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The Normal distribution is therefore said to have light-tail behaviour. e. {Ws + bs, s ≥ 0} reaches the positive level a > 0. ∗∗ ) φIG (u; a, b) = exp −a( −2iu + b2 − b) . The density function of the IG(a, b) law is explicitly known: a fIG (x; a, b) = √ exp(ab)x −3/2 exp(−(a 2 x −1 + b2 x)/2), 2π x > 0. 4 describes the characteristics of the IG distribution. Moreover, the IG distribution satisfies the √ following scaling property: if X is IG(a, b), for any positive √ c, cX is IG( ca, b/ c). The IG distribution is infinitely divisible and we can thus define the IG process X(IG) = {Xt(IG) , t ≥ 0}, with parameters a, b > 0 as the process which starts at zero, has independent and stationary increments, and is such that its characteristic function is given by: E[exp(iuXt(IG) )] = φIG (u; at, b) = exp −at ( −2iu + b2 − b) .

To be precise, the following theorem describes the one-to-one relationship between L´evy processes and infinitely divisible distributions. 1 (Infinite Divisibility of L´evy Processes) Let X = {Xt , t ≥ 0} be a L´evy process. Then X = {Xt , t ≥ 0} has an infinitely divisible distribution F for every t. Conversely if F is an infinitely divisible distribution there exists a L´evy process X = {Xt , t ≥ 0} such that the distribution of X1 is given by F . 6). 2 To be mathematically precise we have to also impose a technical condition, namely that Xt is a stochastically continuous process: ∀ε > 0 lim P (|Xt+h − Xt | ≥ ε) = 0.

Moreover, it seems reasonable that the amount of new information released should not be affected by the amount already released, in other words, the information process should have independent increments. 10 A sample path of a VG process stationary. This leads to modelling the information process by a non-decreasing process with stationary and independent increments. 6 as the sum of an idiosyncratic component (σ 2 ), coming from the Brownian motion, and an exogenous component (νθ 2 ) arising from the time change.

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