arXiv subjects

Gilles Pagès

Publications and source records attributed to Gilles Pagès.

At least 19 recordsLinked to original sources

arXiv2026

Efficient simulation of a new class of Volterra-type SDEs

We propose a new theoretical framework that exploits convolution kernels to transform a Volterra-type path-dependent (non-Markovian) stochastic process into a standard (Markovian) diffusion process. The transformation is reversible. We discuss existence and path-wise regularity of solutions for our class of stochastic differential equations. In the fractional-kernel case, when $H\in(0,\frac12)$, where $H$ is the Hurst coefficient, we propose a numerical simulation scheme which exhibits a strong convergence rate of order $1/2$, improving upon the rate typically obtained by Euler schemes for stochastic Volterra equations with comparably rough trajectories. This improvement is made possible by the distinctive structure of the proposed class, characterized by a non-Markovian process whose coefficients are driven by an associated Markovian one.

q-fin.MF↗

arXiv2026

On a Stationarity Theory for Stochastic Volterra Integral Equations with Affine Drift

This paper investigate the properties of solutions to forward Stochastic Volterra Integral Equations (SVIEs for short) with affine drift, specifically their stationarity, both over a finite horizon and in the long run. We demonstrate that it is possible to induce a $\textit{fake stationary regime}$, in the sense that all marginal distributions share the same expectation and variance. This phenomenon can be achieved either through explicit closed-form specifications of the deterministic initial condition $ϕ$ and the mean-reversion function $μ$ appearing in the drift, or by introducing a deterministic stabilizing factor $ς$ in the diffusion coefficient, associated with the kernel, while keeping the function $μ$ otherwise fully flexible. We further look at the $L^p$-confluence properties $p>0$ of such processes as time goes to infinity, namely we investigate whether the marginals of solutions associated with different initial values become asymptotically confluent in $L^p$. We finally study the functional weak long-run asymptotics for some classes of diffusion coefficients. More precisely, we establish that, in both settings, the time-shifted solutions of such SVIEs converge weakly, in the functional sense, toward a family of $L^2$-stationary processes sharing the same covariance function. These results are then applied to a class of Exponential-Fractional Stochastic Volterra Integral Equations driven by an $α$-gamma fractional integration kernel,in the particular case $α\in (0,1]$, which corresponds to the $\textit{rough-path}$ regime. Building on these fake stationary Volterra processes, we finally introduce a family of stabilized Rough volatility models.

math.PR↗

arXiv2026

On (fake) Stationarity in Stochastic Volterra Equations with Affine Drift and Regular Kernels

We investigate the fake stationarity properties of solutions to forward Stochastic Volterra Integral Equations (SVIEs) with affine drift and long-memory (regular) kernels, both on finite horizons and in the long-run regime. By either deriving explicit closed-form specifications for the deterministic initial condition $ϕ$ and the mean-reversion function $μ$ appearing in the drift, or by introducing a deterministic stabilizing factor $ς$ in the diffusion coefficient associated with the kernel while keeping $μ$ fully flexible, we show that it is possible to induce a \textit{fake stationary} regime, in the sense that all marginal distributions share the same mean and variance. Afterwards, using a refined asymptotic analysis, we further establish that, in both frameworks, the time-shifted solutions of these long-memory SVIEs converge weakly, in the functional sense, toward a family of $L^2$-stationary processes sharing the same covariance structure, for suitable classes of diffusion coefficients. These results are applied to a class of exponential-fractional Stochastic Volterra Integral Equations driven by an $α$-gamma fractional integration kernel, in the particular regime $α\geq 1$, which regularizes diffusion paths and invoke textit{ long-term memory}, persistence or long range dependence.

math.PR↗

arXiv2026

A note on the $\mathcal{W}_2$-convergence rate of the empirical measure of an ergodic $\mathbb{R}^d$-valued diffusion

In this note, we consider a Stochastic Differential Equation under a strong confluence and Lipschitz continuity assumption of the coefficients. For the unique stationary solution, we study the rate of convergence of its empirical measure toward the invariant probability measure. We provide rate for the Wasserstein distance in the mean quadratic and almost sure sense.

math.PR↗

arXiv2026

Convergence rate of the occupation measure of classes of ergodic processes toward their invariant distribution in mean Wasserstein distance

N. Fournier and A. Guillin obtained in their 2015 PTRF paper some bounds of the L^p-mean rate of convergence in Wasserstein distance of empirical distributions for a class of stationary mixing processes. In this paper, we propose to extend their strategy of proof and provide general criterions which allow to keep similar rates for a larger class of processes. These results (which do not require regularization techniques) lead to various applications to occupation measures of ergodic processes which may be not stationary or not Markovian under an assumption of {\em conditional} convergence to equilibrium in Total Variation or Wasserstein distance. We then provide explicit conditions which lead to these rates for Brownian diffusions and additive SDEs driven by fractional Brownian Motions {or by Gaussian processes with stationary increments}.

