# On the Padé and Laguerre–Tricomi–Weeks Moments Based Approximations of the Scale Function W and of the Optimal Dividends Barrier for Spectrally Negative Lévy Risk Processes

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## Abstract

**:**

## 1. Introduction

- Optimizing dividends starts by optimizing the so-called “barrier function”$${H}_{D}\left(b\right):=\frac{1}{{W}_{q}^{\prime}\left(b\right)},\phantom{\rule{0.277778em}{0ex}}b\ge 0,$$$${H}_{D}^{\prime}\left(0\right)>0\iff {W}_{q}^{\u2033}\left(0\right)<0$$$$V\left(x\right):=su{p}_{b\ge 0}{V}^{b]}\left(x\right)={V}^{{b}^{*}]}\left(x\right),$$
- The challenge of multiple inflection points. In the presence of several inflection points, however, the optimal policy is multiband Azcue and Muler (2005); Schmidli (2007); Avram et al. (2015); Loeffen (2008). The first numerical examples of multiband policies were produced in Azcue and Muler (2005); Loeffen (2008), with Erlang claims $Er{l}_{2,1}$. However, it was shown in Loeffen (2008) that multibands cannot occur when ${W}_{q}^{\prime}\left(x\right)$ is increasing after its last global minimum ${b}^{*}$ (i.e., when no local minima are allowed after the global minimum).Loeffen (2008) further made the interesting observation that for Erlang claims $E{R}_{2,1}$ (which are non-monotone), multiband policies may occur for volatility parameters $\sigma $ smaller than a threshold value, but barrier policies (with a non-concave value function) will occur when $\sigma $ is large enough.Figure 1 displays the first derivative ${W}_{q}^{\prime}\left(x\right),$ for ${\sigma}^{2}/2\in \{\frac{1}{2},1,\frac{3}{2},2\}$. The last two values yield barrier policies with a non-concave value function, due to the presence of an inflection point in the interior of the interval $[0,{b}^{*}]$.

- including known values of ${W}_{q}\left(0\right),{W}_{q}^{\prime}\left(0\right)$ (using thus two-point Padé approximations).
- shifting the approximations around ${\mathsf{\Phi}}_{q}$ specified in Equation (4), which transforms ${W}_{q}\left(x\right)$ into a survival probability. As a consequence, we end up using a certain judicious choice of the Laguerre exponential decay parameter of Equation (40), which is usually left to be tuned by the user in the Laguerre–Tricomi–Weeks method Weideman (1999).

## 2. A Short Review of Classical Ruin Theory

- First passage times below and above a level a$${T}_{a,-(+)}:=inf\{t>0:X\left(t\right)<(>)a\}.$$
- The first first passage quantity to be studied historically was the eventual ruin probability:$$\mathsf{\Psi}\left(x\right):=P[{T}_{0,-}<\infty |X\left(0\right)=x].$$In order that the eventual ruin probability not be identically 1, the parameter$$p:=c-\lambda {m}_{1}={\kappa}^{\prime}\left(0\right),\phantom{\rule{4.pt}{0ex}}\mathrm{where}\phantom{\rule{4.pt}{0ex}}{m}_{1}={\int}_{0}^{\infty}zF\left(dz\right),$$The Laplace transform of the ruin probability is explicit, given by the so-called Pollaczek–Khinchine formula, which states that the Laplace transform of $\overline{\mathsf{\Psi}}\left(x\right)=1-\mathsf{\Psi}\left(x\right)$ is:$$\widehat{\overline{\mathsf{\Psi}}}\left(s\right)={\int}_{0}^{\infty}{e}^{-sx}\overline{\mathsf{\Psi}}\left(x\right)dx=\frac{c-\lambda {m}_{1}}{\kappa \left(s\right)}=\frac{{\kappa}^{\prime}\left(0\right)}{\kappa \left(s\right)}.$$

**Laplace transform inversion**. As explained here, the first passage theory for Lévy processes with one-sided jumps reduces essentially to inverting the Laplace transform Equation (3). This applies not only to the well-known ruin probabilities, but also to intricate optimization problems involving dividends, capital gains, liquidation of subsidiary companies, etc.

**Remark**

**1.**

- 1.
- non-matrix exponential claims,
- 2.
- when ${\gamma}_{1}$ does not exist (the non-Cramér case), and
- 3.
- when not all moments exist, which we will call “heavy tails”.

