A stochastic solution is constructed for a fractional generalization of the KPP (Kolmogorov, Petrovskii, Piskunov) equation. The solution uses a fractional generalization of the branching exponential process and propagation processes which are spectral integrals of Levy processes.

Let X = {X(t), t ∈ R+} be a real-valued symmetric Lévy process with continuous local times {L x t , (t, x) ∈ R+ × R} and characteristic function Ee iλX(t) = e −tψ(λ). Let σ 2 0 (x − y) = 4 π ∞ 0 sin 2 (λ(x − y)/2) ψ(λ) dλ. If σ 2 0 (h) is concave, and satisfies some additional very weak regularity conditions, then for any p ≥ 1, and all t ∈ R+, lim h↓0 b a L x+h t − L x t σ0(h) p dx = 2 p/2 E|η...

In this paper the problem of European option valuation in a Levy process setting is analysed. In our model the underlying asset follows a geometric Levy process. The jump part of the log-price process, which is a linear combination of Poisson processes, describes upward and downward jumps in price. The proposed pricing method is based on stochastic analysis and the theory of fuzzy sets. We assu...