math.PR↗

arXiv2026

Kolmogorov-Smirnov distance and discrepancies versus Wasserstein distances

We establish inequalities that compare the p-Wasserstein distance to distances which are built as suprema of box measures. More precisely, when the measures are supported on $[0,1]^d$, we obtain sharp upper-bounds of the $p$-Wasserstein distance by (powers of) the (uniform) discrepancy. As an application, we retrieve the Pro\''inov Theorem. When the two distributions are supported {by the whole} $R^d$, {their} $p$-Wasserstein distance is upper bounded by the product of a (power of) their Kolmogorov-Smirnov (KS) distance with the sum of their $p$-moments. Reverse inequalities are established when one of the two distributions has a density, depending on its ${\cal L}^s$-integrability with respect to the Lebesgue measure for some $s>1$.

math.PR↗

arXiv2026

Asymptotic Error Analysis of Multilevel Stochastic Approximations for the Value-at-Risk and Expected Shortfall

Crépey, Frikha, and Louzi (2025) introduced a nested stochastic approximation algorithm and its multilevel acceleration to compute the value-at-risk and expected shortfall of a random financial loss. We hereby establish central limit theorems for the renormalized estimation errors associated with both algorithms as well as their averaged versions. Our findings are substantiated through a numerical example.

q-fin.RM↗

arXiv2026

On Path-dependent Volterra Integral Equations: Strong Well-posedness and Stochastic Numerics

The aim of this paper is to provide a comprehensive analysis of the path-dependent Stochastic Volterra Integral Equations (SVIEs), in which both the drift and the diffusion coefficients are allowed to depend on the whole trajectory of the process up to the current time. We investigate the existence and uniqueness (aka the strong well-posedness) of solutions to such equations in the $L^p$ setting, $p>0$, locally in time and their properties specifically their path regularity and flows. Then, we introduce a numerical approximation method based on an interpolated $K-$integrated Euler-Maruyama scheme to simulate numerically the process, and we prove the convergence, with an explicit rate, of this scheme towards the strong solution in the $L^p$ norm.

math.PR↗

arXiv2026

Zador Theorem for optimal quantization with respect to Bregman divergences

We establish a Zador like theorem for $L^r$-optimal vector quantization when the similarity measure is a twice differentiable Bregman divergence of a strictly convex function. On our way we also prove a similar result when the Bregman divergence is replaced by a continuous matrix-valued vector field having values in the set of positive definite matrices. We adopt the strategy of the first fully rigorous proof of the original Zador' theorem (when the similarity measure is the power of a norm). We have to overcome several difficulties which are specific to this framework especially concerning the so-called firewall lemma.

math.FA↗

arXiv2026

Fake stationary rough Heston volatility: Microstructure-inspired foundations

This paper investigates the asymptotic behavior of suitably time-modulated Hawkes processes with heavy-tailed kernels in a nearly unstable regime. We show that, under appropriate scaling, both the intensity processes and the rescaled Hawkes processes converge to a mean-reverting, time-inhomogeneous rough fractional square-root process and its integrated counterpart, respectively. In particular, when the original Hawkes process has a stationary first moment (constant marginal mean), the limiting process takes the form of a time-inhomogeneous rough fractional Cox-Ingersoll-Ross (CIR) equation with a constant mean-reversion parameter and a time-dependent diffusion coefficient. This class of equations is particularly appealing from a practical perspective, especially for the so-called $\textit{fake stationary rough Heston}$ model. We further investigate the properties of such limiting scaled time-inhomogeneous Volterra equations, including moment bounds, path regularity and maximal inequality in the $L^p$ setting for every $p>0$.

math.PR↗

arXiv2026

Strong Solutions and Quantization-Based Numerical Schemes for a Class of Non-Markovian Volatility Models

We investigate a class of non-Markovian processes that hold particular relevance in the realm of mathematical finance. This family encompasses path-dependent volatility models, including those pioneered by [Platen and Rendek, 2018] and, more recently, by [Guyon and Lekeufack, 2023]. Our study unfolds in two principal phases. In the first phase, we introduce a functional quantization scheme based on an extended version of the Lamperti transformation that we propose to handle the presence of a memory term incorporated into the diffusion coefficient. In the second phase, we study the problem of existence and uniqueness of a strong solution for the SDEs related to the examples that motivate our study, in order to provide a theoretical basis to correctly apply the proposed numerical schemes.

q-fin.MF↗

arXiv2025

On Inhomogeneous Affine Volterra Processes: Stationarity and Applications to the Volterra Heston Model

