Related papers: Path-by-path uniqueness of infinite-dimensional st…
This paper studies the existence and uniqueness of solution of It\^o type stochastic differential equation $dx(t)=b(t, x(t), \om)dt+\si(t,x(t), \om) d B(t)$, where $B(t)$ is a fractional Brownian motion of Hurst parameter $H>1/2$ and…
We consider singular SDEs like \begin{equation} \label{ss} dX_t = b(t, X_t) dt + A X_t dt + \sigma(t) d{L}_t , \;\; t \in [0,T], \;\; X_0 =x \in {\mathbb R}^n, \end{equation} where $A$ is a real $n \times n $ matrix, i.e., $A \in {{\mathbb…
In this paper we study the pathwise uniqueness of solution to the following stochastic partial differential equation (SPDE) with H\"older continuous coefficient: \begin{eqnarray*} \frac{\partial X_t(x)}{\partial t}=\frac{1}{2} \Delta X_t(x)…
We prove pathwise uniqueness for a class of stochastic differential equations (SDE) on a Hilbert space with cylindrical Wiener noise, whose nonlinear drift parts are sums of the sub-differential of a convex function and a bounded part. This…
We show that any stochastic differential equation (SDE) driven by Brownian motion with drift satisfying the Krylov-R\"ockner condition has exactly one solution in an ordinary sense for almost every trajectory of the Brownian motion.…
Pathwise uniqueness is established for a class of one-dimensional stochastic Volterra equations driven by Brownian motion with singular kernels and H\"older continuous diffusion coefficients. Consequently, the existence of unique strong…
We present a new approach to Davie's theorem on the uniqueness of solutions to the equation $dX_t = b(t, X_t)\,dt + dW_t$ for almost all Brownian paths. A generalization of this result and a discussion of some close problems are given.
We prove path-by-path uniqueness of solution to hyperbolic stochastic partial differential equations when the drift coefficient is the difference of two componentwise monotone Borel measurable functions of spatial linear growth. The…
The Tanaka equation $dX_t={\operatorname{sign}}(X_t)\,dB_t$ is an example of a stochastic differential equation (SDE) without strong solution. Hence pathwise uniqueness does not hold for this equation. In this note we prove that if we…
We prove pathwise uniqueness for stochastic differential equations driven by non-degenerate symmetric $\alpha$-stable L\'evy processes with values in $\R^d$ having a bounded and $\beta$-H\"older continuous drift term. We assume $\beta > 1 -…
We establish the existence and uniqueness for a one-dimensional stochastic differential equation driven by a Brownian motion and a pure jump {\levy} process. It is shown that under fairly general conditions on the coefficients, pathwise…
Pathwise uniqueness for multi-dimensional stochastic McKean--Vlasov equation is established under moderate regularity conditions on the drift and diffusion coefficients. Both drift and diffusion depend on the marginal measure of the…
We study a one-dimensional stochastic differential equation driven by a stable L\'evy process of order $\alpha$ with drift and diffusion coefficients $b,\sigma$. When $\alpha\in (1,2)$, we investigate pathwise uniqueness for this equation.…
We present a condition for a stochastic differential equation dX_{t}={\mu}(t,X_{t})dt+{\sigma}(t,X_{t})dB_{t} to have a unique functional solution of the form Z(t,B_{t}). The condition expresses a relation between {\mu} and {\sigma}. A…
The theory of one-dimensional stochastic differential equations driven by Brownian motion is classical and has been largely understood for several decades. For stochastic differential equations with jumps the picture is still incomplete,…
This paper extends the results of Ma, Wu, Zhang, Zhang [11] to the context of path-dependent multidimensional forward-backward stochastic differential equations (FBSDE). By path-dependent we mean that the coefficients of the…
These notes rigorously construct the stochastic integral of a Hilbert Space valued process driven by a Cylindrical Brownian Motion. We expand upon this stochastic calculus to present an introduction to stochastic differential equations in…
One introduces a new variational concept of solution for the stochastic differential equation $dX+A(t)X\,dt+\lambda X\,dt=X\,dW,$ $t\in(0,T)$; $X(0)=x$ in a real Hilbert space where $A(t)=\partial\varphi(t)$, $t\in(0,T)$, is a maximal…
Let $U,H$ be two separable Hilbert spaces. The main goal of this paper is to study the weak uniqueness of the Stochastic Differential Equation evolving in $H$ \begin{align*} dX(t)=AX(t)dt+\mathcal{V}B(X(t))dt+GdW(t), \quad t>0, \quad X(0)=x…
We establish in this paper the existence of weak solutions of infinite-dimensional shift invariant stochastic differential equations driven by a Brownian term. The drift function is very general, in the sense that it is supposed to be…