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Arturo Kohatsu-Higa

Publications and source records attributed to Arturo Kohatsu-Higa.

At least 19 recordsLinked to original sources
arXiv2026

Ergodic approximation for the invariant distribution: An abstract framework for law-dependent dynamics

This paper studies the approximation of invariant distributions for a broad class of law-dependent dynamics, including McKean-Vlasov stochastic differential equations and Boltzmann-type equations. We consider discrete-time approximation schemes with decreasing time steps and analyse the convergence of their associated ergodic (or occupation) measure towards the invariant distribution of the underlying continuous-time process. Under a general coupling assumption, we prove convergence in the expected $p$-Wasserstein distance ($p \ge 1$) and derive explicit convergence rates. Our approach combines estimates on ergodic averages, regularization techniques for discrete measures, and a generalized discrete Gronwall lemma to control the error between the self-interacting scheme and the target invariant measure. We show that our framework applies to a wide range of models: McKean-Vlasov SDEs, a Boltzmann type equation, and a neuronal model.

math.PR↗
arXiv2026

Euler-type approximation for the invariant measure: An abstract framework

We establish a general framework to study the rate of convergence of a Euler type approximation scheme with decreasing time steps to the invariant measure, for a general class of stochastic systems. The error is measured in general Wasserstein distances, which enables to encompass cases with non global contractivity conditions. Our main assumption is a coupling property which is expressed in terms of the one-step approximation. We show that the proposed set-up can be applied to a wide range of equations that may be law dependent, such as Langevin equations, reflected equations, Boltzmann type equations and for a recent McKean Vlasov type model for neuronal activity.

math.PR↗
arXiv2026

Representation and Integration by Parts Formulas for Affine Processes

Affine processes play an important role in mathematical finance and other applied areas due to their tractable structure. In the present article, we derive probabilistic representations and integration by parts (IBP) formulas for expectations involving affine processes. These formulas are expressed in terms of expectations of affine processes with modified parameters and are derived using Fourier analytic techniques and characteristic functions. Notably, our method does not require pathwise differentiability, allowing us to handle models with square-root diffusion coefficients for a large set of parameters. The methodology can be applied to the classic Cox--Ingersoll--Ross (CIR) model, a model for interest rates in mathematical finance, where the initial value derivative corresponds to one of the ``Greeks'' used in option pricing in mathematical finance. Furthermore, we illustrate the theory with an application to a population evolution model arising as a scaling limit of discrete branching processes. Our approach offers a unified and robust framework for sensitivity analysis in models where classical Malliavin calculus techniques are difficult to apply.

math.PR↗
arXiv2024

A probabilistic representation of the derivative of a one dimensional killed diffusion semigroup

We provide a probabilistic representation for the derivative of the semigroup corresponding to a diffusion process killed at the boundary of a half interval. In particular, we show that the derivative of the semi-group can be expressed as the expected value of a functional of a reflected diffusion process. Furthermore, as an application, we obtain a Bismut-Elworthy-Li formula which is also valid at the boundary.

math.PR↗
arXiv2024

Probabilistic representation of the gradient of a killed diffusion semigroup: The half-space case

We introduce a probabilistic representation of the derivative of the semigroup associated to a multidimensional killed diffusion process defined on the half-space. The semigroup derivative is expressed as a functional of a process that is normally reflected when it hits the hyperplane. The representation of the derivative also involves a matrix-valued process which replaces the Jacobian of the underlying process that appears in the traditional pathwise derivative of a classical diffusion. The components of this matrix-valued process become zero except for those on the first row every time the reflected process touches the boundary. The results in this paper extend those in recent work of the authors, where the one-dimensional case was studied.

math.PR↗
arXiv2023

Integration by parts formula for exit times of one dimensional diffusions

In line with the methodology introduced in our recent article for formulating probabilistic representations of integration by parts involving killed diffusion, we establish an integration by parts formula for the first exit time of one-dimensional diffusion processes. However, our approach diverges from the conventional differential calculus applied to the associated space Markov chain; instead, we employ calculus techniques that focus on the underlying time variables.

math.PR↗
arXiv2022

Simulation of Reflected Brownian motion on two dimensional wedges

We study a correlated Brownian motion in two dimensions, which is reflected, stopped or killed in a wedge represented as the intersection of two half spaces. First, we provide explicit density formulas, hinted by the method of images. These explicit expressions rely on infinite oscillating sums of Bessel functions and may demand computationally costly procedures. We propose suitable recursive algorithms for the simulation of the laws of reflected and stopped Brownian motion which are based on generalizations of the reflection principle in two dimensions. We study and give bounds for the complexity of the proposed algorithms.

math.PR↗
arXiv2021

Density estimates for jump diffusion processes

We consider a real-valued diffusion process with a linear jump term driven by a Poisson point process and we assume that the jump amplitudes have a centered density with finite moments. We show upper and lower estimates for the density of the solution in the case that the jump amplitudes follow a Gaussian or Laplacian law. The proof of the lower bound uses a general expression for the density of the solution in terms of the convolution of the density of the continuous part and the jump amplitude density. The upper bound uses an upper tail estimate in terms of the jump amplitude distribution and techniques of the Malliavin calculus in order to bound the density by the tails of the solution. We also extend the lower bounds to the multidimensional case.

