By Paul Lévy

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**Extra resources for Calcul des probabilités**

**Example text**

Since the BDLP (via z(1) ) jumps inﬁnitely often in any ﬁnite (time) interval, the IG-OU process also jumps inﬁnitely often in every interval. The cumulant of the BDLP (at time 1) is given by k(u) = −uab−1 (1 + 2ub−2 )−1/2 . t In the IG-OU case the characteristic function of the intOU process Yt = 0 ys ds can also be given explicitly. 20) −2 κ = −2b iu/ϑ. 14). Fast simulation of the BDLP is achieved by recalling that the BDLP is the sum of two independent L´evy processes. 08. 1 CREDIT DEFAULT SWAPS Credit Default Swaps (CDSs) are the simplest and most popular credit derivatives.

11). In line with the property of the VG(C, G, M) distribution, a VG process can be expressed as the difference of two independent Gamma processes (Madan et al. 1998) as follows: (2) XtVG = G(1) t − Gt , where G(1) = {G(1) t , t ≥ 0} is a Gamma process with parameters a = C and b = M and G(2) = {G(2) t , t ≥ 0} is a Gamma process with parameters a = C and b = G. This characterization allows the L´evy measure to be determined: νVG (dx) = C exp(Gx)|x|−1 dx C exp(−Mx)x −1 dx x<0 x >0 . 13) The L´evy measure has inﬁnite mass, and hence a VG process has inﬁnitely +∞ many jumps in any ﬁnite time interval.

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) . 1). Note that the IG process is a non-decreasing L´evy process. To simulate an IG process with parameters a and b in the time points {n t, n = 0, 1, . }, we can use the following typical scheme: X0IG = 0, IG XnIG t = X(n−1) t + in where {in , n ≥ 1} is a sequence of IG(a t, b) random variables.