As we have remarked, dependence is very common in time series observations. To model this time series dependence, we start with univariate ARMA models. To motivate the model, basically we can track two lines of thinking. First, for a series xt, we can model that the level of its current observations depends on the level of its lagged observations. For example, if we observe a high GDP realizati...

Let us consider the insurance company having an initial capital a > 0 which insures two kind of risks. The i-th risk makes the stream of insurance premiums with constant rate ci and pays out successive claims, which are representing by i.i.d. random variables Xi,1, Xi,2, . . . with cumulative distribution function Hi. The losses related to the i-th risk occur according to the renewal process {N...

In this paper, we define and study free Jacobi processes of parameters λ > 0 and 0 < θ ≤ 1, as the limit of the complex version of the matrix Jacobi process already defined by Y. Doumerc. In the first part, we focus on the stationary case for which we compute the law (that does not depend on time) and derive, for λ ∈]0, 1] and 1/θ ≥ λ + 1 a free SDE analogous to the classical one. In the second...