Processus de Lévy en Finance - Laboratoire de Probabilités et ...

172 CHAPTER 5. APPLICATIONS OF LEVY COPULAS
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Figure 5.2: Contour plots of Lévy density with 1-stable margins and Lévy copula (5.4) with
η = 1. Top left: θ = 0.2. Top right: θ = 0.7. Bottom left: θ = 1. Bottom right: θ = 5. On each
curve the Lévy density ν(x, y) is constant and it increases from outer curves to inner curves.
values of η, positive jumps in one component can correspond to both positive and negative
jumps in the other component. The parameter θ is responsible for the dependence of absolute
values of jumps in different components: from Figure 5.2 it is seen that as θ grows from 0.2 to
5, the dependence structure changes from independence to almost complete dependence.
Theorem 5.2 can be used to construct parsimonious models of dependence; this is typically
useful when one has little information about the dependence structure of the problem. If a more
precise vision is necessary, a possible strategy is to model the dependence in each corner of the
Lévy measure separately. A d-dimensional Lévy copula can be constructed from 2 d positive