Related papers: Weak solutions for stochastic differential equatio…
We investigate properties of the (conditional) law of the solution to SDEs driven by fractional Brownian noise with a singular, possibly distributional, drift. Our results on the law are twofold: i) we quantify the spatial regularity of the…
The aim of this paper is to study the $d$-dimensional stochastic heat equation with a multiplicative Gaussian noise which is white in space and it has the covariance of a fractional Brownian motion with Hurst parameter $% H\in (0,1)$ in…
In the setting of stochastic Volterra equations, and in particular rough volatility models, we show that conditional expectations are the unique classical solutions to path-dependent PDEs. The latter arise from the functional It\^o formula…
In this article, the existence of a unique solution in the variational approach of the stochastic evolution equation $$\dX(t) = F(X(t)) \dt + G(X(t)) \dL(t)$$ driven by a cylindrical L\'evy process $L$ is established. The coefficients $F$…
We find an explicit expression for the cross-covariance between stochastic integral processes with respect to a $d$-dimensional fractional Brownian motion (fBm) $B_t$ with Hurst parameter $H>1/2$, where the integrands are vector fields…
This paper presents a new approach to the analysis of mixed processes \[X_t=B_t+G_t,\qquad t\in[0,T],\] where $B_t$ is a Brownian motion and $G_t$ is an independent centered Gaussian process. We obtain a new canonical innovation…
Let $B^H$ be a fractional Brownian motion with Hurst index $0<H<1$ and the weighted local time ${\mathscr L}^H(\cdot,t)$. In this paper, we consider the integral functional $$ {\mathcal C}^H_t(a):=\lim_{\varepsilon\downarrow…
We investigate, in the setting of UMD Banach spaces E, the continuous dependence on the data A, F, G and X_0 of mild solutions of semilinear stochastic evolution equations with multiplicative noise of the form dX(t) = [AX(t) + F(t,X(t))]dt…
We prove an It\^o-Wentzell formula for the fractional Brownian motion. As an application we derive an existence and uniqueness result for a class of stochastic differential equations driven by this stochastic process.
In this paper we introduce a definition of a multi-dimensional fractional Brownian motion of Hurst index $H \in (0, 1)$ under volatility uncertainty (in short G-fBm). We study the properties of such a process and provide first results about…
This article is devoted to the existence and uniqueness of pathwise solutions to stochastic evolution equations, driven by a H\"older continuous function with H\"older exponent in $(1/2,1)$, and with nontrivial multiplicative noise. As a…
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 derive estimates for the solutions to differential equations driven by a H\"older continuous function of order $\beta>1/2$. As an application we deduce the existence of moments for the solutions to stochastic partial differential…
Here, we provide a unified framework for numerical analysis of stochastic nonlinear fractional diffusion equation driven by fractional Gaussian noise with Hurst index $H\in(0,1)$. A novel estimate of the second moment of the stochastic…
In this work we study fractal properties of rough differential equations driven by a fractional Brownian motions with Hurst parameter $H>\frac{1}{4}$. In particular, we show that the Hausdorff dimension of the sample paths of the solution…
In this paper we study a singular stochastic differential equation driven by an additive fractional Brownian motion with Hurst parameter $H>\frac 12$. Under some assumptions on the drift, we show that there is a unique solution, which has…
In this paper, we establish the existence of weak solutions for distribution-dependent stochastic differential equations (DDSDEs) driven by a broad class of L\'{e}vy noises, where the drift coefficients satisfy specific integrability…
The Landau-Lifshitz-Gilbert equation perturbed by a multiplicative space-dependent noise is considered for a ferromagnet filling a bounded three-dimensional domain. We show the existence of weak martingale solutions taking values in a…
Within the rough path framework we prove the continuity of the solution to random differential equations driven by fractional Brownian motion with respect to the Hurst parameter $H$ when $H \in (1/3, 1/2]$.
We use the framework of the first-order differential structure in metric measure spaces introduced by Gigli to define a notion of weak solutions to gradient flows of convex, lower semicontinuous and coercive functionals. We prove their…