Results 1  10
of
40
A quantization algorithm for solving multidimensional Optimal Stopping problems
 Bernoulli
, 2001
"... A new grid method for computing the Snell envelop of a function of a R valued Markov chain (X k ) 0#k#n is proposed. (This problem is typically non linear and cannot be solved by the standard Monte Carlo method.) Every X k is replaced by a "quantized approximation" X k taking its values in a gr ..."
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Cited by 33 (2 self)
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A new grid method for computing the Snell envelop of a function of a R valued Markov chain (X k ) 0#k#n is proposed. (This problem is typically non linear and cannot be solved by the standard Monte Carlo method.) Every X k is replaced by a "quantized approximation" X k taking its values in a grid # k of size N k . The n grids and their transition probability matrices make up a discrete tree on which a pseudoSnell envelop is devised by mimicking the regular dynamic programming formula. We show, using Quantization Theory of probability distributions the existence of a set of optimal grids, given the total number N of elementary R valued vector quantizers. A recursive stochastic algorithm, based on some simulations of (X k ) 0#k#n , yields the optimal grids and their transition probability matrices. Some a priori error estimates based on the quantization errors are established. These results are applied to the computation of the Snell envelop of a di#usion (assuming that it can be directly simulated or using its Euler scheme). We show how this approach yields a discretization method for Reflected Backward Stochastic Di#erential Equation. Finally, some first numerical tests are carried out on a 2dimensional American option pricing problem.
Weighted norm inequalities, offdiagonal estimates and elliptic operators, Part II: Offdiagonal estimates on spaces of homogeneous type
, 2005
"... Abstract. This is the fourth article of our series. Here, we apply the results of [AM1] to study weighted norm inequalities for the Riesz transform of the LaplaceBeltrami operator on Riemannian manifolds and of subelliptic sum of squares on Lie groups, under the doubling volume property and Poincar ..."
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Cited by 23 (6 self)
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Abstract. This is the fourth article of our series. Here, we apply the results of [AM1] to study weighted norm inequalities for the Riesz transform of the LaplaceBeltrami operator on Riemannian manifolds and of subelliptic sum of squares on Lie groups, under the doubling volume property and Poincaré inequalities. 1. Introduction and
Schauder estimates, Harnack inequality and Gaussian lower bound for Kolmogorov type operators in nondivergence form
 Advances in Differential Equations
, 2006
"... We prove some Schauder type estimates and an invariant Harnack inequality for a class of degenerate evolution operators of Kolmogorov type. We also prove a Gaussian lower bound for the fundamental solution of the operator and a uniqueness result for the Cauchy problem. The proof of the lower bound i ..."
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Cited by 12 (4 self)
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We prove some Schauder type estimates and an invariant Harnack inequality for a class of degenerate evolution operators of Kolmogorov type. We also prove a Gaussian lower bound for the fundamental solution of the operator and a uniqueness result for the Cauchy problem. The proof of the lower bound is obtained by solving a suitable optimal control problem and using the invariant Harnack inequality.
Probabilistic interpretation and random walk on spheres algorithms for the PoissonBoltzmann equation in molecular dynamics
, 2010
"... Abstract. Motivated by the development of efficient Monte Carlo methods for PDE models in molecular dynamics, we establish a new probabilistic interpretation of a family of divergence form operators with discontinuous coefficients at the interface of two open subsets of R d. This family of operators ..."
