## Computer approaches to mathematical problems by Jurg Nievergelt, etc.

By Jurg Nievergelt, etc.

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One then defines, for every such infinitely divisible distribution, a stochastic process, X = {X t , t ≥ 0}, called a Lévy process, which starts at zero and has independent and stationary increments such that the distribution of an increment X t+s − X s over the time period [s, s + t],s, t ≥ 0 has φ(x)t as its characteristic function. The function log φ(x) is often called the characteristic exponent, and one can prove that it satisfies the following Lévy–Khintchine formula (see Schoutens, 2003) 1 log φ(x) = iγ u − σ 2 u 2 + 2 +∞ −∞ exp(iux) − 1 − iux|x|<1 ν(dx), with γ ∈ R, σ > 0 and ν(dx) a measure on the real line.

The price of a security paying a dollar if the event occurs. If that price is paid at event resolution, the event price we speak of is then a forward price. These forward prices are non-negative, being claims to non-negative cash flows. The prices for disjoint or mutually exclusive events are additive by no-arbitrage. They sum to unity over a set of mutually exclusive and exhaustive events as the forward price of a dollar for certain is a dollar. Hence, prices of events behave like probabilities (pricing world) and the mathematics of probability applies to them, but they are not probabilities; they are forward prices.

1, that exp(−M x) , if x > 0 x exp(Gx) =C , if x < 0, |x| k(x) = C where we work with the C G M-parametrization of the VG distribution. 3 Probabilities conditional on a jump. 30) Therefore the A j and B j , j = 1, . . 3). 15, the real and imaginary parts of the multinomial characteristic function (φ (u)) N and the target VG characteristic function φ(u), we see that the approximation is not really good yet. 18). 16. 3 Numerical Techniques In this chapter we discuss several numerical implementation issues.