Remarks on compositions of some random integral mappings

Abstract

The random integral mappings (some type of functionals of Lévy processes) are continuous homomorphisms between convolution subsemigroups of the semigroup of all infinitely divisible measures. Compositions of those random integrals (mappings) can be always expressed as another single random integral mapping. That fact is illustrated by some old and new examples.

Exponential bounds of ruin probabilities for non-homogeneous risk models

“…that is, those classes correspond to the compositions of k + 1 mappings I and J , respectively; see [19] for the general theory of compositions of random integral mappings.…”

“…where h is a real function, r (a time change) is a monotone, non-negative function and D h,r (a,b] denotes the domain of the random integral I h,r (a,b] ; for details see, for instance, [13,14,15,17,19]. To Y (resp.…”

Urbanik type subclasses of the free-infinitely divisible transforms

“…(0,1] (I t,t (0,1] (...(I t,t (0,1] (ν))), (k-times); see Jurek (2004), Proposition 4 and Corollary 2 and for more general theory of compositions of random integrals see Jurek (2018).…”

“…where h is a real function, r (a time change) is a monotone, nonnegative function and D h,r (a,b] denotes the domain of a random integral I h,r (a,b] ; for details see for instance Jurek (1988Jurek ( ) or (1989Jurek ( ) or (2004Jurek ( ) or (2007Jurek ( ) or (2018. To Y (to µ) we refer as the background driving Lévy process (BDLP) (background driving probability distribution(BDPD)) of the measure ρ.…”

“…* ) = I(ID log ), L k+1 = I(I e −t ,r k (t) k−1 dx;that is, those classes correspond to compositions of k + 1 mappings I and J , respectively; seeJurek (2018) for a general theory of compositions of random integral mappings.where a ∈ R and G is a finite Borel measure on (−2, 0) ∪ (0, 2]. Let use the identification (4) ThenV ν (it) = it 2 ∞ 0 log φ ν (−u)e −tu du = |x|)t 1−|x| ie iπx/2 + x G(dx) log(u)e −u du = 1 − γ; (Euler's constant);which gives part (i) of Proposition 4.…”