The earliest model of stock prices based on Brownian diffusion is the Bachelier model. In this paper we propose an extension of the Bachelier model, which reflects the subdiffusive nature of the underlying asset dynamics. The subdiffusive property is manifested by the random (infinitely divisible) periods of time, during which the asset price does not change. We introduce a subdiffusive arithme...

If α is an irreducible nonexpansive ergodic automorphism of a compact abelian group X (such as an irreducible nonhyperbolic ergodic toral automorphism), then α has no finite or infinite state Markov partitions, and there are no nontrivial continuous embeddings of Markov shifts in X. In spite of this we are able to construct a symbolic space V and a class of shift-invariant probability measures ...

Let Γ be a finitely generated, amenable group. Using an idea ofÉ. Ghys, we prove that if Γ has a nontrivial, orientation-preserving action on the real line, then Γ has an infinite, cyclic quotient. (The converse is obvious.) This implies that if Γ has a faithful action on the circle, then some finite-index subgroup of Γ has the property that all of its nontrivial, finitely generated subgroups h...

In this paper, we propose a multivariate market model with returns assumed to follow a multivariate normal tempered stable distribution. This distribution, defined by a mixture of the multivariate normal distribution and the tempered stable subordinator, is consistent with two stylized facts that have been observed for asset distributions: fat-tails and an asymmetric dependence structure. Assum...

The notion of a successful coupling of Markov processes, based on the idea that both components of the coupled system “intersect” in finite time with probability one, is extended to cover situations when the coupling is unnecessarily Markovian and its components are only converging (in a certain sense) to each other with time. Under these assumptions the unique ergodicity of the original Markov...

Almost every, essentially: Given a Lebesgue measure space (X,B, μ), a property P (x) predicated of elements of X is said to hold for almost every x ∈ X, if the set X \ {x : P (x) holds} has zero measure. Two sets A,B ∈ B are essentially disjoint if μ(A ∩B) = 0. Conservative system: Is an infinite measure preserving system such that for no set A ∈ B with positive measure are A,T−1A,T−2A, . . . p...