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We study the numerical approximation of the stochastic heat equation with a distributional reaction term. Under a condition on the Besov regularity of the reaction term, it was proven recently that a strong solution exists and is unique in the pathwise sense, in a class of H\"older continuous processes. For a suitable choice of sequence $(b^k)_{k\in \mathbb{N}}$ approximating $b$, we prove that the error between the solution $u$ of the SPDE with reaction term $b$ and its tamed Euler finite-difference scheme with mollified drift $b^k$, converges to $0$ in $L^m(\Omega)$ with a rate that depends on the Besov regularity of $b$. In particular, one can consider two interesting cases: first, even when $b$ is only a (finite) measure, a rate of convergence is obtained. On the other hand, when $b$ is a bounded measurable function, the (almost) optimal rate of convergence $(\frac{1}{2}-\varepsilon)$-in space and $(\frac{1}{4}-\varepsilon)$-in time is achieved. Stochastic sewing techniques are used in the proofs, in particular to deduce new regularising properties of the discrete Ornstein-Uhlenbeck process.

We investigate the problem of joint statistical estimation of several parameters for a stochastic differential equation driven by an additive fractional Brownian motion. Based on discrete-time observations of the model, we construct an estimator of the Hurst parameter, the diffusion parameter and the drift, which lies in a parametrised family of coercive drift coefficients. Our procedure is based on the assumption that the stationary distribution of the SDE and of its increments permits to identify the parameters of the model. Under this assumption, we prove consistency results and derive a rate of convergence for the estimator. Finally, we show that the identifiability assumption is satisfied in the case of a family of fractional Ornstein-Uhlenbeck processes and illustrate our results with some numerical experiments.

We study the well-posedness and numerical approximation of multidimensional stochastic differential equations (SDEs) with distributional drift, driven by a fractional Brownian motion. First, we prove weak existence for such SDEs. This holds under a condition that relates the Hurst parameter $H$ of the noise to the Besov regularity of the drift. Then under a stronger condition, we study the error between a solution $X$ of the SDE with drift $b$ and its tamed Euler scheme with mollified drift $b^n$. We obtain a rate of convergence in $L^m(\Omega)$ for this error, which depends on the Besov regularity of the drift. This result covers the critical case of the regime of strong existence and pathwise uniqueness. When the Besov regularity increases and the drift becomes a bounded measurable function, we recover the (almost) optimal rate of convergence $1/2-\varepsilon$. As a byproduct of this convergence, we deduce that pathwise uniqueness holds in a class of H\"older continuous solutions and that any such solution is strong. The proofs rely on stochastic sewing techniques, especially to deduce new regularising properties of the discrete-time fractional Brownian motion. We also present several examples and numerical simulations that illustrate our results.

The fractional Brownian motion can be considered as a Gaussian field indexed by $(t,H)\in { \mathbb{R}_{+} \times (0,1)}$, where $H$ is the Hurst parameter. On compact time intervals, it is known to be almost surely jointly H\"older continuous in time and Lipschitz continuous in $H$. First, we extend this result to the whole time interval $\mathbb{R}_{+}$ and consider both simple and rectangular increments. Then we consider SDEs driven by fractional Brownian motion with contractive drift. The solutions and their ergodic means are proven to be almost surely H\"older continuous in $H$, uniformly in time. This result is used in a separate work for statistical applications. The proofs are based on variance estimates of the increments of the fractional Brownian motion and fractional Ornstein-Uhlenbeck processes, multiparameter versions of the Garsia-Rodemich-Rumsey lemma and a combinatorial argument to estimate the expectation of a product of Gaussian variables.

We study existence and uniqueness of solutions to the equation $dX_t=b(X_t)dt + dB_t$, where $b$ is a distribution in some Besov space and $B$ is a fractional Brownian motion with Hurst parameter $H\leqslant 1/2$. First, the equation is understood as a nonlinear Young equation. This involves a nonlinear Young integral constructed in the space of functions with finite $p$-variation, which is well suited when $b$ is a measure. Depending on $H$, a condition on the Besov regularity of $b$ is given so that solutions to the equation exist. The construction is deterministic, and $B$ can be replaced by a deterministic path $w$ with a sufficiently smooth local time. Using this construction we prove the existence of weak solutions (in the probabilistic sense). We also prove that solutions coincide with limits of strong solutions obtained by regularisation of $b$. This is used to establish pathwise uniqueness and existence of a strong solution. In particular when $b$ is a finite measure, weak solutions exist for $H<\sqrt{2}-1$, while pathwise uniqueness and strong existence hold when $H\leqslant 1/4$. The proofs involve fine properties of the local time of the fractional Brownian motion, as well as new regularising properties of this process which are established using the stochastic sewing Lemma.

