Asymptotic Expansion of Laplace Convolutions for Large Argument and Tail Densities for Certain Sums of Random Variables

Value of the future: Discounting in random environments

“…Long-time asymptotic expressions of D(t) are easily derived using Tauberian theorems which relate the small σ behavior ofD(σ ) with the long-time behavior of D(t) [33]. In Appendix C we show that as t → ∞…”

“…In order to find asymptotic expressions of D(t) for large values of t we will use the so-called Tauberian theorems which under rather general conditions relate the long-time behavior of any function with the small σ behavior of its Laplace transform [33]. Looking at Eq.…”

“…(C33) Tauberian theorems [33] allow us to obtain the asymptotic long-time behavior of D(t) by means of the Laplace inversion of the approximate expression (C32). This yields the exponential decay:…”

The continuous time random walk formalism in financial markets

“…(49) in terms of the incomplete gamma function: U(a, a, x) = e x Γ(1 − a, x) (Magnus et al 1966). Therefore, Now, using Tauberian theorems (Handelsman and Lew 1974), the behavior as t → 0 of ψ(t) will be given by the behavior ofψ(s) as s → ∞ while the behavior of ψ(t) as t → ∞ is determined byψ(s) as s → 0.…”

“…One can also obtain, by using Tauberian theorems (Handelsman and Lew 1974), the asymptotic behavior of ψ(t) as t → ∞ and also the short time behavior t → 0. It is worth to mention that Kotulski gives an alternative method based on renewal theory and limit theorems for random sum of jumps.…”