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Eulalia Nualart

Publications and source records attributed to Eulalia Nualart.

At least 19 recordsLinked to original sources
arXiv2026

On the local well-posedness of randomly forced reaction-diffusion equations with $L^2$ initial data and a superlinear reaction term

We consider a parabolic stochastic partial differential equation (SPDE) on $[0\,,1]$ that is forced with multiplicative space-time white noise with a bounded and Lipschitz diffusion coefficient and a drift coefficient that is locally Lipschitz and satisfies an $L\log L$ growth condition. We prove that the SPDE is well posed when the initial data is in $L^2[0\,,1]$. This solves a strong form of an open problem.

math.PR↗
arXiv2025

Fast convergence rates for estimating the stationary density in SDEs driven by a fractional Brownian motion with semi-contractive drift

We study the estimation of the invariant density of additive fractional stochastic differential equations with Hurst parameter $H \in (0,1)$. We first focus on continuous observations and develop a kernel-based estimator achieving faster convergence rates than previously available. This result stems from a martingale decomposition combined with new bounds on the (conditional) convergence in total variation to equilibrium of fractional SDEs. For $H<1/2$, we further refine the rates based on recent bounds on the marginal density. We then extend the methodology to discrete observations, showing that the same convergence rates can be attained. Moreover, we establish concentration inequalities for the estimator and introduce a data-driven bandwidth selection procedure that adapts to unknown smoothness. Numerical experiments for the fractional Ornstein-Uhlenbeck process illustrate the estimator's practical performance. Finally, our results weaken the usual convexity assumptions on the drift component, allowing us to consider settings where strong convexity only holds outside a compact set.

math.ST↗
arXiv2025

Convergence of continuous-time stochastic gradient descent with applications to deep neural networks

We study a continuous-time approximation of the stochastic gradient descent process for minimizing the population expected loss in learning problems. The main results establish general sufficient conditions for the convergence, extending the results of Chatterjee (2022) established for (nonstochastic) gradient descent. We show how the main result can be applied to the case of overparametrized neural network training.

cs.LG↗
arXiv2025

On the well-posedness of SPDEs with locally Lipschitz coefficients

We consider the stochastic partial differential equation, $\partial_t u = \tfrac12 \partial^2_x u + b(u) + σ(u) \dot{W},$ where $u=u(t\,,x)$ is defined for $(t\,,x)\in(0\,,\infty)\times\mathbb{R}$, and $\dot{W}$ denotes space-time white noise. We prove that this SPDE is well posed solely under the assumptions that the initial condition $u(0)$ is bounded and measurable, and $b$ and $σ$ are locally Lipschitz continuous functions and have at most linear growth. Our method is based on a truncation argument together with moment bounds and tail estimates of the truncated solution. The results naturally generalize to the case where $b$ and $σ$ are time dependent with uniform-in-time growth and oscillation properties. Additionally, our method can be extended to the stochastic wave equation.

math.PR↗
arXiv2025

On the implied volatility of Inverse options under stochastic volatility models

In this paper we study short-time behavior of the at-the-money implied volatility for Inverse European options with fixed strike price. The asset price is assumed to follow a general stochastic volatility process. Using techniques of the Malliavin calculus such as the anticipating It^o's formula we first compute the level of the implied volatility of the option when the maturity converges to zero. Then, we find a short maturity asymptotic formula for the skew of the implied volatility that depends on the roughness of the volatility model. We also show that our results extend easily to Quanto-Inverse options. We apply our general results to the SABR and fractional Bergomi models, and provide some numerical simulations that confirm the accurateness of the asymptotic formula for the skew. Finally, we provide an empirical application using Bitcoin options traded on Debirit to show how our theoretical formulas can be used to model real market data of such options.

