Related papers: Stochastic selection problem for a Stratonovich SD…
In this paper we study the Stratonovich stochastic differential equation $\mathrm{d} X=|X|^{\alpha}\circ\mathrm{d} B$, $\alpha\in(-1,1)$, which has been introduced by Cherstvy et al. [New Journal of Physics 15:083039 (2013)] in the context…
In this work, we introduce a new method to prove the existence and uniqueness of a variational solution to the stochastic nonlinear diffusion equation $dX(t)={\rm div} [\frac{\nabla X(t)}{|\nabla X(t)|}]dt+X(t)dW(t) in…
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…
The solution $X_n$ to a nonlinear stochastic differential equation of the form $dX_n(t)+A_n(t)X_n(t)\,dt-\tfrac12\sum_{j=1}^N(B_j^n(t))^2X_n(t)\,dt=\sum_{j=1}^N B_j^n(t)X_n(t)d\beta_j^n(t)+f_n(t)\,dt$, $X_n(0)=x$, where $\beta_j^n$ is a…
We introduce a new method of proving pathwise uniqueness, and we apply it to the degenerate stochastic differential equation \[dX_t=|X_t|^{\alpha} dW_t,\] where $W_t$ is a one-dimensional Brownian motion and $\alpha\in(0,1/2)$. Weak…
A recent paper of Melbourne & Stuart, A note on diffusion limits of chaotic skew product flows, Nonlinearity 24 (2011) 1361-1367, gives a rigorous proof of convergence of a fast-slow deterministic system to a stochastic differential…
This paper is concerned with the existence and uniqueness of the solution for the stochastic fast logarithmic equation with Stratonovich multiplicative noise in $\mathbb{R}^{d}$ for $d\geqslant 3$. It provides an answer to a critical case…
We study existence and uniqueness of a variational solution in terms of stochastic variational inequalities (SVI) to stochastic nonlinear diffusion equations with a highly singular diffusivity term and multiplicative Stratonovich…
Pathwise non-uniqueness is established for non-negative solutions of the parabolic stochastic pde $$\frac{\partial X}{\partial t}=\frac{\Delta}{2}X+X^p\dot W+\psi,\ X_0\equiv 0$$ where $\dot W$ is a white noise, $\psi\ge 0$ is smooth,…
We study existence and uniqueness of distributional solutions to the stochastic partial differential equation $dX - ( \nu \Delta X + \Delta \psi (X) ) dt = \sum_{i=1}^N \langle b_i, \nabla X \rangle \circ d\beta_i$ in $]0,T[ \times…
In this article, we construct a Stratonovich solution for the stochastic wave equation in spatial dimension $d \leq 2$, with time-independent noise and linear term $\sigma(u)=u$ multiplying the noise. The noise is spatially homogeneous and…
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…
In this paper, we study the diffusion approximation for slow-fast stochastic differential equations with state-dependent switching, where the slow component $X^{\varepsilon}$ is the solution of a stochastic differential equation with…
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 a two-dimensional stochastic differential equation that has a unique weak solution but no strong solution. We show that this SDE shares notable properties with Tsirelson's example of a one-dimensional SDE with no strong solution.…
We prove that a solution, in a variational framework, to the Stratonovich stochastic partial differential equation with noise $G\left(t, \Psi_t\right) \circ dW_t$ is given by a solution to the It\^{o} equation with It\^{o}-Stratonovich…
We study the convergence of a Zakharov system driven by a time white noise, colored in space, to a multiplicative stochastic nonlinear Schr{\"o}dinger equation, as the ion-sound speed tends to infinity. In the absence of noise, the…
In this article, we construct a global martingale solution to a general nonlinear Schr\"{o}dinger equation with linear multiplicative noise in the Stratonovich form. Our framework includes many examples of spatial domains like…
The zero-noise limit of differential equations with singular coefficients is investigated for the first time in the case when the noise is an $\alpha $-stable process. It is proved that extremal solutions are selected and the respective…
In this paper, we are interested in the following one dimensional forward stochastic differential equation (SDE) \[ d X_{t}=b(t,X_{t},\omega)d t +\sigma d B_{t},\quad 0\leq t\leq T,\quad X_{0}=\,x\in \mathbb{R}, \] where the driving noise…