Calcul des probabilités by Paul Lévy

By Paul Lévy

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For 0 ≤ s < t the random variable Wt − Ws follows a Normal distribution N (0, t − s). The paths of a Brownian motion are continuous but very erratic. 1 Finally, the following scaling property holds: if W = {Wt , t ≥ 0} is a Brownian motion then for any c = 0, Wˆ = {Wˆ t = cWt/c2 , t ≥ 0} is also a Brownian motion. A Brownian motion can be easily simulated by discretizing time using a very small step t. The value of a Brownian motion at time points {n t, n = 1, 2, . } is obtained by sampling a series of Standard Normal N (0, 1) random numbers {νn , n = 1, 2, .

A CDS is a bilateral agreement where the protection buyer transfers the credit risk of a reference entity to the protection seller for a determined amount of time T . The buyer of this protection makes predetermined payments to the seller. The payments continue until the maturity date T of the contract, or until default occurs, whichever is earlier. In the case of default of the reference entity, the contract provides that the protection buyer pays to the protection seller a determined amount. 1 presents schematically the building blocks of a CDS contract.

In other words, ν(A) is the expected number of jumps per unit time, whose size belongs to A. +1 If σ 2 = 0 and −1 |x|ν(dx) < ∞, it follows from standard L´evy process theory (Bertoin 1996, Sato 1999, Kyprianou 2006) that the process is of finite variation. Moreover, there is a finite number of jumps in any finite interval and the process is said to be of finite activity. Because the Brownian motion is of infinite variation, a L´evy process with a Brownian component is of infinite variation. e. with no Brownian component (σ 2 = 0), is of infinite variation if and only if +1 −1 |x|ν(dx) = ∞.

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