The W,Z/ν,δ Paradigm for the First Passage of Strong Markov Processes without Positive Jumps
Abstract
:1. A Brief Review of First Passage Theory for Strong Markov Processes without Positive Jumps and Their Draw-Downs
- The function may be obtained by integrating the fundamental law (Mijatovic and Pistorius 2012, Thm 1), (Landriault et al. 2017a, Thm 3.1)2
2. Geometric Considerations Concerning the Joint Evolution of a Lévy Process and Its Draw-Down in a Rectangle
- If , it is impossible for the process to leave R through the upper boundary of and for these parameter values reduces to . Here it suffices to know the functions (1) to obtain the Laplace transform of .
- If , it is impossible for the process to leave R through the left boundary of , and reduces to . Here it suffices to apply the spectrally negative drawdown formulas provided in Landriault et al. (2017a); Mijatovic and Pistorius (2012).
- In the remaining case , both drawdown and classic exits are possible. For the latter case, see Figure 1. The key observation here is that drawdown [classic] exits occur iff does [does not] cross the line . The final answers will combine these two cases.
3. The Three Laplace Transforms of the Exit Time out of a Rectangle for Lévy Processes without Positive Jumps
- the middle case may happen only if visits a before ;
- the first case (exit through b) and the third case (drawdown exit) may happen only if visits first , with the drawdown barrier being invisible, and that subsequently the lower first passage barrier a becomes invisible.
4. Generalized Draw-Down Stopping for Processes without Positive Jumps
- 1.
- Due to creeping, is a product of infinitesimal eventsTaking product, with , yields (24).
- 2.
- Informally, we condition on the density . The integrand of is obtained multiplying survival infinitesimal events up to level y by an infinitesimal termination event in . The probability of this event, conditioned on survival up to y, is given by the deficit formulaFor a rigorous (rather intricate) proof, see Avram et al. (2018b).
5. The Three Laplace Transforms of the Exit Time out of a Curved Trapezoid, for Processes without Positive Jumps
6. de Finetti’s Optimal Dividends for Spectrally Negative Markov Processes with Generalized Draw-Down Stopping
7. Example: Affine Draw-Down Stopping for Brownian Motion
Author Contributions
Funding
Acknowledgments
Conflicts of Interest
References
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1 | The fact that the survival probability has the multiplicative structure (2) is equivalent to the absence of positive jumps, by the strong Markov property. |
2 | Please note that (Mijatovic and Pistorius 2012, Thm. 1) give a more complicated “sextuple law” with two cases, and that (Landriault et al. 2017a, Thm 3.1) use an alternative to the function , so that some computing is required to get (11) and (14). |
3 | Choosing optimally in various control problems involving optimal dividends and capital injections should be of interest, and will be pursued in further work. |
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Avram, F.; Grahovac, D.; Vardar-Acar, C. The W,Z/ν,δ Paradigm for the First Passage of Strong Markov Processes without Positive Jumps. Risks 2019, 7, 18. https://doi.org/10.3390/risks7010018
Avram F, Grahovac D, Vardar-Acar C. The W,Z/ν,δ Paradigm for the First Passage of Strong Markov Processes without Positive Jumps. Risks. 2019; 7(1):18. https://doi.org/10.3390/risks7010018
Chicago/Turabian StyleAvram, Florin, Danijel Grahovac, and Ceren Vardar-Acar. 2019. "The W,Z/ν,δ Paradigm for the First Passage of Strong Markov Processes without Positive Jumps" Risks 7, no. 1: 18. https://doi.org/10.3390/risks7010018
APA StyleAvram, F., Grahovac, D., & Vardar-Acar, C. (2019). The W,Z/ν,δ Paradigm for the First Passage of Strong Markov Processes without Positive Jumps. Risks, 7(1), 18. https://doi.org/10.3390/risks7010018