Related papers: Reflected rough differential equations via control…
This paper presents existence and uniqueness results for reflected system of quasilinear stochastic partial differential equations in a convex domain D from Rk. The method is based on the probabilistic interpretation of the solution by…
We discuss regular and weak solutions to rough partial differential equations (RPDEs), thereby providing a (rough path-)wise view on important classes of SPDEs. In contrast to many previous works on RPDEs, our definition gives honest…
In this paper, we introduce a new method to study the doubly reflected backward stochastic differential equation driven by G-Brownian motion (G-BSDE). Our approach involves approximating the solution through a family of penalized reflected…
In this note, we consider an optimal control problem associated to a differential equation driven by a H\"{o}lder continuous function g of index greater than 1/2. We split our study in two cases. If the coefficient of dg\_t does not depend…
We study the problem of the existence, uniqueness and stability of solutions of reflected stochastic differential equations (SDEs) with a minimality condition depending on the law of the solution (and not on the paths). We require that some…
We investigate existence, uniqueness and regularity for solutions of rough parabolic equations of the form $\partial _tu-A_tu-f=(\dot X_t(x) \cdot \nabla + \dot Y_t(x))u$ on $[0,T]\times\mathbb{R}^d.$ To do so, we introduce a concept of…
We consider stochastic differential equations with (oblique) reflection in a $2$-dimensional domain that has a cusp at the origin, i..e. in a neighborhood of the origin has the form $\{(x_1,x_2):0<x_1\leq\delta_0,\psi_1(x_1)<x_2<\psi_…
We propose a new scheme for the long time approximation of a diffusion when the drift vector field is not globally Lipschitz. Under this assumption, regular explicit Euler scheme --with constant or decreasing step-- may explode and implicit…
We construct new stationary weak solutions of the 3D Euler equation with compact support. The solutions, which are piecewise smooth and discontinuous across a surface, are axisymmetric with swirl. The range of solutions we find is different…
We study parabolic quasi-variational inequalities (QVIs) of obstacle type. Under appropriate assumptions on the obstacle mapping, we prove the existence of solutions of such QVIs by two methods: one by time discretisation through elliptic…
We use Perron's method to construct viscosity solutions of fully nonlinear degenerate parabolic pathwise (rough) partial differential equations. This provides an intrinsic method for proving the existence of solutions that relies only on a…
The universal limit theorem is a central result in rough path theory, which has been proved for: (i) rough paths with roughness $\frac{1}{3}< \alpha \leq \frac{1}{2}$; (ii) geometric rough paths with roughness $0< \alpha \leq 1$; (iii)…
In this paper, we study the reflected solutions of one-dimensional backward stochastic differential equations driven by G-Brownian motion (RGBSDE for short). The reflection keeps the solution above a given stochastic process. In order to…
In this paper we address the existence, uniqueness and approximation of solutions of delay differential equations (DDEs) with Carath\'eodory type right-hand side functions. We provide construction of randomized Euler scheme for DDEs and…
This article focuses on parabolic equations with rough diffusion coefficients which are ill-posed in the classical sense of distributions due to the presence of a singular forcing. Inspired by the philosophy of rough paths and regularity…
We develop a rough-path framework for two-parameter rough differential equations on rectangular and simplicial domains, motivated by the signature kernel and Schwinger--Dyson kernel equations. The theory is formulated in spaces of jointly…
We apply duality theory to discretized convex minimization problems to obtain computable guaranteed upper bounds for the distance of given discrete functions and the exact discrete minimizer. Furthermore, we show that the discrete duality…
We consider the Cauchy problem for semilinear parabolic equation in divergence form with obstacle. We show that under natural conditions on the right-hand side of the equation and mild conditions on the obstacle the problem has a unique…
We consider the well-posedness problem of multi-dimensional reflected backward stochastic differential equations driven by $G$-Brownian motion ($G$-BSDEs) with diagonal generators. Two methods, i.e., the penalization method and the Picard…
We present modifications of the second-order Douglas stabilizing corrections method, which is a splitting method based on the implicit trapezoidal rule. Inclusion of an explicit term in a forward Euler way is straightforward, but this will…