arXiv subjects

Benjamin Jourdain

Publications and source records attributed to Benjamin Jourdain.

At least 55 records · Page 3Linked to original sources

arXiv2020

Strong solutions to a beta-Wishart particle system

The purpose of this paper is to study the existence and uniqueness of solutions to a Stochastic Differential Equation (SDE) coming from the eigenvalues of Wishart processes. The coordinates are non-negative, evolve as Cox-Ingersoll-Ross (CIR) processes and repulse each other according to a Coulombian like interaction force. We show the existence of strong and pathwise unique solutions to the system until the first multiple collision, and give a necessary and sufficient condition on the parameters of the SDE for this multiple collision not to occur in finite time.

math.PR↗

arXiv2019

Squared quadratic Wasserstein distance : optimal couplings and Lions differentiability

In this paper, we remark that any optimal coupling for the quadratic Wasserstein distance $W^2_2(μ,ν)$ between two probability measures $μ$ and $ν$ with finite second order moments on $\mathbb{R}^d$ is the composition of a martingale coupling with an optimal transport map ${\mathcal T}$. We check the existence of an optimal coupling in which this map gives the unique optimal coupling between $μ$ and ${\mathcal T}\#μ$. Next, we give a direct proof that $σ\mapsto W_2^2(σ,ν)$ is differentiable at $μ$ in the Lions sense iff there is a unique optimal coupling between $μ$ and $ν$ and this coupling is given by a map. It was known combining results by Ambrosio, Gigli and Savaré and Ambrosio and Gangbo that, under the latter condition, geometric differentiability holds. Moreover, the two notions of differentiability are equivalent according to the recent paper of Gangbo and Tudorascu. Besides, we give a self-contained probabilistic proof that mere Fréchet differentiability of a law invariant function $F$ on $L^2(Ω,\mathbb{P};\mathbb{R}^d)$ is enough for the Fréchet differential at $X$ to be a measurable function of $X$.

math.PR↗

arXiv2019

Convergence of metadynamics: discussion of the adiabatic hypothesis

By drawing a parallel between metadynamics and self interacting models for polymers, we study the longtime convergence of the original metadynamics algorithm in the adiabatic setting, namely when the dynamics along the collective variables decouples from the dynamics along the other degrees of freedom. We also discuss the bias which is introduced when the adiabatic assumption does not holds.

math.PR↗

arXiv2019

A new family of one dimensional martingale couplings

In this paper, we exhibit a new family of martingale couplings between two one-dimensional probability measures $μ$ and $ν$ in the convex order. This family is parametrised by two dimensional probability measures on the unit square with respective marginal densities proportional to the positive and negative parts of the difference between the quantile functions of $μ$ and $ν$. It contains the inverse transform martingale coupling which is explicit in terms of the associated cumulative distribution functions. The integral of $\vert x-y\vert$ with respect to each of these couplings is smaller than twice the $W_1$ distance between $μ$ and $ν$. When $μ$ and $ν$ are in the decreasing (resp. increasing) convex order, the construction is generalised to exhibit super (resp. sub) martingale couplings.

math.PR↗

arXiv2019

Sampling of probability measures in the convex order by Wasserstein projection

In this paper, for $μ$ and $ν$ two probability measures on $\mathbb{R}^d$ with finite moments of order $ρ\ge 1$, we define the respective projections for the $W_ρ$-Wasserstein distance of $μ$ and $ν$ on the sets of probability measures dominated by $ν$ and of probability measures larger than $μ$ in the convex order. The $W_2$-projection of $μ$ can be easily computed when $μ$ and $ν$ have finite support by solving a quadratic optimization problem with linear constraints. In dimension $d=1$, Gozlan et al.~(2018) have shown that the projections do not depend on $ρ$. We explicit their quantile functions in terms of those of $μ$ and $ν$. The motivation is the design of sampling techniques preserving the convex order in order to approximate Martingale Optimal Transport problems by using linear programming solvers. We prove convergence of the Wasserstein projection based sampling methods as the sample sizes tend to infinity and illustrate them by numerical experiments.

math.PR↗

arXiv2018

Non-asymptotic error bounds for The Multilevel Monte Carlo Euler method applied to SDEs with constant diffusion coefficient

