A Review of First-Passage Theory for the Segerdahl-Tichy Risk Process and Open Problems
Abstract
:1. Introduction and Brief Review of First Passage Theory
- with phase-type jumps, there is Asmussen’s embedding into a regime switching diffusion Asmussen (1995)—see Section 5, and the complex integral representations of Jacobsen and Jensen (2007); Jiang et al. (2019).
- for Lévy driven Langevin-type processes, renewal equations have been provided in Czarna et al. (2017) —see Section 2
- for processes with affine operator, an explicit integrating factor for the Laplace transform may be found in Avram and Usabel (2008)—see Section 3.
- for the Segerdahl process, the direct IDE solving approach is successful (Paulsen and Gjessing (1997)) —see Section 4.
2. The Renewal Equation for the Scale Derivative of Lévy Driven Langevin Processes Czarna et al. (2017)
The Linear Case
3. The Laplace transform-Integrating Factor Approach for Jump-Diffusions with Affine Operator Avram and Usabel (2008)
- A penalty at a stopping time T,
- A reward for survival after t years: .
Segerdahl’s Process via the Laplace Transform Integrating Factor
- for , when it holds that
4. Direct Conversion to an Ode of Kolmogorov’S Integro-Differential Equation for the Discounted Ruin Probability with Phase-Type Jumps
Paulsen’s Result for Segerdahl’s Process with Exponential Jumps Paulsen and Gjessing (1997), ex. 2.1
5. Asmussen’s Embedding Approach for Solving Kolmogorov’s Integro-Differential Equation with Phase-Type Jumps
Exit Problems for the Segerdahl-Tichy process, with
6. Revisiting Segerdahl’s Process via the Scale Derivative/Integrating Factor Approach, When
6.1. Laplace Transforms of the Eventual Ruin and Survival Probabilities
6.2. The Eventual Ruin and survival probabilities
7. Further Details on the Identities Used in the Proof of Theorem 2
8. Conclusions and Future Work
Author Contributions
Funding
Conflicts of Interest
References
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1 | |
2 | |
3 | may be more useful than in the spectrally negative Markov framework Avram and Goreac (2019). |
4 | For some background first passage results on these processes, see for example Borovkov and Novikov (2008); Loeffen and Patie (2010). |
5 | Note that when , this function reduces to a power: |
6 | More generally, for any phase-type jumps with Laplace transform , it may be checked that in the sense that , thus removing the convolution by applying the denominator . |
7 | this is implied by the Kolmogorov integro-differential equation |
8 | and are the increasing/decreasing solutions of the to Weiler’s canonical form of Kummer equation , which is obtained via the substitution with . Some computer systems use instead of M the Laguerre function defined by , which yields for natural the Laguerre polynomial of degree . |
9 | Note that we have corrected Paulsen’s original denominator by using the identity (Abramowitz and Stegun 1965, 13.4.18) |
10 | Putting we must solve the equation
|
11 | See also (Borodin and Salminen 2012, p. 640), where however the first formula has a typo. |
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Avram, F.; Perez-Garmendia, J.-L. A Review of First-Passage Theory for the Segerdahl-Tichy Risk Process and Open Problems. Risks 2019, 7, 117. https://doi.org/10.3390/risks7040117
Avram F, Perez-Garmendia J-L. A Review of First-Passage Theory for the Segerdahl-Tichy Risk Process and Open Problems. Risks. 2019; 7(4):117. https://doi.org/10.3390/risks7040117
Chicago/Turabian StyleAvram, Florin, and Jose-Luis Perez-Garmendia. 2019. "A Review of First-Passage Theory for the Segerdahl-Tichy Risk Process and Open Problems" Risks 7, no. 4: 117. https://doi.org/10.3390/risks7040117