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Cited by 10 (5 self)
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Abstract. Motivated by the development of efficient Monte Carlo methods for PDE models in molecular dynamics, we establish a new probabilistic interpretation of a family of divergence form operators with discontinuous coefficients at the interface of two open subsets of R d. This family of operators includes the case of the linearized PoissonBoltzmann equation used to compute the electrostatic free energy of a molecule. More precisely, we explicitly construct a Markov process whose infinitesimal generator belongs to this family, as the solution of a SDE including a non standard local time term related to the interface of discontinuity. We then prove an extended FeynmanKac formula for the PoissonBoltzmann equation. This formula allows us to justify various probabilistic numerical methods to approximate the free energy of a molecule. We analyse the convergence rate of these simulation procedures and numerically compare them on idealized molecules models. Résumé. Motivés par le développement de méthodes de MonteCarlo efficaces pour des équations aux dérivées partielles en dynamique moléculaire, nous établissons une nouvelle interprétation probabiliste d’une famille d’opérateurs sous forme divergence et à coefficients discontinus le long de l’interface de deux ouverts de R d. Cette famille d’opérateurs inclut le cas de l’équation de PoissonBoltzmann
An invariance principle for diffusion in turbulence
 ANN. OF PROB
, 1999
"... We prove an almost sure invariance principle for diffusion driven by velocities with unbounded stationary vector potentials. The result generalizes to multiple particles motion, driven by a common velocity field and independent molecular Brownian motions. ..."
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Cited by 8 (6 self)
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We prove an almost sure invariance principle for diffusion driven by velocities with unbounded stationary vector potentials. The result generalizes to multiple particles motion, driven by a common velocity field and independent molecular Brownian motions.
Vessella Doubling properties of caloric functions
"... ”Science is the great antidote to the poison of enthusiasm and superstition ” (Adam Smith, ..."
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Cited by 8 (2 self)
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”Science is the great antidote to the poison of enthusiasm and superstition ” (Adam Smith,
AUXILIARY SDES FOR HOMOGENIZATION OF QUASILINEAR PDES WITH PERIODIC COEFFICIENTS
, 2004
"... We study the homogenization property of systems of quasilinear PDEs of parabolic type with periodic coefficients, highly oscillating drift and highly oscillating nonlinear term. To this end, we propose a probabilistic approach based on the theory of forward–backward stochastic differential equation ..."
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Cited by 7 (1 self)
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We study the homogenization property of systems of quasilinear PDEs of parabolic type with periodic coefficients, highly oscillating drift and highly oscillating nonlinear term. To this end, we propose a probabilistic approach based on the theory of forward–backward stochastic differential equations and introduce the new concept of “auxiliary SDEs.” 1. Introduction and
A Gaussian upper bound for the fundamental solutions of a class of ultraparabolic equations
, 2003
"... We prove Gaussian estimates from above of the fundamental solutions to a class of ultraparabolic equations. These estimates are independent of the modulus of continuity of the coefficients and generalize the classical upper bounds by Aronson for uniformly parabolic equations. ..."
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Cited by 5 (5 self)
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We prove Gaussian estimates from above of the fundamental solutions to a class of ultraparabolic equations. These estimates are independent of the modulus of continuity of the coefficients and generalize the classical upper bounds by Aronson for uniformly parabolic equations.
On uniformly subelliptic operators and stochastic area
, 2006
"... We consider uniformly subelliptic operators on certain unimodular Lie groups of polynomial growth. It was shown by SaloffCoste and Stroock that classical results of De Giorgi, Nash, Moser, Aronson extend to this setting. It was then observed by Sturm that many proofs extend naturally to the setting ..."
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Cited by 5 (4 self)
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We consider uniformly subelliptic operators on certain unimodular Lie groups of polynomial growth. It was shown by SaloffCoste and Stroock that classical results of De Giorgi, Nash, Moser, Aronson extend to this setting. It was then observed by Sturm that many proofs extend naturally to the setting of locally compact Dirichlet spaces. We relate these results to what is known as rough path theory by showing that they provide a natural and powerful analytic machinery for construction and study of (random) geometric Hölder rough paths. (In particular, we obtain a simple construction of the LyonsStoica stochastic area for a diffusion process with uniformly elliptic generator in divergence form.) Our approach then enables us to establish a number of farreaching generalizations of classical theorems in diffusion theory including WongZakai approximations, FreidlinWentzell sample path large deviations and the StroockVaradhan support theorem. The latter was conjectured by T. Lyons in his recent St. Flour lecture. 1