delft
Colloids and Surfaces A: Physicochemical and Engineering Aspects | Year: 2011

In foam EOR, complex dynamics of bubble creation and destruction controls foam properties. Here we reconsider whether and when non-equilibrium effects are important, focusing specifically on the entrance region, where injected gas and liquid are transformed into foam. We solve for water saturation and foam texture in the entrance region using the population-balance foam model of Kam (2008), which features three steady states (no foam, strong foam, and an unstable intermediate state) at some injection rates, as seen in experiments. We derive and solve equations for water saturation and foam properties along the entrance region at steady state. Mathematical conditions on the entrance region itself can control which of the several possible steady states is ultimately taken downstream by foam. For instance, if foam is not pre-generated, and capillary-pressure gradients are neglected, as in many published simulation studies, the final steady state downstream is the one with highest water saturation - the weakest foam. Simulations neglecting capillary pressure therefore may lead to inference of the wrong foam state in the formation or core. In some cases, in the presence of capillary pressure, analysis of the asymptotic dynamic behavior in the vicinity of possible downstream steady states may rule out some possible steady states. We show that the apparent length of entrance region can be quite different if one measures water saturation or pressure gradient. Finally, we fit foam kinetic parameters to the length of the entrance region seen in some experiments; a companion paper [18] investigates the effect of these parameters on the traveling wave at the shock front downstream. © 2011 Elsevier B.V.


Ashoori E.,Technical University of Delft | Marchesin D.,Instituto Nacional Of Matematica Pura E Aplicada | Rossen W.R.,Technical University of Delft
Colloids and Surfaces A: Physicochemical and Engineering Aspects | Year: 2011

In foam EOR, complex dynamics of bubble creation and destruction controls foam properties. Recently, there has been consensus that local equilibrium between bubble creation and destruction adequately describes foam displacements. We assume that local equilibrium applies throughout a foam displacement on the field scale, with the exception of an entrance region and at shock fronts, where saturations and texture (bubble size) change abruptly. We find a range of conditions in which the local-equilibrium condition applies even within the shock front. In a waterflood, the width of a shock transition zone is determined by capillary-pressure gradients. For foam, this equation is joined by one for evolving foam texture. One expects that slow foam dynamics widens the traveling wave at the shock considerably. Theory and simulations show that the width of and mobility inside a shock front can affect foam sweep. If there is no gas ahead of the foam, as is common in published simulations, we prove that foam texture is everywhere at local equilibrium within the shock, regardless of the foam model, as previously observed for one dynamic foam model. If there is gas initially in the formation, slow foam generation and coalescence processes can actually narrow the shock from that assuming local equilibrium. In other cases, the dynamics of the traveling wave leads to oscillations near the shock; these are not numerical artifacts, but reflections of the models. Multiple steady states seen in experiment for some injection rates can be predicted by certain foam models. The approach of solving for the traveling wave can rule out some of these states for certain displacements. © 2011 Elsevier B.V.


Mailybaev A.A.,Moscow State University | Bruining J.,Technical University of Delft | Marchesin D.,Instituto Nacional Of Matematica Pura E Aplicada
Combustion and Flame | Year: 2011

We study one-dimensional flows, when air is injected into a porous medium filled with inert gas, medium or high viscosity oil and water, giving rise to a combustion wave in a process known as high-temperature oxidation (HTO). In the oil we distinguish three pseudo-components: asphaltenes, medium and light oil. At high temperatures, the heaviest components (" precoke" ) are converted to coke, which undergoes combustion. Medium oil components are cracked at intermediate temperatures releasing gaseous oil. Light oil components and water are vaporized. The oxidation rate of gaseous oil components is negligible. Combustion regimes are described in the form of a sequence of waves. We develop a simple mathematical pathway based on Zeldovich's approach to provide analytical formulae for parameters in these waves. It is shown that there is a combustion regime in which either coke or oxygen are partially consumed in the combustion as well as a regime in which both are consumed completely. Each of the regimes can be subdivided in two regimes, where the reaction is either trailing or leading with respect to the thermal wave. Explicit conditions for each combustion regime are given. The structure of the oil cracking layer is investigated. Stability of the solutions is studied. We analyse our formulae for typical in situ combustion data and compare the results with numerical simulations. © 2010 The Combustion Institute.


