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We show the existence of unique global strong solutions of a class of stochastic differential equations on the cone of symmetric positive definite matrices. Our result includes affine diffusion processes and therefore extends considerably the known statements concerning Wishart processes, which have recently been extensively employed in financial… (More)

We show that the only dynamic risk measure which is law invariant, time consistent and relevant is the entropic one. Moreover, a real valued function c on L ∞ (a, b) is normalized, strictly monotone, continuous, law invariant, time consistent and has the Fatou property if and only if it is of the form c(X) = u −1 • E [u(X)], where u : (a, b) → R is a… (More)

We revisit affine diffusion processes on general and on the canonical state space in particular. A detailed study of theoretic and applied aspects of this class of Markov processes is given. In particular, we derive admissibility conditions and provide a full proof of existence and uniqueness through stochastic invariance of the canonical state space.… (More)

We introduce a concept of causality in the framework of generalized pseudo-Riemannian Geometry in the sense of Colombeau. This is motivated by a generalized point value characterization of generalized pseudo-Riemannian metrics due to M. Kunzinger et al. We prove an appropriate version of the inverse Cauchy-Schwarz inequality. As an application, we establish… (More)

We prove local unique solvability of the wave equation for a large class of weakly singular, locally bounded space-time metrics in a suitable space of generalised functions.

We show spherical completeness of the ring of Colombeau generalized real numbers endowed with the sharp norm. As an application, we establish a Hahn-Banach extension theorem for ultra-pseudo-normed modules of generalized functions in the sense of Colombeau.

We introduce a concept of causality in the framework of generalized pseudo-Riemannian Geometry in the sense of J.F. Colombeau and establish the inverse Cauchy-Schwarz inequality in this context. As an application, we prove a dominant energy condition for some energy tensors as put forward in Hawking and Ellis's book " The large scale structure of space-time… (More)

We show that contrary to recent papers by S. Albeverio, A. Yu. Khrennikov and V. Shelkovich, point values do not determine elements of the so-called p-adic Colombeau-Egorov algebra G(Q n p) uniquely. We further show in a more general way that for an Egorov algebra G(M, R) of generalized functions on a locally compact ultrametric space (M, d) taking values… (More)

- Clemens Hanel, Eberhard Mayerhofer, Stevan Pilipović, Hans Vernaeve
- 2007

We investigate homogeneity in the special Colombeau algebra on R d as well as on the pierced space R d \ {0}. It is shown that strongly scaling invariant functions on R d are simply the constants. On the pierced space, strongly homogeneous functions of degree α admit tempered representatives, whereas on the whole space, such functions are polynomials with… (More)

We present recent developments concerning Lorentzian geometry in algebras of generalized functions. These have, in particular, raised a new interest in refined regularity theory for the wave equation on singular space-times.