In this paper, we are interested in deriving non-asymptotic error bounds for the multilevel Monte Carlo method. As a first step, we deal with the explicit Euler discretization of stochastic differential equations with a constant diffusion coefficient. We prove that, as long as the deviation is below an explicit threshold, a Gaussian-type concentration inequality optimal in terms of the variance holds for the multilevel estimator. To do so, we use the Clark-Ocone representation formula and derive bounds for the moment generating functions of the squared difference between a crude Euler scheme and a finer one and of the squared difference of their Malliavin derivatives.

math.PR↗

arXiv2018

Bias behaviour and antithetic sampling in mean-field particle approximations of SDEs nonlinear in the sense of McKean

In this paper, we prove that the weak error between a stochastic differential equation with nonlinearity in the sense of McKean given by moments and its approximation by the Euler discretization with time-step h of a system of N interacting particles is O(1/N + h). We provide numerical experiments confirming this behaviour and showing that it extends to more general mean-field interaction and study the efficiency of the antithetic sampling technique on the same examples.

math.PR↗

arXiv2017

Convergence and efficiency of adaptive importance sampling techniques with partial biasing

We consider a generalization of the discrete-time Self Healing Umbrella Sampling method, which is an adaptive importance technique useful to sample multimodal target distributions. The importance function is based on the weights (namely the relative probabilities) of disjoint sets which form a partition of the space. These weights are unknown but are learnt on the fly yielding an adaptive algorithm. In the context of computational statistical physics, the logarithm of these weights is, up to a multiplicative constant, the free energy, and the discrete valued function defining the partition is called the collective variable. The algorithm falls into the general class of Wang-Landau type methods, and is a generalization of the original Self Healing Umbrella Sampling method in two ways: (i) the updating strategy leads to a larger penalization strength of already visited sets in order to escape more quickly from metastable states, and (ii) the target distribution is biased using only a fraction of the free energy, in order to increase the effective sample size and reduce the variance of importance sampling estimators. The algorithm can also be seen as a generalization of well-tempered metadynamics. We prove the convergence of the algorithm and analyze numerically its efficiency on a toy example.

math.PR↗

arXiv2017

Existence of a calibrated regime switching local volatility model and new fake Brownian motions

By Gyongy's theorem, a local and stochastic volatility (LSV) model is calibrated to the market prices of all European call options with positive maturities and strikes if its local volatility function is equal to the ratio of the Dupire local volatility function over the root conditional mean square of the stochastic volatility factor given the spot value. This leads to a SDE nonlinear in the sense of McKean. Particle methods based on a kernel approximation of the conditional expectation, as presented by Guyon and Henry-Labordère (2011), provide an efficient calibration procedure even if some calibration errors may appear when the range of the stochastic volatility factor is very large. But so far, no global existence result is available for the SDE nonlinear in the sense of McKean. In the particular case where the local volatility function is equal to the inverse of the root conditional mean square of the stochastic volatility factor multiplied by the spot value given this value and the interest rate is zero, the solution to the SDE is a fake Brownian motion. When the stochastic volatility factor is a constant (over time) random variable taking finitely many values and the range of its square is not too large, we prove existence to the associated Fokker-Planck equation. Thanks to Figalli (2008), we then deduce existence of a new class of fake Brownian motions. We then extend these results to the special case of the LSV model called regime switching local volatility, where the stochastic volatility factor is a jump process taking finitely many values and with jump intensities depending on the spot level. Under the same condition on the range of its square, we prove existence to the associated Fokker-Planck PDE. Finally, we deduce existence of the calibrated model by extending the results in Figalli (2008).

math.PR↗

arXiv2016

Ninomiya-Victoir scheme : strong convergence properties and discretization of the involved Ordinary Differential Equations

In this paper, we summarize the results about the strong convergence rate of the Ninomiya-Victoir scheme and the stable convergence in law of its normalized error that we obtained in previous papers. We then recall the properties of the multilevel Monte Carlo estimators involving this scheme that we introduced and studied before. Last, we are interested in the error introduced by discretizing the ordinary differential equations involved in the Ninomiya-Victoir scheme. We prove that this error converges with strong order 2 when an explicit Runge-Kutta method with order 4 (resp. 2) is used for the ODEs corresponding to the Brownian (resp. Stratonovich drift) vector fields. We thus relax the order 5 for the Brownian ODEs needed by Ninomiya and Ninomiya (2009) to obtain the same order of strong convergence. Moreover, the properties of our multilevel Monte-Carlo estimators are preserved when these Runge-Kutta methods are used.

