arXiv subjects

Clément Rey

Publications and source records attributed to Clément Rey.

9 recordsLinked to original sources

arXiv2026

Malliavin calculus for signatures with applications to finance

Malliavin calculus is a powerful and general framework for the analysis of square-integrable random variables, but it often suffers from a lack of tractability and explicit representations. To address this limitation, we focus on a subclass of random variables given by finite linear combinations of time-extended Brownian motion signatures. The class remains rich due to the universal approximation properties of signatures. Leveraging the algebraic structure of signatures, we first derive explicit formulas for the Malliavin derivative of signatures of continuous Itô processes. As a consequence, we obtain closed-form expressions for the Clark--Ocone representation, the Ornstein--Uhlenbeck semigroup and its generator, as well as the integration-by-parts formula within the class of Brownian signature variables. These results provide purely algebraic formulations of the classical operators of Malliavin calculus. As an application, we compute Greeks for general path-dependent options under signature volatility models, and numerically compare different choices of Malliavin weights.

math.PR↗

arXiv2025

Discretization of the Ergodic Functional Central Limit Theorem

In this paper, we study the discretization of the ergodic Functional Central Limit Theorem (CLT) established by Bhattacharya (see \cite{Bhattacharya_1982}) which states the following: Given a stationary and ergodic Markov process $(X_t)_{t \geqslant 0}$ with unique invariant measure $ν$ and infinitesimal generator $A$, then, for every smooth enough function $f$, $(n^{1/2} \frac{1}{n}\int_0^{nt} Af(X_s)ds)_{t \geqslant 0}$ converges in distribution towards the distribution of the process $(\sqrt{-2 \langle f, Af \rangle_ν} W_{t})_{t \geqslant 0}$ with $(W_{t})_{t \geqslant 0}$ a Wiener process. In particular, we consider the marginal distribution at fixed $t=1$, and we show that when $\int_0^{n} Af(X_s)ds$ is replaced by a well chosen discretization of the time integral with order $q$ ($e.g.$ Riemann discretization in the case $q=1$), then the CLT still holds but with rate $n^{q/(2q+1)}$ instead of $n^{1/2}$. Moreover, our results remain valid when $(X_t)_{t \geqslant 0}$ is replaced by a $q$-weak order approximation (not necessarily stationary). This paper presents both the discretization method of order $q$ for the time integral and the $q$-order ergodic CLT we derive from them. We finally propose applications concerning the first order CLT for the approximation of Markov Brownian diffusion stationary regimes with Euler scheme (where we recover existing results from the literature) and the second order CLT for the approximation of Brownian diffusion stationary regimes using Talay's scheme \cite{Talay_1990} of weak order two.

math.PR↗

arXiv2024

Deterministic computation of quantiles in a Lipschitz framework

In this article, we focus on computing the quantiles of a random variable $f(X)$, where $X$ is a $[0,1]^d$-valued random variable, $d \in \mathbb{N}^{\ast}$, and $f:[0,1]^d\to \mathbb{R}$ is a deterministic Lipschitz function. We are particularly interested in scenarios where the cost of a single function evaluation is high, while the law of $X$ is assumed to be known. In this context, we propose a deterministic algorithm to compute deterministic lower and upper bounds for the quantile of $f(X)$ at a given level $α\in (0,1)$. With a fixed budget of $N$ function calls, we demonstrate that our algorithm achieves an exponential deterministic convergence rate for $d=1$ ($\mathcal{O}( ρ^N)$ with $ρ\in (0,1)$) and a polynomial deterministic convergence rate for $d>1$ ($\mathcal{O}(N^{-\frac{1}{d-1}})$) and show the optimality of those rates. Furthermore, we design two algorithms, depending on whether the Lipschitz constant of $f$ is known or unknown.

math.PR↗

arXiv2024

High order weak approximation of Stochastic Differential Equations for bounded and measurable test functions

We present a method for approximating solutions of Stochastic Differential Equations (SDEs) with arbitrary rates. This approximation is derived for bounded and measurable test functions. Specifically, we demonstrate that, leveraging the standard weak approximation properties of numerical schemes for smooth test functions (such as first-order weak convergence for the Euler scheme) we can achieve convergence for simply bounded and measurable test functions at any desired rate by constructing a tailored approximation for the semigroup of the SDE. This is achieved by evaluating the scheme (e.g., Euler) on a random time grid. To establish convergence, we exploit the regularization properties of the scheme, which hold under a weak uniform Hörmander condition.

math.PR↗

arXiv2024

Hörmander properties of discrete time Markov processes

We present an abstract framework for establishing smoothing properties within a specific class of inhomogeneous discrete-time Markov processes. These properties, in turn, serve as a basis for demonstrating the existence of density functions for our processes or more precisely for regularized versions of them. They can also be exploited to show the total variation convergence towards the solution of a Stochastic Differential Equation as the time step between two observations of the discrete time Markov processes tends to zero. The distinctive feature of our methodology lies in the exploration of smoothing properties under some local weak Hörmander type conditions satisfied by the discrete-time Markov processes. Our Hörmander properties are demonstrated to align with the standard local weak Hörmander properties satisfied by the coefficients of the Stochastic Differential Equations which are the total variation limits of our discrete time Markov processes.

math.PR↗

arXiv2021

Arbitrary order total variation Convergence of Markov semigroups using random grids

We provide a abstract framework to prove total variation convergence result with arbitrary rate for numerical scheme for SDE. In particular we show that under standard weak approximation properties of scheme such as Euler we can obtain total variation convergence with any desired rate by building a specific approximation for the SDE.

math.PR↗

arXiv2018

Recursive computation of the invariant distribution of Markov and Feller processes

This paper provides a general and abstract approach to approximate ergodic regimes of Markov and Feller processes. More precisely, we show that the recursive algorithm presented in Lamberton & Pages (2002) and based on simulation algorithms of stochastic schemes with decreasing step can be used to build invariant measures for general Markov and Feller processes. We also propose applications in three different configurations: Approximation of Markov switching Brownian diffusion ergodic regimes using Euler scheme, approximation of Markov Brownian diffusion ergodic regimes with Milstein scheme and approximation of general diffusions with jump components ergodic regimes.

math.PR↗

arXiv2018

Recursive computation of the invariant distributions of Feller processes: Revisited examples and new applications

In this paper, we show that the abstract framework developed in Pages & Rey (2017) and inspired by Lamberton & Pages (2002) can be used to build invariant distributions for Brownian diffusion processes using the Milstein scheme and for diffusion processes with censored jump using the Euler scheme. Both studies rely on a weakly mean reverting setting for both cases. For the Milstein scheme we prove the convergence for test functions with polynomial (Wasserstein convergence) and exponential growth. For the Euler scheme of diffusion processes with censored jump we prove the convergence for test functions with polynomial growth.

math.PR↗

arXiv2016

Maximum Likelihood Estimation for Wishart processes

In the last decade, there has been a growing interest to use Wishart processes for modelling, especially for financial applications. However, there are still few studies on the estimation of its parameters. Here, we study the Maximum Likelihood Estimator (MLE) in order to estimate the drift parameters of a Wishart process. We obtain precise convergence rates and limits for this estimator in the ergodic case and in some nonergodic cases. We check that the MLE achieves the optimal convergence rate in each case. Motivated by this study, we also present new results on the Laplace transform that extend the recent findings of Gnoatto and Grasselli and are of independent interest.

math.ST↗