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We introduce an explicit adaptive Milstein method for stochastic differential equations (SDEs) with no commutativity condition. The drift and diffusion are separately locally Lipschitz and together satisfy a monotone condition. This method…
We develop two novel couplings between general pure-jump L\'evy processes in $\R^d$ and apply them to obtain upper bounds on the rate of convergence in an appropriate Wasserstein distance on the path space for a wide class of L\'evy…
We establish a general theory of optimal strong error estimation for numerical approximations of a second-order parabolic stochastic partial differential equation with monotone drift driven by a multiplicative infinite-dimensional Wiener…
In this work, we investigate the existence and properties of Gaussian-like densities for weak solutions of multidimensional stochastic differential equations driven by a mixture of completely correlated fractional Brownian motions. We…
Contraction theory for dynamical systems on Euclidean spaces is well-established. For contractive (resp. semi-contractive) systems, the distance (resp. semi-distance) between any two trajectories decreases exponentially fast. For partially…
In this paper we develop a metastability theory for a class of stochastic reaction-diffusion equations exposed to small multiplicative noise. We consider the case where the unperturbed reaction-diffusion equation features multiple…
Estimating parameters of drift and diffusion coefficients for multidimensional stochastic delay equations with small noise are considered. The delay structure is written as an integral form with respect to a delay measure. Our contrast…
We present a theoretical analysis of some popular adaptive Stochastic Gradient Descent (SGD) methods in the small learning rate regime. Using the stochastic modified equations framework introduced by Li et al., we derive effective…
We give a new take on the error analysis of approximations of stochastic differential equations (SDEs), utilizing and developing the stochastic sewing lemma of L\^e (2020). This approach allows one to exploit regularization by noise effects…
We propose a new tamed Milstein-type scheme for stochastic differential equation with Markovian switching when drift coefficient is assumed to grow super-linearly. The strong rate of convergence is shown to be equal to $1.0$ under mild…
We study the convergence of semilinear parabolic stochastic evolution equations, posed on a sequence of Banach spaces approximating a limiting space and driven by additive white noise projected onto the former spaces. Under appropriate…
We study the long time statistics of a class of semi--linear damped wave equations with polynomial nonlinearities and perturbed by additive Gaussian noise in dimensions 2 and 3. We find that if sufficiently many directions in the phase…
We consider a generic and explicit tamed Euler--Maruyama scheme for multidimensional time-inhomogeneous stochastic differential equations with multiplicative Brownian noise. The diffusive coefficient is uniformly elliptic, H\"older…
This work focuses on the quantitative contraction rates for McKean-Vlasov stochastic differential equations (SDEs) with multiplicative noise. Under suitable conditions on the coefficients of the SDE, this paper derives explicit quantitative…
In this paper, we derive exponential ergodicity in relative entropy for general kinetic SDEs under a partially dissipative condition. It covers non-equilibrium situations where the forces are not of gradient type and the invariant measure…
Discrete time analogues of ergodic stochastic differential equations (SDEs) are one of the most popular and flexible tools for sampling high-dimensional probability measures. Non-asymptotic analysis in the $L^2$ Wasserstein distance of…
In this paper, we consider a new approach for semi-discretization in time and spatial discretization of a class of semi-linear stochastic partial differential equations (SPDEs) with multiplicative noise. The drift term of the SPDEs is only…
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…
In this paper we investigate the well-posedness of backward or forward stochastic differential equations whose law is constrained to live in an a priori given (smooth enough) set and which is reflected along the corresponding ''normal''…
In this work, we analyze the properties of the solution to the covariance steering problem for discrete time Gaussian linear systems with a squared Wasserstein distance terminal cost. In our previous work, we have shown that by utilizing…