math.PR↗

arXiv2016

Evolution of the Wasserstein distance between the marginals of two Markov processes

In this paper, we are interested in the time derivative of the Wasserstein distance between the marginals of two Markov processes. As recalled in the introduction, the Kantorovich duality leads to a natural candidate for this derivative. Up to the sign, it is the sum of the integrals with respect to each of the two marginals of the corresponding generator applied to the corresponding Kantorovich potential. For pure jump processes with bounded intensity of jumps, we prove that the evolution of the Wasserstein distance is actually given by this candidate. In dimension one, we show that this remains true for Piecewise Deterministic Markov Processes. We apply the formula to estimate the exponential decrease rate of the Wasserstein distance between the marginals of two birth and death processes with the same generator in terms of the Wasserstein curvature.

math.PR↗

arXiv2016

Asymptotic error distribution for the Ninomiya-Victoir scheme in the commutative case

In a previous work, we proved strong convergence with order $1$ of the Ninomiya-Victoir scheme $X^{NV}$ with time step $T/N$ to the solution $X$ of the limiting SDE when the Brownian vector fields commute. In this paper, we prove that the normalized error process $N \left(X - X^{NV}\right)$ converges to an affine SDE with source terms involving the Lie brackets between the Brownian vector fields and the drift vector field. This result ensures that the strong convergence rate is actually $1$ when the Brownian vector fields commute, but at least one of them does not commute with the drift vector field. When all the vector fields commute the limit vanishes. Our result is consistent with the fact that the Ninomiya-Victoir scheme solves the SDE in this case.

math.PR↗

arXiv2016

Asymptotics for the normalized error of the Ninomiya-Victoir scheme

In a previous work, we proved strong convergence with order $1/2$ of the Ninomiya-Victoir scheme $X^{NV,η}$ with time step $T/N$ to the solution $X$ of the limiting SDE. In this paper we check that the normalized error defined by $\sqrt{N}\left(X - X^{NV,η}\right)$ converges to an affine SDE with source terms involving the Lie brackets between the Brownian vector fields. The limit does not depend on the Rademacher random variables $η$. This result can be seen as a first step to adapt to the Ninomiya-Victoir scheme the central limit theorem of Lindeberg Feller type, derived by M. Ben Alaya and A. Kebaier for the multilevel Monte Carlo estimator based on the Euler scheme. When the Brownian vector fields commute, the limit vanishes. This suggests that the rate of convergence is greater than $1/2$ in this case and we actually prove strong convergence with order $1$.

math.PR↗

arXiv2015

Optimal convergence rate of the multitype sticky particle approximation of one-dimensional diagonal hyperbolic systems with monotonic initial data

Brenier and Grenier [SIAM J. Numer. Anal., 1998] proved that sticky particle dynamics with a large number of particles allow to approximate the entropy solution to scalar one-dimensional conservation laws with monotonic initial data. In [arXiv:1501.01498], we introduced a multitype version of this dynamics and proved that the associated empirical cumulative distribution functions converge to the viscosity solution, in the sense of Bianchini and Bres-san [Ann. of Math. (2), 2005], of one-dimensional diagonal hyperbolic systems with monotonic initial data of arbitrary finite variation. In the present paper, we analyse the L 1 error of this approximation procedure, by splitting it into the discretisation error of the initial data and the non-entropicity error induced by the evolution of the particle system. We prove that the error at time t is bounded from above by a term of order (1 + t)/n, where n denotes the number of particles, and give an example showing that this rate is optimal. We last analyse the additional error introduced when replacing the multitype sticky particle dynamics by an iterative scheme based on the typewise sticky particle dynamics, and illustrate the convergence of this scheme by numerical simulations.

math.AP↗

arXiv2015

Ninomiya-Victoir scheme: strong convergence, antithetic version and application to multilevel estimators