We study Euler-type discrete-time schemes for the rough Heston model, which can be described by a stochastic Volterra equation (with non-Lipschtiz coefficient functions), or by an equivalent integrated variance formulation. Using weak convergence techniques, we prove that the limits of the discrete-time schemes are solution to some modified Volterra equations. Such modified equations are then proved to share the same unique solution as the initial equations, which implies the convergence of the discrete-time schemes. Numerical examples are also provided in order to evaluate different derivative options prices under the rough Heston model.

In this work, we study the convergence of the empirical measure of moderately interacting particle systems with singular interaction kernels. First, we prove quantitative convergence of the time marginals of the empirical measure of particle positions towards the solution of the limiting nonlinear Fokker-Planck equation. Second, we prove the well-posedness for the McKean-Vlasov SDE involving such singular kernels and the convergence of the empirical measure towards it (propagation of chaos). Our results only require very weak regularity on the interaction kernel, including the Biot-Savart kernel, and attractive kernels such as Riesz and Keller-Segel kernels in arbitrary dimension. For some of these important examples, this is the first time that a quantitative approximation of the PDE is obtained by means of a stochastic particle system. In particular, this convergence still holds (locally in time) for PDEs exhibiting a blow-up in finite time. The proofs are based on a semigroup approach combined with a fine analysis of the regularity of infinite-dimensional stochastic convolution integrals.

In this work we obtain rates of convergence for two moderately interacting stochastic particle systems with singular kernels associated to the viscous Burgers and Keller-Segel equations. The main novelty of this work is to consider a non-locally integrable kernel. Namely for the viscous Burgers equation in $\mathbb{R}$, we obtain almost sure convergence of the mollified empirical measure to the solution of the PDE in some Bessel space with a rate of convergence of order $N^{-1/8}$, on any time interval. The same holds for the genuine empirical measure in Wasserstein distance. In the case of the Keller-Segel equation on a $d$-dimensional torus, we obtain almost sure convergence of the mollified empirical measure to the solution of the PDE in some $L^q$ space with a rate of order $N^{-\frac{1}{2d+1}}$. The result holds up to the maximal existence time of the PDE, for any value of the chemo-attractant sensitivity $\chi$., Comment: New version with emphasis on Burgers equation and results in Wasserstein distance

We study discrete-time simulation schemes for stochastic Volterra equations, namely the Euler and Milstein schemes, and the corresponding Multi-Level Monte-Carlo method. By using and adapting some results from Zhang [22], together with the Garsia-Rodemich-Rumsey lemma, we obtain the convergence rates of the Euler scheme and Milstein scheme under the supremum norm. We then apply these schemes to approximate the expectation of functionals of such Volterra equations by the (Multi-Level) Monte-Carlo method, and compute their complexity.

In this paper we solve real-valued rough differential equations (RDEs) reflected on an irregular boundary. The solution $Y$ is constructed as the limit of a sequence $(Y^n)_{n\in\mathbb{N}}$ of solutions to RDEs with unbounded drifts $(\psi_n)_{n\in\mathbb{N}}$. The penalisation $\psi_n$ increases with $n$. Along the way, we thus also provide an existence theorem and a Doss-Sussmann representation for RDEs with a drift growing at most linearly. In addition, a speed of convergence of the sequence of penalised paths to the reflected solution is obtained. \\ We finally use the penalisation method to prove that the law at time $t>0$ of some reflected Gaussian RDE is absolutely contiuous with respect to the Lebesgue measure.

This paper examines the Root solution of the Skorohod embedding problem given full marginals on some compact time interval. Our results are obtained by limiting arguments based on finitely-many marginals Root solution of Cox, Obl\'oj and Touzi. Our main result provides a characterization of the corresponding potential function by means of a convenient parabolic PDE.

The convergence to the stationary regime is studied for Stochastic Differential Equations driven by an additive Gaussian noise and evolving in a semi-contractive environment, i.e. when the drift is only contractive out of a compact set but does not have repulsive regions. In this setting, we develop a synchronous coupling strategy to obtain sub-exponential bounds on the rate of convergence to equilibrium in Wasserstein distance. Then by a coalescent coupling close to terminal time, we derive a similar bound in total variation distance.

