Related papers: Malliavin Calculus for rough stochastic differenti…

We consider measurable $F: \Omega \times \mathbb{R}^d \to \mathbb{R}$ where $F(\cdot, x)$ belongs for any $x$ to the Malliavin Sobolev space $\mathbb{D}_{1,2}$ (with respect to a L\'evy process) and provide sufficient conditions on $F$ and…

This paper establishes results on the existence and uniqueness of solutions to McKean-Vlasov equations, also called mean-field stochastic differential equations, in an infinite-dimensional Hilbert space setting with irregular drift. Here,…

In this article, we consider elliptic diffusion problems on random domains with non-smooth diffusion coefficients. We start by illustrating the problems that arise from a non-smooth diffusion coefficient by recapitulating the corresponding…

We study fully nonlinear second-order (forward) stochastic partial differential equations (SPDEs). They can also be viewed as forward path-dependent PDEs (PPDEs) and will be treated as rough PDEs (RPDEs) under a unified framework. We…

We consider a one-dimensional jumping Markov process $\{X^x_t\}_{t \geq 0}$, solving a Poisson-driven stochastic differential equation. We prove that the law of $X^x_t$ admits a smooth density for $t>0$, under some regularity and…

We study counterfactual stochastic optimization of conditional loss functionals under misspecified and noisy gradient information. The difficulty is that when the conditioning event has vanishing or zero probability, naive Monte Carlo…

In the pathwise stochastic calculus framework, the paper deals with the general study of equations driven by an additive Gaussian noise, with a drift function having an infinite limit at point zero. An ergodic theorem and the convergence of…

Consider the stochastic evolution equation in a separable Hilbert space with a nice multiplicative noise and a locally Dini continuous drift. We prove that for any initial data the equation has a unique (possibly explosive) mild solution.…

We consider a reaction--diffusion equation perturbed by noise (not necessarily white). We prove an integral inequality for the invariant measure $\nu$ of a stochastic reaction--diffusion equation. Then we discuss some consequences as an…

The purpose of these lectures is threefold: We first give a short survey of the Hida white noise calculus, and in this context we introduce the Hida-Malliavin derivative as a stochastic gradient with values in the Hida stochastic…

In this work we prove Malliavin differentiability for the solution to an SDE with locally Lipschitz and semi-monotone drift. To this end we construct a sequence of SDEs with globally Lipschitz drifts. We show that the solutions of these…

In this paper we consider a general class of second order stochastic partial differential equations on $\mathbb{R}^d$ driven by a Gaussian noise which is white in time and it has a homogeneous spatial covariance. Using the techniques of…

In this article, we consider diffusion approximations for a general class of stochastic recursions. Such recursions arise as models for population growth, genetics, financial securities, multiplicative time series, numerical schemes and…

In this paper, by virtue of Malliavin calculus, we establish a relationship between backward doubly stochastic differential equations with random coefficients and quasilinear stochastic PDEs, and thus extend the well-known nonlinear…

In this paper, we investigate reflected backward stochastic differential equations driven by rough paths (rough RBSDEs), which can be viewed as probabilistic representations of nonlinear rough partial differential equations (rough PDEs) or…

By using the Malliavin calculus and solving a control problem, Bismut type derivative formulae are established for a class of degenerate diffusion semigroups with non-linear drifts. As applications, explicit gradient estimates and Harnack…

We combine the rough path theory and stochastic backward error analysis to develop a new framework for error analysis on numerical schemes. Based on our approach, we prove that the almost sure convergence rate of the modified Milstein…

Determining evolution equations governing the probability density function (pdf) of non-Markovian responses to random differential equations (RDEs) excited by coloured noise, is an important issue arising in various problems of stochastic…

This paper presents a novel generic asymptotic expansion formula of expectations of multidimensional Wiener functionals through a Malliavin calculus technique. The uniform estimate of the asymptotic expansion is shown under a weaker…

We extend the functional Breuer-Major theorem by Nourdin and Nualart (2020) to the space of rough paths. The proof of tightness combines the multiplication formula for iterated Malliavin divergences, due to Furlan and Gubinelli (2019), with…