Related papers: Stochastic differential equations with time-delaye…
In this paper we focus on the stochastic Euler-Poincar\'{e} equations with pseudo-differential/multiplicative noise. We first establish two new cancellation properties on pseudo-differential operators, which play a key role in energy…
We discuss the global existence of solutions to a system of stochastic Schr\"odinger equations with multiplicative noise. Our setting of the quadratic nonlinear terms in dimension 4 is $L^2$-critical. We treat the solutions under the ground…
We consider stochastic partial differential equations on $\mathbb{R}^{d}, d\geq 1$, driven by a Gaussian noise white in time and colored in space, for which the pathwise uniqueness holds. By using the Skorokhod representation theorem we…
We consider a stochastic partial differential equation with logarithmic (or negative power) nonlinearity, with one reflection at 0 and with a constraint of conservation of the space average. The equation, driven by the derivative in space…
We provide tight finite-time convergence bounds for gradient descent and stochastic gradient descent on quadratic functions, when the gradients are delayed and reflect iterates from $\tau$ rounds ago. First, we show that without stochastic…
We consider a general multidimensional stochastic differential delay equation (SDDE) with state-dependent colored noises. We approximate it by a stochastic differential equation (SDE) system and calculate its limit as the time delays and…
In this article, we consider the following stochastic fractional diffusion equation \begin{equation*} \left(\partial^{\beta}+\dfrac{\nu}{2}\left(-\Delta\right)^{\alpha / 2}\right) u(t, x)= \lambda\: I_{0_+}^{\gamma}\left[u(t, x) \dot{W}(t,…
In this work, we shall consider the existence and uniqueness of stationary solutions to stochastic partial functional differential equations with additive noise in which a neutral type of delay is explicitly presented. We are especially…
In this paper, we study a class of multi-dimensional reflected backward stochastic differential equations when the noise is driven by a Brownian motion and an independent Poisson point process, and when the solution is forced to stay in a…
In this paper, we established a quadratic transportation cost inequality for solutions of stochastic reaction diffusion equations driven by multiplicative space-time white noise based on a new inequality we proved for the moments (under the…
In this paper we discuss backward stochastic differential equations with Markov chain noise, having continuous drivers. We obtain the existence of a solution which is possibly not unique. Moreover, we show there is a minimal solution for…
We consider a reaction-diffusion equation on a network subjected to dynamic boundary conditions, with time delayed behaviour, also allowing for multiplicative Gaussian noise perturbations. Exploiting semigroup theory, we rewrite the…
We prove existence of infinitely many stationary solutions as well as ergodic stationary solutions for the stochastic Navier-Stokes equations on $\mathbb{T}^2$ \begin{align*} \dif u+\div(u\otimes u)\dif t+\nabla p\dif t&=\Delta u\dif t +…
In this paper, we establish existence and uniqueness of strong solutions for a stochastic differential equation driven by an additive noise given by the sum of two correlated fractional Brownian sheets with different Hurst parameters. Our…
This paper is a survey of uniqueness results for stochastic differential equations with jumps and regularity results for the corresponding harmonic functions.
We study a class of backward doubly stochastic differential equations (BDSDEs) involving martingales with spatial parameters, and show that they provide probabilistic interpretations (Feynman-Kac formulae) for certain semilinear stochastic…
In this paper we are concerned with one-dimensional backward stochastic differential equations (BSDE in short) of the following type: \[Y_t=\xi -\int_{t\wedge \tau}^{\tau}Y_r|Y_r|^q dr-\int_{t\wedge \tau}^{\tau}Z_r dB_r,\qquad t\geq 0,\]…
In this article spatial and temporal regularity of the solution process of a stochastic partial differential equation (SPDE) of evolutionary type with nonlinear multiplicative trace class noise is analyzed.
Covariance of the resulting probabilities requires the "anti-Ito" sense. The corresponding Fokker-Planck equation is simplified and preserves important features of the case with a constant diffusion. Multiplicative noise can always be…
We consider one-dimensional stochastic differential equations with a boundary condition, driven by a Poisson process. We study existence and uniqueness of solutions and the absolute continuity of the law of the solution. In the case when…