Weak error for Continuous Time Markov Chains related to fractional in time P(I)DEs
We provide sharp error bounds for the difference between the transition densities of some multidimensional Continuous Time Markov Chains (CTMC) and the fundamental solutions of some fractional in time Partial (Integro) Differential Equations (P(I)DEs). Namely, we consider equations involving a time fractional derivative of Caputo type and a spatial operator corresponding to the generator of a non degenerate Brownian or stable driven Stochastic Differential Equation (SDE).
In this paper, we consider a multidimensional time-changed stochastic process in the context of asset-pricing modeling. The proposed model is constructed from stable processes, and its construction is based on two popular concepts:multivariate subordination and Lévy copulas. From a theoretical point of view, our main result is Theorem 1, which yields a simulation method from the considered class of processes. Our empirical study shows that the model represents the correlation between asset returns quite well. Moreover, we provide some evidence that this model is more appropriate for describing stock prices than classical time-changed Brownian motion, at least if the cumulative amount of transactions is used for a stochastic time change.
This paper is concerned with Random walk approximations of the Brownian motion on the Affine group Aff(R). We are in particular interested in the case where the innovations are discrete. In this framework, the return probabilities of the walk have fractional exponential decay in large time, as opposed to the polynomial one of the continuous object. We prove that in tegrating those return probabilities on a suitable neighborhood of the origin, the expected polynomial decay is restored. This is what we call a Quasi-local theorem.
In this paper, we consider some properties of the model constructed from the one-dimensional stable processes by changing the time to a non-decreasing Lévy process. Our first result reveals a relation between this class of processes and the class of time-changed Brownian motions. Moreover, we describe the CGMY (Carr-Geman-Madan-Yor) model as subordinated stable process, and show the representation of the Lévy density of the corresponding subordinator via the Mellin–Barnes integral.
Let k be a field of characteristic zero, let G be a connected reductive algebraic group over k and let g be its Lie algebra. Let k(G), respectively, k(g), be the field of k- rational functions on G, respectively, g. The conjugation action of G on itself induces the adjoint action of G on g. We investigate the question whether or not the field extensions k(G)/k(G)^G and k(g)/k(g)^G are purely transcendental. We show that the answer is the same for k(G)/k(G)^G and k(g)/k(g)^G, and reduce the problem to the case where G is simple. For simple groups we show that the answer is positive if G is split of type A_n or C_n, and negative for groups of other types, except possibly G_2. A key ingredient in the proof of the negative result is a recent formula for the unramified Brauer group of a homogeneous space with connected stabilizers. As a byproduct of our investigation we give an affirmative answer to a question of Grothendieck about the existence of a rational section of the categorical quotient morphism for the conjugating action of G on itself.
Let G be a connected semisimple algebraic group over an algebraically closed field k. In 1965 Steinberg proved that if G is simply connected, then in G there exists a closed irreducible cross-section of the set of closures of regular conjugacy classes. We prove that in arbitrary G such a cross-section exists if and only if the universal covering isogeny Ĝ → G is bijective; this answers Grothendieck's question cited in the epigraph. In particular, for char k = 0, the converse to Steinberg's theorem holds. The existence of a cross-section in G implies, at least for char k = 0, that the algebra k[G]G of class functions on G is generated by rk G elements. We describe, for arbitrary G, a minimal generating set of k[G]G and that of the representation ring of G and answer two Grothendieck's questions on constructing generating sets of k[G]G. We prove the existence of a rational (i.e., local) section of the quotient morphism for arbitrary G and the existence of a rational cross-section in G (for char k = 0, this has been proved earlier); this answers the other question cited in the epigraph. We also prove that the existence of a rational section is equivalent to the existence of a rational W-equivariant map T- - - >G/T where T is a maximal torus of G and W the Weyl group.