Cumulant representations of Lundberg’s root for semicontinuous processes. (English. Ukrainian original) Zbl 1235.60045

Gusak, D. V.

Theory Probab. Math. Stat. 82, 1-10 (2011); translation from Teor. Jmovirn. Mat. Stat. No. 82, 21-29.

Summary: For the case of homogeneous processes \( \xi (t)\), \( \xi (0)=0\), \( t\geq 0\), with independent increments and negative jumps, A.V. Skorokhod [Stochastic processes with independent increments (Russian). Moskva: Izdatel’stvo “Nauka”. Glavnaya Redaktsiya Fiziko-Matematicheskoj Literatury (1986; Zbl 0622.60082)] proved that the functional \[ \tau ^+(x)=\inf\left\{t\geq 0\colon \xi (t)>x\right\}, \qquad x\geq 0, \] is a nondecreasing process with independent increments with respect to \( x\), and its moment generating function is expressed via the cumulant that satisfies the corresponding Lundberg equation. The corresponding representations of this cumulant are specified and its Lévy characteristics (namely, \( \gamma \) and Lévy’s integral measure \( N(x)\)) are evaluated by using results of the author’s earlier work [Boundary value problems for processes with independent increments in risk theory (Ukrainian). Kyïv: Instytut Matematyky NAN Ukraïny (2007; Zbl 1199.60001)] on the processes under consideration.

MSC:

60G50	Sums of independent random variables; random walks
60K05	Renewal theory

Keywords:

independent increments; negative jumps; Lundberg equation

Citations:

Zbl 0622.60082; Zbl 1199.60001

References:

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