Related papers: Malliavin Calculus for the one-dimensional Stochas…
We study the one-dimensional stochastic partial differential equation \begin{equation*} \frac{\partial u}{\partial t}(t,x) = -\kappa \frac{\partial^4 u}{\partial x^4}(t,x) + \rho \frac{\partial^2 u}{\partial x^2}(t,x) + b(u(t,x)) +…
We study existence and regularity of the density for the solution $u(t,x)$ (with fixed $t > 0$ and $x \in D$) of the heat equation in a bounded domain $D \subset \mathbb R^d$ driven by a stochastic inhomogeneous Neumann boundary condition…
It is well known that Malliavin calculus can be applied to a stochastic differential equation with Lipschitz continuous coefficients in order to clarify the existence and the smoothness of the solution. In this paper, we apply Malliavin…
For a mixed stochastic differential driven by independent fractional Brownian motions and Wiener processes, the existence and integrability of the Malliavin derivative of its solution are established. It is also proved that the solution…
By means of the Malliavin calculus, integral representations for the likelihood function and for the derivative of the log-likelihood function are given for a model based on discrete time observations of the solution to equation…
In this work we present a condition for the regularity, in both space and Malliavin sense, of strong solutions to SDEs driven by Brownian motion. We conjecture that this condition is optimal. As a consequence, we are able to improve the…
We study the supercooled Stefan problem in arbitrary dimensions. First, we study general solutions and their irregularities, showing generic fractal freezing and nucleation, based on a novel Markovian gluing principle. In contrast, we then…
We prove the existence of a unique Malliavin differentiable strong solution to a stochastic differential equation on the plane with merely integrable coefficients driven by the fractional Brownian sheet with Hurst parameters less than 1/2.…
Malliavin calculus provides a characterization of the centered model in regularity structures that is stable under removing the small-scale cut-off. In conjunction with a spectral gap inequality, it yields the stochastic estimates of the…
We prove uniqueness of the maximal weak solutions to the supercooled Stefan problem in 1 dimension. This follows by showing that in 1 dimension, the optimal solution of the corresponding free target optimal transport problem given in…
We study the adapted solution, numerical methods, and related convergence analysis for a unified backward stochastic partial differential equation (B-SPDE). The equation is vector-valued, whose drift and diffusion coefficients may involve…
We study solutions of a class of one-dimensional continuous reflected backward stochastic Volterra integral equations driven by Brownian motion, where the reflection keeps the solution above a given stochastic process (lower obstacle). We…
We establish a sharp estimate on the negative moments of the smallest eigenvalue of the Malliavin matrix $\gamma_Z$ of $Z := (u(s, y), u(t, x) - u(s, y))$, where $u$ is the solution to system of $d$ non-linear stochastic heat equations in…
We consider the Stefan problem with surface tension, also known as the Stefan-Gibbs-Thomson problem, in an ambient space of arbitrary dimension. Assuming the radial symmetry of the initial data we introduce a novel "probabilistic" notion of…
We consider the stochastic wave equation with multiplicative noise, which is fractional in time with index $H>1/2$, and has a homogeneous spatial covariance structure given by the Riesz kernel of order $\alpha$. The solution is interpreted…
We formulate and solve a free target optimal Brownian stopping problem from a given distribution while the target distribution is free and is conditioned to satisfy a given density height constraint. The free target optimization problem…
This work establishes a scaling limit theorem for the Stefan problem incorporating a mushy region, demonstrating that solutions to stochastic variants with turbulent transport terms converge to the solution to a deterministic partial…
We investigate the existence of densities for finite-dimensional distributions of Hermite processes of order \(q \ge 1\) and self-similarity parameter \(H\in(\frac12,1)\). Whereas the Gaussian case \(q=1\) (fractional Brownian motion) is…
We study the nonlocal Stefan problem, where the phase transition is described by a nonlocal diffusion as well as the change of enthalpy functions. By using a stochastic optimization approach introduced for the local case, we construct…
We consider the stochastic continuity equation driven by Brownian motion. We use the techniques of the Malliavin calculus to show that the law of the solution has a density with respect to the Lebesgue measure. We also prove that the…