True Volterra equations are inherently non stationary and therefore do not admit $\textit{genuine stationary regimes}$ over finite horizons. This motivates the study of the finite-time behavior of the solutions to scaled inhomogeneous affine Stochastic Volterra equations through the lens of a weaker notion of stationarity referred to as $\textit{fake stationary regime}$ in the sense that all marginal distributions share the same expectation and variance. As a first application, we introduce the $\textit{Fake stationary Volterra Heston model}$ and derive a closed-form expression for its characteristic function. Having established this finite-time proxy for stationarity, we then investigate the asymptotic (long-time) behavior to assess whether genuine stationary regimes emerge in the limit. Using an extension of the exponential-affine transformation formula for those processes, we establish in the long run the existence of limiting distributions, which (unlike in the case of classical affine diffusion processes) may depend on the initial state of the process, unless the Volterra kernel coincides with the $α-$ fractional integration kernel, for which the dependence on the initial state vanishes. We then proceed to the construction of stationary processes associated with these limiting distributions. However, the dynamics in this long-term regime are analytically intractable, and the process itself is not guaranteed to be stationary in the classical sense over finite horizons. This highlights the relevance of finite-time analysis through the lens of the aforementioned $\textit{fake stationarity}$, which offers a tractable approximation to stationary behavior in genuinely non-stationary Volterra systems.

math.PR↗

arXiv2025

Optimized Multi-Level Monte Carlo Parametrization and Antithetic Sampling for Nested Simulations

Estimating risk measures such as large loss probabilities and Value-at-Risk is fundamental in financial risk management and often relies on computationally intensive nested Monte Carlo methods. While Multi-Level Monte Carlo (MLMC) techniques and their weighted variants are typically more efficient, their effectiveness tends to deteriorate when dealing with irregular functions, notably indicator functions, which are intrinsic to these risk measures. We address this issue by introducing a novel MLMC parametrization that significantly improves performance in practical, non-asymptotic settings while maintaining theoretical asymptotic guarantees. We also prove that antithetic sampling of MLMC levels enhances efficiency regardless of the regularity of the underlying function. Numerical experiments motivated by the calculation of economic capital in a life insurance context confirm the practical value of our approach for estimating loss probabilities and quantiles, bridging theoretical advances and practical requirements in financial risk estimation.

q-fin.CP↗

arXiv2025

Volterra equations with affine drift: looking for stationarity

We investigate the properties of the solutions of scaled Volterra equations (i.e. with an affine mean-reverting drift) in terms of stationarity at both a finite horizon and on the long run. In particular we prove that such an equation never has a stationary regime, except if the kernel is constant (i.e. the equation is a standard Brownian diffusion) or in some fully degenerate pathological settings. We introduce a deterministic stabilizer $ ς$ associated to the kernel which produces a {\em fake stationary regime} in the sense that all the marginals share the same expectation and variance. We also show that the marginals of such a process starting from when starting various initial values are confluent in $L^2$ as time goes to infinity. We establish that for some classes of diffusion coefficients (square root of positive quadratic polynomials) the time shifted solutions of such Volterra equations weakly functionally converges toward a family of $L^2$-stationary processes sharing the same covariance function. We apply these results to (stabilized) rough volatility models (when the kernel $K(t)= t^{H-\frac 12}$, $0<H<\frac 12$) which leads to produce a fake stationary quadratic rough Heston model.

math.PR↗

arXiv2025

Convex ordering for stochastic control: the (path dependent) swing contracts case

We investigate propagation of convexity and convex ordering on a typical discrete-time stochastic optimal control problem, namely the pricing of swing option. The dynamics of the underlying asset is modelled by the Euler scheme of a Brownian diffusion with affine drift, and convex volatility. We prove that the value function associated to the stochastic optimal control problem is a convex function of the underlying asset price. We also introduce a domination criterion offering insights into the functional monotonicity of the value function with respect to parameters of the underlying dynamics. We particularly focus on the one-dimensional setting where, by means of Stein's formula and regularization techniques, we show that the convexity assumption for the volatility dynamics can be relaxed with a semi-convexity assumption. Finally, to validate our results, we also conduct numerical illustrations.

q-fin.MF↗

arXiv2025

Optimal Bregman quantization : existence and uniqueness of optimal quantizers revisited

In this paper we revisit the exsistence theorem for $L^r$-optimal quantization, $r\ge 2$, with respect to a Bregman divergence: we establish the existence of optimal quantizaers under lighter assumptions onthe strictly convex function which generates the divergence, espcially in the quadratic case ($r=2$). We then prove a uniqueness theorem ``à la Trushkin'' in one dimension for strongly unimodal distributions and divergences gerated by strictly convex functions whiose thire dervative is either stictly $\log$-convex or $\log$-concave.

math.PR↗

arXiv2025

Locally optimal Functional Quantization

In this note we demonstrate that locally optimal functional quantizers for probability distributions on a Banach space lying in the support of $P$ behave exactly like globally optimal functional quantizers in terms of stationarity/self-consistency.

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↗