q-fin.MF↗
arXiv2025

Differential Machine Learning for Time Series Prediction

Accurate time series prediction is challenging due to the inherent nonlinearity and sensitivity to initial conditions. We propose a novel approach that enhances neural network predictions through differential learning, which involves training models on both the original time series and its differential series. Specifically, we develop a differential long short-term memory (Diff-LSTM) network that uses a shared LSTM cell to simultaneously process both data streams, effectively capturing intrinsic patterns and temporal dynamics. Evaluated on the Mackey-Glass, Lorenz, and Rössler chaotic time series, as well as a real-world financial dataset from ACI Worldwide Inc., our results demonstrate that the Diff- LSTM network outperforms prevalent models such as recurrent neural networks, convolutional neural networks, and bidirectional and encoder-decoder LSTM networks in both short-term and long-term predictions. This framework offers a promising solution for enhancing time series prediction, even when comprehensive knowledge of the underlying dynamics of the time series is not fully available.

cs.LG↗
arXiv2025

On the implied volatility of European and Asian call options under the stochastic volatility Bachelier model

In this paper we study the short-time behavior of the at-the-money implied volatility for European and arithmetic Asian call options with fixed strike price. The asset price is assumed to follow the Bachelier model with a general stochastic volatility process. Using techniques of the Malliavin calculus such as the anticipating Ito's formula we first compute the level of the implied volatility when the maturity converges to zero. Then, we find a short maturity asymptotic formula for the skew of the implied volatility that depends on the roughness of the volatility model. We apply our general results to the SABR and fractional Bergomi models, and provide some numerical simulations that confirm the accurateness of the asymptotic formula for the skew.

q-fin.MF↗
arXiv2025

Multivariate Poisson approximation of joint subgraph counts in random graphs via size-biased couplings

Using Chen-Stein method in combination with size-biased couplings, we obtain the multivariate Poisson approximation in terms of the Wasserstein distance. As applications, we study the multivariate Poisson approximation of the distribution of joint subgraph counts in an Erdős-Rényi random graph and the multivariate hypergeometric distribution giving explicit convergence rates.

math.PR↗
arXiv2024

On the implied volatility of Asian options under stochastic volatility models

In this paper we study the short-time behavior of the at-the-money implied volatility for arithmetic Asian options with fixed strike price. The asset price is assumed to follow the Black-Scholes model with a general stochastic volatility process. Using techniques of the Malliavin calculus such as the anticipating Ito's formula we first compute the level of the implied volatility of the option when the maturity converges to zero. Then, we find and short maturity asymptotic formula for the skew of the implied volatility that depends on the roughness of the volatility model. We apply our general results to the SABR model and the rough Bergomi model, and provide some numerical simulations that confirm the accurateness of the asymptotic formula for the skew.

q-fin.MF↗
arXiv2023

Instantaneous everywhere-blowup of parabolic SPDEs

We consider the following stochastic heat equation \begin{equation*} \partial_t u(t\,,x) = \tfrac12 \partial^2_x u(t\,,x) + b(u(t\,,x)) + σ(u(t\,,x)) \dot{W}(t\,,x), \end{equation*} defined for $(t\,,x)\in(0\,,\infty)\times\mathbb{R}$, where $\dot{W}$ denotes space-time white noise. The function $σ$ is assumed to be positive, bounded, globally Lipschitz, and bounded uniformly away from the origin, and the function $b$ is assumed to be positive, locally Lipschitz and nondecreasing. We prove that the Osgood condition \[ \int_1^\infty\frac{\mathrm{d} y}{b(y)}<\infty \] implies that the solution almost surely blows up everywhere and instantaneously, In other words, the Osgood condition ensures that $\mathbb{P}\{ u(t\,,x)=\infty\quad\text{for all $t>0$ and $x\in\mathbb{R}$}\}=1.$ The main ingredients of the proof involve a hitting-time bound for a class of differential inequalities (Remark 4.3), and the study of the spatial growth of stochastic convolutions using techniques from the Malliavin calculus and the Poincaré inequalities that were developed in Chen et al [3,4].

