Band 32, Heft 2

A one-dimensional generalized backward stochastic differential equation with jumps and two barriers is the main objective of this paper. When the generators are monotone and the barriers are right continuous with left limits and completely separated, we prove the existence and uniqueness of a solution. As in application, we provide a probabilistic interpretation of a solution of a double obstacle problem of second-order parabolic integral-partial differential equations with nonlinear Neumann boundary conditions.

In this paper, random optimization problems are investigated. Some sufficient conditions ensuring the existence of random solutions to random optimization problems are proposed.

In this paper, we have studied stochastic differential equations with unbounded delay in fractional power spaces perturbed by fractional Ornstein–Uhlenbeck process Y H , ξ ⁢ ( t ) {{Y^{H,\xi}}(t)} with H ∈ ( 1 2 , 1 ) {H\in(\frac{1}{2},1)} . Subsequently, the existence and uniqueness of mild solution of the considered equation have been proved with fixed-point theorem. Finally, we obtain the global attracting set of the considered equations by some stochastic analysis and inequality technique.

In this paper, we study the differentiability of solutions of neutral stochastic differential equations driven by G -Brownian motion with respect to parameter. Under suitable assumptions, we show that solutions are differentiable with respect to the parameter which appears in the initial data. In addition, the stochastic differential equation of the derivative is given and the existence-uniqueness of solution is proved. Moreover, an example to illustrate the theoretically obtained results is presented.

We prove a Wegner estimate for alloy-type models merely assuming that the single site potential is lower bounded by a characteristic function of a thick set (a particular class of sets of positive measure). The proof exploits on one hand recently proven unique continuation principles or uncertainty relations for linear combinations of eigenfunctions of the Laplacian on cubes and on the other hand the well developed machinery for proving Wegner estimates. We obtain a Wegner estimate with optimal volume dependence at all energies, and localization near the minimum of the spectrum, even for some non-stationary random potentials. We complement the result by showing that a lower bound on the potential by the characteristic function of a thick set is necessary for a Wegner estimate to hold. Hence, we have identified a sharp condition on the size for the support of random potentials that is sufficient and necessary for the validity of Wegner estimates.

We study a family of representations of the canonical commutation relations (CCR)-algebra, which we refer to as “admissible,” with an infinite number of degrees of freedom. We establish a direct correlation between each admissible representation and a corresponding Gaussian stochastic calculus. Moreover, we derive the operators of Malliavin’s calculus of variation using an algebraic approach, which differs from the conventional methods. The Fock-vacuum representation leads to a maximal symmetric pair. This duality perspective offers the added advantage of resolving issues related to unbounded operators and dense domains much more easily than with alternative approaches.