In this paper, we are interested in the strong convergence properties of the Ninomiya-Victoir scheme which is known to exhibit weak convergence with order 2. We prove strong convergence with order $1/2$. This study is aimed at analysing the use of this scheme either at each level or only at the finest level of a multilevel Monte Carlo estimator: indeed, the variance of a multilevel Monte Carlo estimator is related to the strong error between the two schemes used on the coarse and fine grids at each level. Recently, Giles and Szpruch proposed a scheme permitting to construct a multilevel Monte Carlo estimator achieving the optimal complexity $O\left(ε^{-2}\right)$ for the precision $ε$. In the same spirit, we propose a modified Ninomiya-Victoir scheme, which may be strongly coupled with order $1$ to the Giles-Szpruch scheme at the finest level of a multilevel Monte Carlo estimator. Numerical experiments show that this choice improves the efficiency, since the order $2$ of weak convergence of the Ninomiya-Victoir scheme permits to reduce the number of discretization levels.

q-fin.CP↗

arXiv2015

Computation of sensitivities for the invariant measure of a parameter dependent diffusion

We consider the solution to a stochastic differential equation with a drift function which depends smoothly on some real parameter $λ$, and admitting a unique invariant measure for any value of $λ$ around $λ$ = 0. Our aim is to compute the derivative with respect to $λ$ of averages with respect to the invariant measure, at $λ$ = 0. We analyze a numerical method which consists in simulating the process at $λ$ = 0 together with its derivative with respect to $λ$ on long time horizon. We give sufficient conditions implying uniform-in-time square integrability of this derivative. This allows in particular to compute efficiently the derivative with respect to $λ$ of the mean of an observable through Monte Carlo simulations.

math.PR↗

arXiv2015

A multitype sticky particle construction of Wasserstein stable semigroups solving one-dimensional diagonal hyperbolic systems with large monotonic data

This article is dedicated to the study of diagonal hyperbolic systems in one space dimension, with cumulative distribution functions, or more generally nonconstant monotonic bounded functions, as initial data. Under a uniform strict hyperbolicity assumption on the characteristic fields, we construct a multitype version of the sticky particle dynamics and obtain existence of global weak solutions by compactness. We then derive a $L^p$ stability estimate on the particle system uniform in the number of particles. This allows to construct nonlinear semigroups solving the system in the sense of Bianchini and Bressan [Ann. of Math. (2), 2005]. We also obtain that these semigroup solutions satisfy a stability estimate in Wasserstein distances of all orders, which encompasses the classical $L^1$ estimate and generalises to diagonal systems the results by Bolley, Brenier and Loeper [J. Hyperbolic Differ. Equ., 2005] in the scalar case. Our results are obtained without any smallness assumption on the variation of the data, and only require the characteristic fields to be Lipschitz continuous and the system to be uniformly strictly hyperbolic.

math.AP↗

arXiv2015

Stochastic particle approximation of the Keller-Segel equation and two-dimensional generalization of Bessel processes

The Keller-Segel partial differential equation is a two-dimensional model for chemotaxis. When the total mass of the initial density is one, it is known to exhibit blow-up in finite time as soon as the sensitivity $χ$ of bacteria to the chemo-attractant is larger than $8π$. We investigate its approximation by a system of $N$ two-dimensional Brownian particles interacting through a singular attractive kernel in the drift term. In the very subcritical case $χ\textless{}2π$, the diffusion strongly dominates this singular drift: we obtain existence for the particle system and prove that its flow of empirical measures converges, as $N\to\infty$ and up to extraction of a subsequence, to a weak solution of the Keller-Segel equation. We also show that for any $N\ge 2$ and any value of $χ\textgreater{}0$, pairs of particles do collide with positive probability: the singularity of the drift is indeed visited. Nevertheless, when $χ\textless{}2πN$, it is possible to control the drift and obtain existence of the particle system until the first time when at least three particles collide. We check that this time is a.s. infinite, so that global existence holds for the particle system, if and only if $χ\leq 8π(N-2)/(N-1)$. Finally, we remark that in the system with $N=2$ particles, the difference between the two positions provides a natural two-dimensional generalization of Bessel processes, which we study in details.

math.PR↗