# The W,Z/ν,δ Paradigm for the First Passage of Strong Markov Processes without Positive Jumps

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## Abstract

**:**

## 1. A Brief Review of First Passage Theory for Strong Markov Processes without Positive Jumps and Their Draw-Downs

**Remark**

**1.**

**Assumption**

**1.**

**Remark**

**2.**

**Remark**

**3.**

- $$Ub{D}_{q,d}^{b}(x):={\mathrm{I}\phantom{\rule{-0.166667em}{0ex}}\mathrm{E}}_{x}\left[{e}^{-q{T}_{b,+}};{T}_{b,+}\le {\tau}_{d}\right]={e}^{-(b-x)\frac{{W}_{q}^{\prime}(d)}{{W}_{q}(d)}},$$
- The function $DbU$ may be obtained by integrating the fundamental law (Mijatovic and Pistorius 2012, Thm 1), (Landriault et al. 2017a, Thm 3.1)2$$\begin{array}{c}{\delta}_{q,\theta}(d,x,s):={\mathrm{I}\phantom{\rule{-0.166667em}{0ex}}\mathrm{E}}_{x}\left[{e}^{-q{\tau}_{d}-\theta ({Y}_{{\tau}_{d}}-d)};{\overline{X}}_{{\tau}_{d}}\in \mathrm{d}s\right]=\left({\nu}_{q}(d)\phantom{\rule{0.277778em}{0ex}}{e}^{-{\nu}_{q}(d){(s-x)}_{+}}\phantom{\rule{0.277778em}{0ex}}\mathrm{d}s\right){\delta}_{q,\theta}(d)\hfill \\ \iff {\mathrm{I}\phantom{\rule{-0.166667em}{0ex}}\mathrm{E}}_{x}\left[{e}^{-q{\tau}_{d}-\theta ({Y}_{{\tau}_{d}}-d)-\vartheta ({\overline{X}}_{{\tau}_{d}}-x)}\right]=\frac{{\nu}_{q}(d)}{\vartheta +{\nu}_{q}(d)}{\delta}_{q,\theta}(d)\hfill \end{array}$$$$\begin{array}{cc}\hfill Db{U}_{q,\theta ,d}^{b}(x)& =\left(1-{e}^{-(b-x)\frac{{W}_{q}^{\prime}(d)}{{W}_{q}(d)}}\right){\delta}_{q,\theta}(d).\hfill \end{array}$$

**Remark**

**4.**

**Contents.**We start in Section 2 by presenting a pedagogic first passage example illustrating the $W,Z$ paradigm: the first time

**Remark**

**5.**

## 2. Geometric Considerations Concerning the Joint Evolution of a Lévy Process and Its Draw-Down in a Rectangle

- If $b\le a+d$, it is impossible for the process to leave R through the upper boundary of $\partial R$ and for these parameter values ${T}_{R}$ reduces to ${T}_{a,-}\wedge {T}_{b,+}$. Here it suffices to know the functions (1) to obtain the Laplace transform of ${T}_{R}$.
- If $a+d\le x$, it is impossible for the process to leave R through the left boundary of $\partial R$, and ${T}_{R}$ reduces to ${T}_{b,+}\wedge {\tau}_{d}$. Here it suffices to apply the spectrally negative drawdown formulas provided in Landriault et al. (2017a); Mijatovic and Pistorius (2012).
- In the remaining case $x\le a+d\le b$, 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 ${X}_{t}$ does [does not] cross the line ${x}_{1}=d+a$. 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

**Theorem**

**1.**

**Proof.**

- the middle case may happen only if ${X}_{t}$ visits a before $a+d$;
- the first case (exit through b) and the third case (drawdown exit) may happen only if ${X}_{t}$ visits first $a+d$, with the drawdown barrier being invisible, and that subsequently the lower first passage barrier a becomes invisible.

**Proof.**

## 4. Generalized Draw-Down Stopping for Processes without Positive Jumps

**Definition**

**1.**

**Example**

**1.**

**Remark**

**6.**

- 1.
- Due to creeping, $UbD$ is a product of infinitesimal events$${\overline{\mathsf{\Psi}}}_{q}^{y+\u03f5}(y,y-d(y))=\frac{{W}_{q}(d(y))}{{W}_{q}(d(y)+\u03f5)}\sim 1-\u03f5{\nu}_{q}(d(y))\sim {e}^{-\u03f5{\nu}_{q}(d(y))}.$$Taking product, with $\u03f5=\mathrm{d}y$, yields (24).
- 2.
- Informally, we condition on the density ${\overline{X}}_{t}\in \mathrm{d}y$. The integrand of $DbU$ is obtained multiplying survival infinitesimal events up to level y by an infinitesimal termination event in $[y,y+\mathrm{d}y]$. The probability of this event, conditioned on survival up to y, is given by the deficit formula$$\begin{array}{c}{\mathsf{\Psi}}_{q,\theta}^{y+\u03f5}(y,y-d(y))={Z}_{q,\theta}(d(y))-{W}_{q}(d(y))\frac{{Z}_{q,\theta}(d(y)+\u03f5)}{{W}_{q}(d(y)+\u03f5)}\hfill \\ \sim \u03f5(-{Z}_{q,\theta}^{\prime}(d(y))+{\nu}_{q}(d(y)){Z}_{q,\theta}(d(y))=\u03f5{\nu}_{q}(d(y)){\delta}_{q,\theta}(d(y))\hfill \end{array}$$For a rigorous (rather intricate) proof, see Avram et al. (2018b).

**Theorem**

**2.**

**Remark**

**7.**

## 5. The Three Laplace Transforms of the Exit Time out of a Curved Trapezoid, for Processes without Positive Jumps

**Theorem**

**3.**

**Proof.**

## 6. de Finetti’s Optimal Dividends for Spectrally Negative Markov Processes with Generalized Draw-Down Stopping

**Remark**

**8.**

## 7. Example: Affine Draw-Down Stopping for Brownian Motion

**Remark**

**9.**

**Remark**

**10.**

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

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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 ${Z}_{q}(x,\theta )$, so that some computing is required to get (11) and (14). |

3 | Choosing $a,b,d$ optimally in various control problems involving optimal dividends and capital injections should be of interest, and will be pursued in further work. |

**Figure 1.**A sample path of $(X,Y)$ with X a spectrally negative Lévy process. The region R has $d=10$, $a=-6$ and $b=7$; the dark boundary shows the possible exit points of $(X,Y)$ from R. The base of the red line separates R in two parts with different behavior.

**Figure 3.**Optimizing dividends with affine drawdown stopping where $\mu =1/2$, $q=1/10$, $\sigma =1$, $\xi =1/3$, $b=20$, $d=1.$ The critical point ${b}^{\ast}=2.12445.$

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**MDPI and ACS Style**

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

**AMA Style**

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 Style**

Avram, 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