Reaiche M.M.C.R.,Instituto Nacional Of Matematica Pura E Aplicada
Operations Research Letters | Year: 2016

We derive a lower bound for the sample complexity of the Sample Average Approximation method for a certain class of multistage stochastic optimization problems. In previous works, upper bounds for such problems were derived. We show that the dependence of the lower bound with respect to the complexity parameters and the problem's data are comparable to the upper bound's estimates. Like previous results, our lower bound presents an additional multiplicative factor showing that it is unavoidable for certain stochastic problems. © 2016 Elsevier B.V. All rights reserved.


Ashoori E.,Technical University of Delft | Marchesin D.,Instituto Nacional Of Matematica Pura E Aplicada | Rossen W.R.,Technical University of Delft
Transport in Porous Media | Year: 2012

The use of foam for mobility control is a promising mean to improve sweep efficiency in EOR. Experimental studies discovered that foam exhibits three different states (weak foam, intermediate foam, and strong foam). The intermediate-foam state is found to be unstable in the lab whereas the weak- and strong-foam states are stable. The model of Kam (Colloids Surf A Physicochem Eng Asp 318(1-3): 62-77, 2008) is the only mechanistic foam model that can fit a variety of steady-state experimental data including multiple steady states. This model is modified from a previous mechanistic foam model to resolve the intrinsic instability of the strong-foam state. Simple finite-difference simulations have found that an arbitrary perturbation grows for the unstable intermediate foam but diminishes for the strong- and weak-foam states. The issue of the stability of foam states, especially the strong-foam state, is a serious concern in application of foam in EOR. Instabilities may rule out one or more states and consequently have considerable effect on reservoir sweep efficiency and injection pressure. Here, for the first time the stability of the various equilibrium foam states is investigated by an analytical stability-analysis method together with numerical simulations. We demonstrate the instability of most intermediate states, consistent with the laboratory observations. However, our analysis reveals an instability of the strong-foam state. We show that the diffusion, whether introduced artificially by the finite-difference scheme or representing physical dispersion, damps this instability. We obtain good agreement with finite-element simulations with and without additional diffusion. We also prove that all states are unconditionally stable for a local-equilibrium-foam model. © 2011 The Author(s).


Morales J.A.C.,Instituto Nacional Of Matematica Pura E Aplicada | Zilber B.,University of Oxford
Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences | Year: 2015

In this paper, we will present an ongoing project that aims to use model theory as a suitable mathematical setting for studying the formalism of quantum mechanics. We argue that this approach provides a geometric semantics for such a formalism by means of establishing a (non-commutative) duality between certain algebraic and geometric objects. © 2015 The Author(s) Published by the Royal Society. All rights reserved.


Jimenez J.C.,Institute Cibernetica | de la Cruz Cancino H.,Instituto Nacional Of Matematica Pura E Aplicada
BIT Numerical Mathematics | Year: 2012

There is a variety of strong Local Linearization (LL) schemes for the numerical integration of stochastic differential equations with additive noise, which differ with respect to the algorithm that is used in the numerical implementation of the strong Local Linear discretization. However, in contrast with the Local Linear discretization, the convergence rate of the LL schemes has not been studied so far. In this paper, two general theorems about this matter are presented and, with their support, additional results are derived for some particular schemes. As a direct application, the convergence rate of some strong LL schemes for SDEs with jumps is briefly expounded as well. © 2011 Springer Science + Business Media B.V.


Oliveira R.I.,Instituto Nacional Of Matematica Pura E Aplicada
IEEE Transactions on Information Theory | Year: 2015

This paper introduces the concept of random context representations for the transition probabilities of a finite-alphabet stochastic process. Processes with these representations generalize context tree processes (also known as variable length Markov chains), and are proved to coincide with processes whose transition probabilities are almost surely continuous functions of the (infinite) past. This is similar to a classical result by Kalikow about continuous transition probabilities. Existence and uniqueness of a minimal random context representation are shown, in the sense that there exists a unique representation that looks into the past as little as possible in order to determine the next symbol. Both this representation and the transition probabilities can be consistently estimated from data, and some finite sample adaptivity properties are also obtained (including an oracle inequality). In particular, the estimator achieves minimax performance, up to logarithmic factors, for the class of binary renewal processes whose arrival distributions have bounded moments of order 2 + γ. © 2015 IEEE.

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