Analysis of the Gerber–Shiu function and dividend barrier problems for a risk process with two classes of claims

2009 ◽  
Vol 45 (3) ◽  
pp. 470-484 ◽  
Author(s):  
Stathis Chadjiconstantinidis ◽  
Apostolos D. Papaioannou
Keyword(s):  
Risk Process ◽  
Barrier Problems

Monotone Stochastic Recursions and their Duals

1996 ◽  
Vol 10 (1) ◽  
pp. 1-20 ◽  
Author(s):  
Søren Asmussen ◽  
Karl Sigman
Keyword(s):  
Markov Processes ◽  
Branching Processes ◽  
Sample Path ◽  
Risk Process ◽  
Dual Function ◽  
Sample Path Analysis ◽  
Reservoir Models ◽  
Stability Issues ◽  
Barrier Problems ◽  
Steady State Probabilities

A duality is presented for real-valued stochastic sequences [Vn] defined by a general recursion of the form Vn+1 = f(Vn, Un), with [Un] a stationary driving sequence and f nonnegative, continuous, and monotone in its first variable. The duality is obtained by defining a dual function g of f, which if used recursively on the time reversal of [Un] defines a dual risk process. As a consequence, we prove that steady-state probabilities for Vn can always be expressed as transient probabilities of the dual risk process. The construction is related to duality of stochastically monotone Markov processes as studied by Siegmund (1976, The equivalence of absorbing and reflecting barrier problems for stochastically monotone Markov processes, Annals of Probability 4: 914–924). Our method of proof involves an elementary sample-path analysis. A variety of examples are given, including random walks with stationary increments and two reflecting barriers, reservoir models, autoregressive processes, and branching processes. Finally, general stability issues of the content process are dealt with by expressing them in terms of the dual risk process.

Expansions for Markov-modulated systems and approximations of ruin probability

1996 ◽  
Vol 33 (01) ◽  
pp. 57-70
Author(s):  
Bartłomiej Błaszczyszyn ◽  
Tomasz Rolski
Keyword(s):  
Ruin Probability ◽  
Marked Point ◽  
Factorial Moment ◽  
Risk Process ◽  
Rate Function ◽  
Marked Point Process ◽  
Probability Of Ruin ◽  
Actual Risk ◽  
Moment Measures ◽  
Markov Modulated

Let N be a stationary Markov-modulated marked point process on ℝ with intensity β ∗ and consider a real-valued functional ψ(N). In this paper we study expansions of the form Eψ(N) = a 0 + β ∗ a 1 + ·· ·+ (β∗ ) nan + o((β ∗) n ) for β ∗→ 0. Formulas for the coefficients ai are derived in terms of factorial moment measures of N. We compute a 1 and a 2 for the probability of ruin φ u with initial capital u for the risk process in the Markov-modulated environment; a 0 = 0. Moreover, we give a sufficient condition for ϕu to be an analytic function of β ∗. We allow the premium rate function p(x) to depend on the actual risk reserve.

scholarly journals On optimality of the barrier strategy for a general Lévy risk process

2011 ◽  
Vol 53 (9-10) ◽  
pp. 1700-1707 ◽  
Author(s):  
Kam Chuen Yuen ◽  
Chuancun Yin
Keyword(s):  
Risk Process ◽  
Barrier Strategy

On a joint distribution for the risk process with constant interest force

2005 ◽  
Vol 36 (3) ◽  
pp. 365-374 ◽  
Author(s):  
Rong Wu ◽  
Guojing Wang ◽  
Chunsheng Zhang
Keyword(s):  
Joint Distribution ◽  
Risk Process ◽  
Interest Force ◽  
Constant Interest Force ◽  
Constant Interest

A general risk process and its properties

1992 ◽  
Vol 29 (1) ◽  
pp. 73-81 ◽  
Author(s):  
Thomas H. Scheike
Keyword(s):  
Risk Process ◽  
Jump Size ◽  
The Past ◽  
Distributional Properties ◽  
Jump Time ◽  
The Law ◽  
Jump Sizes

We construct a risk process, where the law of the next jump time or jump size can depend on the past through earlier jump times and jump sizes. Some distributional properties of this process are established. The compensator is found and some martingale properties are discussed.

scholarly journals An optimization of a continuous time risk process

2009 ◽  
Vol 33 (11) ◽  
pp. 4062-4068 ◽  
Author(s):  
Mi Ock Jeong ◽  
Kyung Eun Lim ◽  
Eui Yong Lee
Keyword(s):  
Continuous Time ◽  
Risk Process

Rate modulation in dams and ruin problems

1996 ◽  
Vol 33 (2) ◽  
pp. 523-535 ◽  
Author(s):  
Søren Asmussen ◽  
Offer Kella
Keyword(s):  
State Space ◽  
Release Rate ◽  
Risk Process ◽  
The State ◽  
Limiting Distribution ◽  
Process Conditions ◽  
Explicit Formulas ◽  
Rate Modulation ◽  
Finite State ◽  
Finite State Space

We consider a dam in which the release rate depends both on the state and some modulating process. Conditions for the existence of a limiting distribution are established in terms of an associated risk process. The case where the release rate is a product of the state and the modulating process is given special attention, and in particular explicit formulas are obtained for a finite state space Markov modulation.

scholarly journals The maximum surplus before ruin in an Erlang(n) risk process and related problems

2006 ◽  
Vol 38 (3) ◽  
pp. 529-539 ◽  
Author(s):  
Shuanming Li ◽  
David C.M. Dickson
Keyword(s):  
Risk Process ◽  
Surplus Before Ruin

Asymptotic theory for a risk process with a high dividend barrier

2007 ◽  
Vol 22 (3) ◽  
pp. 253-258
Author(s):  
Zhaojun Zong ◽  
Feng Hu
Keyword(s):  
Asymptotic Theory ◽  
Risk Process
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