Abstract:
This paper is concerned with the strong solution to the Cauchy-Dirichlet problem for backward stochastic partial differential equations of parabolic type. Existence and uniqueness theorems are obtained, due to an application of the continuation method under fairly weak conditions on variable coefficients and $C^2$ domains. The problem is also considered in weighted Sobolev spaces which allow the derivatives of the solutions to blow up near the boundary. As applications, a comparison theorem is obtained and the semi-linear equation is discussed in the $C^2$ domain.

Abstract:
Regularity of solutions is studied for backward stochastic parabolic Ito equations. An analog of the second energy inequality and the related existence theorem are obtained for domains with boundary.

Abstract:
In this paper we study the maximum principle, the existence of eigenvalue and the existence of solution for the Dirichlet problem for operators which are fully-nonlinear, elliptic but presenting some singularity or degeneracy which are similar to those of the p-Laplacian, the novelty resides in the fact that we consider the equations in bounded domains which only satisfy the exterior cone condition.

Abstract:
We consider a controlled state equation of parabolic type on the halfline $(0,+\infty)$ with boundary conditions of Dirichlet type in which the unknown is equal to the sum of the control and of a white noise in time. We study finite horizon and infinite horizon optimal control problem related by menas of backward stochastic differential equations.

Abstract:
In this paper we are concerned with the maximum principle for quasi-linear backward stochastic partial differential equations (BSPDEs for short) of parabolic type. We first prove the existence and uniqueness of the weak solution to quasi-linear BSPDE with the null Dirichlet condition on the lateral boundary. Then using the De Giorgi iteration scheme, we establish the maximum estimates and the global maximum principle for quasi-linear BSPDEs. To study the local regularity of weak solutions, we also prove a local maximum principle for the backward stochastic parabolic De Giorgi class.

Abstract:
We prove two assumptions made in an article by Ya.A. Butko, M. Grothaus, O.G. Smolyanov concerning the existence of a strongly continuous operator semigroup solving a Cauchy-Dirichlet problem for an elliptic differential operator in a bounded domain and the existence of a smooth contractive embedding of a core of the generator of the semigroup into the space $C_c^{2,\alpha}(\R^n)$. Based on these assumptions a Feynman formula for the solution of the Cauchy-Dirichlet problem is constructed in the article mentioned above. In this article we show that the assumptions are fulfilled for domains with $C^{4,\alpha}$-smooth boundary and coefficients in $C^{2,\alpha}$.

Abstract:
In this paper we introduce a model describing diffusion of species by a suitable regularization of a "forward-backward" parabolic equation. In particular, we prove existence and uniqueness of solutions, as well as continuous dependence on data, for a system of partial differential equations and inclusion, which may be interpreted, e.g., as evolving equation for physical quantities such as concentration and chemical potential. The model deals with a constant mobility and it is recovered from a possibly non-convex free-energy density. In particular, we render a general viscous regularization via a maximal monotone graph acting on the time derivative of the concentration and presenting a strong coerciveness property.

Abstract:
We consider the Cauchy-Dirichlet problem in for a class of linear parabolic partial differential equations. We assume that is an unbounded, open, connected set with regular boundary. Our hypotheses are unbounded and locally Lipschitz coefficients, not necessarily differentiable, with continuous data and local uniform ellipticity. We construct a classical solution to the nonhomogeneous Cauchy-Dirichlet problem using stochastic differential equations and parabolic differential equations in bounded domains. 1. Introduction In this paper, we study the existence and uniqueness of a classical solution to the Cauchy-Dirichlet problem for a linear parabolic differential equation in a general unbounded domain. Let be the differential operator where , , and . The Cauchy-Dirichlet problem is where is an unbounded, open, connected set with regular boundary. In the case of bounded domains, the Cauchy-Dirichlet problem is well understood (see [1, 2] for a detailed description of this problem). Moreover, when the domain is unbounded and the coefficients are bounded, the existence of a classical solution to (1.2) is well known. For a survey of this theory see [3, 4] where the problem is studied with analytical methods and [5] for a probabilistic approach. In the last years, parabolic equations with unbounded coefficients in unbounded domains have been studied in great detail. For the particular case when , there exist many papers in which the existence, uniqueness, and regularity of the solution is studied under different hypotheses on the coefficients; see for example, [6–17]. In the case of general unbounded domains, Fornaro et al. in [18] studied the homogeneous, autonomous Cauchy-Dirichlet problem. They proved, using analytical methods in semigroups, the existence and uniqueness of a solution to the Cauchy-Dirichlet problem when the coefficients are locally , with bounded, and functions with a Lyapunov type growth; that is, there exists a function such that and for some , It is also assumed that has a boundary. Schauder-type estimates were obtained for the gradient of the solution in terms of the data. Bertoldi and Fornaro in [19] obtained analogous results for the Cauchy-Neumann problem for an unbounded convex domain. Later, in [20] Bertoldi et al. generalized the method to nonconvex sets with boundary. They studied the existence, uniqueness, and gradient estimates for the Cauchy-Neumann problem. For a survey of this results, see [21]. Using the theory of semigroups, Da Prato and Lunardi studied, in [22, 23], the realization of the elliptic operator , in the

Abstract:
Backward stochastic partial differential equations of parabolic type in bounded domains are studied in the setting where the coercivity condition is not necessary satisfied and the equation can be degenerate. Some generalized solutions based on the representation theorem are suggested. In addition to problems with a standard Cauchy condition at the terminal time, problems with special non-local boundary conditions are considered. These non-local conditions connect the terminal value of the solution with a functional over the entire past solution. Uniqueness, solvability and regularity results are obtained. Some applications to portfolio selection problem are considered.

Abstract:
We investigate the regularity of linear stochastic parabolic equations with zero Dirichlet boundary condition on bounded Lipschitz domains $O \subset R^d$ with both theoretical and numerical purpose. We use N.V. Krylov's framework of stochastic parabolic weighted Sobolev spaces $\mathfrak{H}^{\gamma,q}_{p,\theta}(O;T)$. The summability parameters p and q in space and time may differ. Existence and uniqueness of solutions in these spaces is established and the H\"older regularity in time is analysed. Moreover, we prove a general embedding of weighted Lp(O)-Sobolev spaces into the scale of Besov spaces $B^\alpha_{\tau,\tau}(O), 1/\tau=\alpha/d+1/p, \alpha > 0$. This leads to a H\"older-Besov regularity result for the solution process. The regularity in this Besov scale determines the order of convergence that can be achieved by certain nonlinear approximation schemes.