Related papers: Stochastic and partial differential equations on n…
We consider the simplest example of a time-dependent first order Hamilton-Jacobi equation, in one space dimension and with a bounded and Lipschitz continuous Hamiltonian which only depends on the spatial derivative. We show that if the…
This paper studies the properties of solutions for a double nonlinear time-dependent parabolic equation with variable density, not in divergence form with a source or absorption. The problem is formulated as a partial differential equation…
This paper establishes the well-posedness of stochastic partial differential equations with reflection in an infinite-dimensional ball, within the fully local monotone framework. Our result is very general, including many important models…
This paper is an attempt to extend the notion of viscosity solution to nonlinear stochastic partial differential integral equations with nonlinear Neumann boundary condition. Using the recently developed theory on generalized backward…
We prove the global-in-time existence of nonnegative weak solutions to a class of fourth order partial differential equations on a convex bounded domain in arbitrary spatial dimensions. Our proof relies on the formal gradient flow structure…
This paper provides an overview of the recently developed notion of viscosity solutions of path-dependent partial di erential equations. We start by a quick review of the Crandall- Ishii notion of viscosity solutions, so as to motivate the…
We consider the Cauchy problem for a stochastic scalar parabolic-hyperbolic equation in any space dimension with nonlocal, nonlinear, and possibly degenerate diffusion terms. The equations are nonlocal because they involve fractional…
In this paper, we investigate the deterministic multidimensional Skorokhod problem with normal reflection in a family of time-dependent convex domains that are c\`adl\`ag with respect to the Hausdorff metric. We then show the existence and…
This work contributes a systematic survey and complementary insights of reflecting Brownian motion and its properties. Extension of the Skorohod problem's solution to more general cases is investigated, based on which a discussion is…
We prove comparison, uniqueness and existence results for viscosity solutions to a wide class of fully nonlinear second order partial differential equations $F(x, u, du, d^{2}u)=0$ defined on a finite-dimensional Riemannian manifold $M$.…
In this paper, we study a system of second order integro-partial differential equations with interconnected obstacles with non-local terms, related to an optimal switching problem with the jump-diffusion model. Getting rid of the…
We extend the notion of viscosity solutions for path-dependent PDEs introduced by Ekren et al. [Ann. Probab. 42 (2014), no. 1, 204-236] to path-dependent integro-differential equations and establish well-posedness, i.e., existence,…
In this work, we introduce a new Skorokhod problem with two reflecting barriers when the trajectories of the driven process and the barriers are right and left limited. We show that this problem has an explicit unique solution in a…
We prove the existence and uniqueness of solutions to the time-dependent incompressible Navier-Stokes equations with a free-boundary governed by surface tension. The solution is found using a topological fixed-point theorem for a nonlinear…
We generalize the notion of pathwise viscosity solutions, put forward by Lions and Souganidis to study fully nonlinear stochastic partial differential equations, to equations set on a sub-domain with Neumann boundary conditions. Under a…
This paper shows how the theory of dynamic risk measures provides viscosity solutions to a family of second-order parabolic partial differential equations, even in the degenerate case. First, motivated by the martingale problem approach of…
We consider reflected generalized backward doubly stochastic differential equations driven by a non-homogeneous L\'evy process. Under stochastic conditions on the coefficients, we prove the existence and uniqueness of a solution.…
We study boundary regularity of viscosity solutions to fully nonlinear degenerate or singular parabolic equations. The gradient-dependent degeneracy or singularity, along with the time derivative, introduces significant challenges beyond…
Sticky diffusion models a Markovian particle experiencing reflection and temporary adhesion phenomena at the boundary. Numerous numerical schemes exist for approximating stopped or reflected stochastic differential equations (SDEs), but…
The notes are an overview of part of the theory of pathwise weak solutions to two classes of scalar fully nonlinear first- and second-order degenerate parabolic partial differential equations with multiplicative rough time dependence, a…