We present an innovating sensitivity analysis for stochastic differential equations: We study the sensitivity, when the Hurst parameter~$H$ of the driving fractional Brownian motion tends to the pure Brownian value, of probability distributions of smooth functionals of the trajectories of the solutions $\{X^H_t\}_{t\in \mathbb{R}_+}$ and of the Laplace transform of the first passage time of $X^H$ at a given threshold. Our technique requires to extend already known Gaussian estimates on the density of $X^H_t$ to estimates with constants which are uniform w.r.t. $t$ in in the whole half-line $\R_+-\{0\}$ and $H$ when $H$ tends to~$\tfrac{1}{2}$.

Long-range dependence (LRD) has been observed in a variety of phenomena in nature, and for several years also in the spiking activity of neurons. Often, this is interpreted as originating from a non-Markovian system. Here we show that a purely Markovian integrate-and-fire (IF) model, with a noisy slow adaptation term, can generate interspike intervals (ISIs) that appear as having LRD. However a proper analysis shows that this is not the case asymptotically. For comparison, we also consider a new model of individual IF neuron with fractional (non-Markovian) noise. The correlations of its spike trains are studied and proven to have LRD, unlike classical IF models. On the other hand, to correctly measure long-range dependence, it is usually necessary to know if the data are stationary. Thus, a methodology to evaluate stationarity of the ISIs is presented and applied to the various IF models. We explain that Markovian IF models may seem to have LRD because of non-stationarities.

Sensitivity analysis w.r.t. the long-range/memory noise parameter for probability distributions of functionals of solutions to stochastic differential equations is an important stochastic modeling issue in many applications. In this paper we consider solutions $\{X^H_t\}_{t\in \mathbb{R}_+}$ to stochastic differential equations driven by fractional Brownian motions. We develop two innovative sensitivity analyses when the Hurst parameter $H$ of the noise tends to the critical Brownian parameter $H=\tfrac{1}{2}$ from above or from below. First, we examine expected smooth functions of $X^H$ at a fixed time horizon $T$. Second, we examine Laplace transforms of functionals which are irregular with regard to Malliavin calculus, namely, first passage times of $X^H$ at a given threshold. In both cases we exhibit the Lipschitz continuity w.r.t. $H$ around the value $\tfrac{1}{2}$. Therefore, our results show that the Markov Brownian model is a good proxy model as long as the Hurst parameter remains close to $\tfrac{1}{2}$.

We are interested in the increment stationarity property for $L^2$-indexed stochastic processes, which is a fairly general concern since many random fields can be interpreted as the restriction of a more generally defined $L^2$-indexed process. We first give a spectral representation theorem in the sense of \citet{Ito54}, and see potential applications on random fields, in particular on the $L^2$-indexed extension of the fractional Brownian motion. Then we prove that this latter process is characterized by its increment stationarity and self-similarity properties, as in the one-dimensional case.

We prove a Chung-type law of the iterated logarithm for a multiparameter extension of the fractional Brownian motion which is not increment stationary. This multiparameter fractional Brownian motion behaves very differently at the origin and away from the axes, which also appears in the Hausdorff dimension of its range and in the measure of its pointwise H\"older exponents. A functional version of this Chung-type law is also provided., Comment: 21 pages. To appear in J. Theoret. Probab

Using structures of Abstract Wiener Spaces, we define a fractional Brownian field indexed by a product space $(0,1/2] \times L^2(T,m)$, $(T,m)$ a separable measure space, where the first coordinate corresponds to the Hurst parameter of fractional Brownian motion. This field encompasses a large class of existing fractional Brownian processes, such as L\'evy fractional Brownian motions and multiparameter fractional Brownian motions, and provides a setup for new ones. We prove that it has satisfactory incremental variance in both coordinates and derive certain continuity and H\"older regularity properties in relation with metric entropy. Also, a sharp estimate of the small ball probabilities is provided, generalizing a result on L\'evy fractional Brownian motion. Then, we apply these general results to multiparameter and set-indexed processes, proving the existence of processes with prescribed local H\"older regularity on general indexing collections.

In this paper, we study the H\"older regularity of set-indexed stochastic processes defined in the framework of Ivanoff-Merzbach. The first key result is a Kolmogorov-like H\"older-continuity Theorem, whose novelty is illustrated on an example which could not have been treated with anterior tools. Increments for set-indexed processes are usually not simply written as $X_U-X_V$, hence we considered different notions of H\"older-continuity. Then, the localization of these properties leads to various definitions of H\"older exponents, which we compare to one another. In the case of Gaussian processes, almost sure values are proved for these exponents, uniformly along the sample paths. As an application, the local regularity of the set-indexed fractional Brownian motion is proved to be equal to the Hurst parameter uniformly, with probability one., Comment: 32 pages