Related papers: On the multi-dimensional skew Brownian motion
Quantum gravity has long remained elusive from an observational standpoint. Developing effective cosmological models motivated by the fundamental aspects of quantum gravity is crucial for bridging theory with observations. One key aspect is…
We consider the Stochastic Differential Equation $X_t = X_0 + \int_0^t b(s,X_s) ds + B_t$, in $\mathbb{R}^d$. We give an example of a drift $b$ such that there does not exist a weak solution, but there exists a solution for almost every…
We prove existence and uniqueness of the solution of a stochastic shell--model. The equation is driven by an infinite dimensional fractional Brownian--motion with Hurst--parameter $H\in (1/2,1)$, and contains a non--trivial coefficient in…
We study some functional inequalities satisfied by the distribution of the solution of a stochastic differential equation driven by fractional Brownian motions. Such functional inequalities are obtained through new integration by parts…
We consider the stochastic continuity equation perturbed by a fractional Brownian motion and the drift is allowed to be discontinuous. We show that for almost all paths of the fractional Brownian motion there exists a solution to the…
We consider the dynamics of the stochastic shadow Gierer-Meinhardt system with one-dimensional standard Brownian motion. We establish the global existence and uniqueness of solutions. We also prove a large deviation result.
We study a multidimensional stochastic differential equation with additive noise: \[ d X_t=b(t, X_t) dt +d \xi_t, \] where the drift $b$ is integrable in space and time, and $\xi$ is either a fractional Brownian motion or a L\'evy process.…
We study pathwise approximation of scalar stochastic differential equations at a single point. We provide the exact rate of convergence of the minimal errors that can be achieved by arbitrary numerical methods that are based (in a…
In this paper, stability theorems for stochastic differential equations and backward stochastic differential equations driven by G-Brownian motion are obtained. We show the existence and uniqueness of solutions to forward-backward…
This paper studies the weak and strong solutions to the stochastic differential equation $ dX(t)=-\frac12 \dot W(X(t))dt+d\mathcal{B}(t)$, where $(\mathcal{B}(t), t\ge 0)$ is a standard Brownian motion and $W(x)$ is a two sided Brownian…
Starting with a Brownian motion, we define and study a novel diffusion process by combining stickiness and oscillation properties. The associated stochastic differential equation, resolvent and semigroup are provided. Also the trivariate…
In this paper we study weak solutions for the following type of stochastic differential equation \[ dX_{t}=dW_{t}+b(t, X_{t})dt, \quad t\ge s, \quad X_{s}=x, \] where $b: [0,\infty) \times \mathbb{R}^{d} \to \mathbb{R}^{d}$ is a measurable…
We study a class of linear first and second order partial differential equations driven by weak geometric $p$-rough paths, and prove the existence of a unique solution for these equations. This solution depends continuously on the driving…
We consider a d-dimensional stochastic differential equation with additive noise and a drift coefficient which is assumed only to be a bounded Borel function. We show that, for almost all choices of the driving Brownian path, the equation…
This paper addresses the question of how Brownian-like motion can arise from the solution of a deterministic differential delay equation. To study this we analytically study the bifurcation properties of an apparently simple differential…
Using elliptic regularity results in weighted spaces, stochastic calculus and the theory of non-symmetric Dirichlet forms, we first show weak existence of non-symmetric distorted Brownian motion for any starting point in some domain $E$ of…
We give a new approach to prove the existence of a weak solution of \[dx_t = f(t,x_t)dt + g(t)dB^H_t\] where $B^H_t$ is a fractional Brownian motion with values in a separable Hilbert space for suitable functions $f$ and $g$. Our idea is to…
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…
In this paper we study the existence and uniqueness of the strong solution of following d dimensional stochastic differential equation (SDE) driven by Brownian motion: dX(t)=b(t,X(t))dt+a(t,X(t))dB(t), X(0)= x, where B is a d-dimensional…
We prove that a stochastic flow of reflected Brownian motions in a smooth multidimensional domain is differentiable with respect to its initial position. The derivative is a linear map represented by a multiplicative functional for…