Operator fractional Brownian motions (OFBMs) are zero mean, operator self-similar (o.s.s.) Gaussian processes with stationary increments. They generalize univariate fractional Brownian motions to the multivariate context. It is well-known that the so-called symmetry group of an o.s.s. process is conjugate to subgroups of the orthogonal group. Moreover, by a celebrated result of Hudson and Mason, the set of all exponents of an operator self-similar process can be related to the tangent space of its symmetry group.

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Exponents, Symmetry Groups and Classification of Operator Fractional Brownian Motions

Operator fractional Brownian motions (OFBMs) are zero mean, operator self-similar (o.s.s.), Gaussian processes with stationary increments. They generalize univariate fractional Brownian motions to the multivariate context. It is well-known that the so-called symmetry group of an o.s.s. process is conjugate to subgroups of the orthogonal group. Moreover, by a celebrated result of Hudson and Mason, the set of all exponents of an operator self-similar process can be related to the tangent space of its symmetry group. 
In this paper, we revisit and study both the symmetry groups and exponent sets for the class of OFBMs based on their spectral domain integral representations. A general description of the symmetry groups of OFBMs in terms of subsets of centralizers of the spectral domain parameters is provided. OFBMs with symmetry groups of maximal and minimal types are studied in any dimension. In particular, it is shown that OFBMs have minimal symmetry groups (as thus, unique exponents) in general, in the topological sense. Finer classification results of OFBMs, based on the explicit construction of their symmetry groups, are given in the lower dimensions 2 and 3. It is also shown that the parametrization of spectral domain integral representations are, in a suitable sense, not affected by the multiplicity of exponents, whereas the same is not true for time domain integral representations.

Exponents, symmetry groups and classification of operator fractional Brownian motions

Random integral representations for free-infinitely divisible and tempered stable distributions☆

The role of the financial mathematician is to find solutions to problems in finance through the application of mathematical theory. The motivation for this work is specification of models to accurately describe the price evolution of a risky asset, a risky asset is for example a security traded on a financial market such as a stock, currency or benchmark index. This thesis makes contributions in two classes of models, namely activity time models and integer valued models, by the discovery of new real valued and integer valued stochastic processes. In both model frameworks applications to option pricing are considered.

Fractal activity time and integer valued models in finance

A new approach to extreme value theory is presented for vector data with heavy tails. The tail index is allowed to vary with direction, where the directions are not necessarily along the coordinate axes. Basic asymptotic theory is developed, using operator regular variation and extremal integrals. A test is proposed to judge whether the tail index varies with direction in any given data set.

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Extreme value theory with operator norming.

The central theme of that monograph are limit distributions in a scheme (1) A n d-valued independent random vectors, A i 's are MATRICES (linear bounded operators) on R d and x i 's are deterministic R d shift vectors. If in (1) one assumes that the triangular array A n X k , 1 ≤ k ≤ n, n ≥ 1, is uniformly infinitesimal then limit distributions are called operator-selfdecomposable. This class includes the classical selfdecomposable measures. For independent and identically distributed X i 's limiting laws are called operator-stable measures. Among them we have the classical and well know stable measures.

J. D. Mason

Papers

15. Operator-Limit Distributions in Probability Theory