math.PR↗
arXiv2019

Integration by parts formula for killed processes: A point of view from approximation theory

In this paper, we establish a probabilistic representation for two integration by parts formulas, one being of Bismut-Elworthy-Li's type, for the marginal law of a one-dimensional diffusion process killed at a given level. These formulas are established by combining a Markovian perturbation argument with a tailor-made Malliavin calculus for the underlying Markov chain structure involved in the probabilistic representation of the original marginal law. Among other applications, an unbiased Monte Carlo path simulation method for both integration by parts formula stems from the previous probabilistic representations.

math.PR↗
arXiv2016

LAN property for an ergodic diffusion with jumps

In this paper, we consider a multidimensional ergodic diffusion with jumps driven by a Brownian motion and a Poisson random measure associated with a pure-jump Lévy process with finite Lévy measure, whose drift coefficient depends on an unknown parameter. Considering the process discretely observed at high frequency, we derive the local asymptotic normality (LAN) property.

math.PR↗
arXiv2016

On the first hitting times of one dimensional elliptic diffusions

In this article, we obtain properties of the law associated to the first hitting time of a threshold by a one-dimensional uniformly elliptic diffusion process and to the associated process stopped at the threshold. Our methodology relies on the parametrix method that we apply to the associated Markov semigroup. It allows to obtain explicit expressions for the corresponding transition densities and to study its regularity properties up to the boundary under mild assumptions on the coefficients. As a by product, we also provide Gaussian upper estimates for these laws and derive a probabilistic representation that may be useful for the construction of an unbiased Monte Carlo path simulation method, among other applications.

math.PR↗
arXiv2015

A probabilistic interpretation of the parametrix method

In this article, we introduce the parametrix technique in order to construct fundamental solutions as a general method based on semigroups and their generators. This leads to a probabilistic interpretation of the parametrix method that is amenable to Monte Carlo simulation. We consider the explicit examples of continuous diffusions and jump driven stochastic differential equations with Hölder continuous coefficients.

math.PR↗
arXiv2015

Optimal transport bounds between the time-marginals of a multidimensional diffusion and its Euler scheme

In this paper, we prove that the time supremum of the Wasserstein distance between the time-marginals of a uniformly elliptic multidimensional diffusion with coefficients bounded together with their derivatives up to the order $2$ in the spatial variables and H{ö}lder continuous with exponent $γ$ with respect to the time variable and its Euler scheme with $N$ uniform time-steps is smaller than $C \left(1+\mathbf{1}\_{γ=1} \sqrt{\ln(N)}\right)N^{-γ}$. To do so, we use the theory of optimal transport. More precisely, we investigate how to apply the theory by Ambrosio, Gigli and Savar{é} to compute the time derivative of the Wasserstein distance between the time-marginals. We deduce a stability inequality for the Wasserstein distance which finally leads to the desired estimation.

math.PR↗
arXiv2014

LAN property for a linear model with jumps

In this paper, we consider a linear model with jumps driven by a Brownian motion and a compensated Poisson process, whose drift and diffusion coefficients as well as its intensity are unknown parameters. Supposing that the process is observed discretely at high frequency we derive the local asymptotic normality (LAN) property. In order to obtain this result, Malliavin calculus and Girsanov's theorem are applied in order to write the log-likelihood ratio in terms of sums of conditional expectations, for which a central limit theorem for triangular arrays can be applied.

math.PR↗
arXiv2012

Local Hölder continuity property of the Densities of Solutions of SDEs with Singular Coefficients

We prove that the weak solution of a uniformly elliptic stochastic differential equation with locally smooth diffusion coefficient and Hölder continuous drift has a Hölder continuous density function. This result complements recent results of Fournier-Printems \cite{F1}, where the density is shown to exist if both coefficients are Hölder continuous and exemplifies the role of the drift coefficient in the regularity of the density of a diffusion.

math.PR↗
arXiv2012

Optimal simulation schemes for Lévy driven stochastic differential equations

We consider a general class of high order weak approximation schemes for stochastic differential equations driven by Lévy processes with infinite activity. These schemes combine a compound Poisson approximation for the jump part of the Lévy process with a high order scheme for the Brownian driven component, applied between the jump times. The overall approximation is analyzed using a stochastic splitting argument. The resulting error bound involves separate contributions of the compound Poisson approximation and of the discretization scheme for the Brownian part, and allows, on one hand, to balance the two contributions in order to minimize the computational time, and on the other hand, to study the optimal design of the approximating compound Poisson process. For driving processes whose Lévy measure explodes near zero in a regularly varying way, this procedure allows to construct discretization schemes with arbitrary order of convergence.

math.PR↗
arXiv2009

An operator approach for Markov chain weak approximations with an application to infinite activity Lévy driven SDEs

Weak approximations have been developed to calculate the expectation value of functionals of stochastic differential equations, and various numerical discretization schemes (Euler, Milshtein) have been studied by many authors. We present a general framework based on semigroup expansions for the construction of higher-order discretization schemes and analyze its rate of convergence. We also apply it to approximate general Lévy driven stochastic differential equations.

math.PR↗