math.PR↗
arXiv2022

Optimal convergence rates for the invariant density estimation of jump-diffusion processes

We aim at estimating the invariant density associated to a stochastic differential equation with jumps in low dimension, which is for $d=1$ and $d=2$. We consider a class of jump diffusion processes whose invariant density belongs to some Hölder space. Firstly, in dimension one, we show that the kernel density estimator achieves the convergence rate $\frac{1}{T}$, which is the optimal rate in the absence of jumps. This improves the convergence rate obtained in [Amorino, Gloter (2021)], which depends on the Blumenthal-Getoor index for $d=1$ and is equal to $\frac{\log T}{T}$ for $d=2$. Secondly, we show that is not possible to find an estimator with faster rates of estimation. Indeed, we get some lower bounds with the same rates $\{\frac{1}{T},\frac{\log T}{T}\}$ in the mono and bi-dimensional cases, respectively. Finally, we obtain the asymptotic normality of the estimator in the one-dimensional case.

math.ST↗
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↗
arXiv2020

The Osgood condition for stochastic partial differential equations

We study the following equation \begin{equation*} \frac{\partial u(t,\,x)}{\partial t}= Δu(t,\,x)+b(u(t,\,x))+σ\dot{W}(t,\,x),\quad t>0, \end{equation*} where $σ$ is a positive constant and $\dot{W}$ is a space-time white noise. The initial condition $u(0,x)=u_0(x)$ is assumed to be a nonnegative and continuous function. We first study the problem on $[0,\,1]$ with homogeneous Dirichlet boundary conditions. Under some suitable conditions, together with a theorem of Bonder and Groisman, our first result shows that the solution blows up in finite time if and only if \begin{equation*} \int_{\cdot}^\infty\frac{1}{b(s)}\,d s<\infty, \end{equation*} which is the well-known Osgood condition. We also consider the same equation on the whole line and show that the above condition is sufficient for the nonexistence of global solutions. Various other extensions are provided; we look at equations with fractional Laplacian and spatial colored noise in $\mathbb{R}^d$.

math.PR↗
arXiv2019

Spatial asymptotics and strong comparison principle for some fractional stochastic heat equations

Consider the following stochastic heat equation, \begin{align*} \frac{\partial u_t(x)}{\partial t}=-ν(-Δ)^{α/2} u_t(x)+σ(u_t(x))\dot{F}(t,\,x), \quad t>0, \; x \in R^d. \end{align*} Here $-ν(-Δ)^{α/2}$ is the fractional Laplacian with $ν>0$ and $α\in (0,2]$, $σ: R\rightarrow R$ is a globally Lipschitz function, and $\dot{F}(t,\,x)$ is a Gaussian noise which is white in time and colored in space. Under some suitable additional conditions, we prove a strong comparison theorem and explore the effect of the initial data on the spatial asymptotic properties of the solution. This constitutes an important extension over a series of recent works.

math.PR↗
arXiv2017

LAN property for stochastic differential equations with additive fractional noise and continuous time observation

We consider a stochastic differential equation with additive fractional noise with Hurst parameter $H>1/2$, and a non-linear drift depending on an unknown parameter. We show the Local Asymptotic Normality property (LAN) of this parametric model with rate $\sqrtτ$ as $τ\rightarrow \infty$, when the solution is observed continuously on the time interval $[0,τ]$. The proof uses ergodic properties of the equation and a Girsanov-type transform. We analyse the particular case of the fractional Ornstein-Uhlenbeck process and show that the Maximum Likelihood Estimator is asymptotically efficient in the sense of the Minimax Theorem.

math.PR↗
arXiv2017

Moment bounds for some fractional stochastic heat equations on the ball

In this paper, we obtain upper and lower bounds for the moments of the solution to a class of fractional stochastic heat equations on the ball driven by a Gaussian noise which is white in time, and with a spatial correlation in space of Riesz kernel type. We also consider the space-time white noise case on an interval.

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↗