## 3. Padé/matrix exponential Approximations of the Scale Function

- Obtain the power series expansion of the Laplace exponent in terms of the moments of the Lévy measure$$\kappa \left(s\right)=s\left(c-\widehat{\overline{\nu}}\left(s\right)+\frac{{\sigma}^{2}}{2}s\right)=(c-{\nu}_{1})\phantom{\rule{0.277778em}{0ex}}s+{\nu}_{2,\sigma}\frac{{s}^{2}}{2}+\sum _{k=3}^{\infty}{\nu}_{k}\frac{{(-s)}^{k}}{k!}.$$
- Construct a Padé approximation$${\widehat{W}}_{q}\left(s\right)=\frac{1}{(c-{\nu}_{1})s+{\nu}_{2,\sigma}{s}^{2}/2-{\nu}_{3}{s}^{3}/6+\cdots -q}\sim \frac{{P}_{n-1}\left(s\right)}{{Q}_{n}\left(s\right)}=\frac{{\sum}_{i=0}^{n-1}{a}_{i}{s}^{i}}{c{s}^{n}+{\sum}_{i=0}^{n-1}{b}_{i}{s}^{i}},$$
- Factor the denominator as$$c{s}^{n}+\sum _{i=0}^{n-1}{b}_{i}{s}^{i}=c(s-{\mathsf{\Phi}}_{q})\prod _{i=1}^{n}(s+{\gamma}_{i}).$$This operation is the essential numerical difficulty, but may be achieved nowadays with arbitrarily high precision.
- Then, partial fractions plus inversion quickly yield an approximate Laplace transform inverse$${W}_{q}\left(x\right)\sim {C}_{0}{e}^{{\mathsf{\Phi}}_{q}x}+\sum _{i=1}^{n}{C}_{i}{e}^{-{\gamma}_{i}x}.$$
- The dividend barrier is obtained by finding the largest nonnegative root of ${W}^{\prime \prime}\left(x\right)=0$.

**Remark**

**2.**

**Remark**

**3.**

## 4. Two-Point Padé Approximations, with Low Order Examples

**Example**

**1.**

**Proposition**

**1.**

- 1.
- To secure both the values of ${W}_{q}\left(0\right)$ and ${W}_{q}^{\prime}\left(0\right)$, take into account Equations (17) and (18), i.e., use the Padé approximation$${\widehat{W}}_{q}\left(s\right)\sim \frac{{\sum}_{i=0}^{n-1}{a}_{i}{s}^{i}}{c{s}^{n}+{\sum}_{i=0}^{n-1}{b}_{i}{s}^{i}},{a}_{n-1}=1,{b}_{n-1}=c{a}_{n-2}-\lambda -q.$$This yields$${\widehat{W}}_{q}\left(s\right)\sim \frac{\frac{1}{{m}_{1}}+s}{c{s}^{2}+s\left(\frac{c}{{m}_{1}}-\lambda -q\right)-\frac{q}{{m}_{1}}}.$$In view of Equation (20), this yields the same result as approximating the claims by exponential claims, with $\mu =\frac{1}{{m}_{1}}$.
- 2.
- To ensure ${W}_{q}\left(0\right)=\frac{1}{c}$, we must only impose the behavior specified in Equation (17), i.e., use the Padé approximation$${\widehat{W}}_{q}\left(s\right)\sim \frac{{\sum}_{i=0}^{n-1}{a}_{i}{s}^{i}}{c{s}^{n}+{\sum}_{i=0}^{n-1}{b}_{i}{s}^{i}},{a}_{n-1}=1.$$For $n=2$, this yields$${\widehat{W}}_{q}\left(s\right)\sim \frac{\frac{2{m}_{1}}{{m}_{2}}+s}{c{s}^{2}+\frac{s\left(2c{m}_{1}-2\lambda {m}_{1}^{2}-{m}_{2}q\right)}{{m}_{2}}-\frac{2{m}_{1}q}{{m}_{2}}}=\frac{\frac{1}{{\tilde{m}}_{1}}+s}{c{s}^{2}+s\left(\frac{c}{{\tilde{m}}_{1}}-\lambda \frac{{m}_{1}}{{\tilde{m}}_{1}}-q\right)-\frac{q}{{\tilde{m}}_{1}}},$$
- 3.
- When the pure Padé approximation is applied, the first step yields$$\begin{array}{c}{\widehat{W}}_{q}\left(s\right)\sim \frac{s+\frac{3{m}_{2}}{{m}_{3}}}{{s}^{2}\left(c+\lambda (\frac{3{m}_{2}^{2}}{2{m}_{3}}-{m}_{1})\right)+s\left(c\frac{3{m}_{2}}{{m}_{3}}-\frac{3{m}_{1}{m}_{2}}{{m}_{3}}\lambda -q\right)-\frac{3{m}_{2}}{{m}_{3}}q}\hfill \\ =\frac{s+\frac{1}{{\widehat{m}}_{3}}}{{s}^{2}\left(c+\lambda (\frac{{\widehat{m}}_{2}}{{\widehat{m}}_{3}}-1)\right)+s\left(c\frac{1}{{\widehat{m}}_{3}}-\frac{{m}_{1}}{{\widehat{m}}_{3}}\lambda -q\right)-\frac{1}{{\widehat{m}}_{3}}q},\hfill \end{array}$$This is the De Vylder A) method (Gerber et al. 2008, (5.2–5.4)), derived therein by assuming exponential claims, with $\mu =\frac{3{m}_{2}}{{m}_{3}}$, and simultaneously modifying both λ and c to fit the first three moments of the risk process.

**Lemma**

**1.**

**Proof.**

## 5. Two Numerical Methods for Computing ${W}_{q}\left(x\right)$: The Tijms–Padé and Laguerre–Tricomi–Weeks Approximations

**Remark**

**4.**

**Remark**

**5.**

#### 5.1. Laguerre–Tricomi–Weeks Laplace Transform Inversion with Prescribed Exponent

**Determining $\alpha $**. Previous work on choosing $\alpha $ involved the radius of convergence of a Laguerre–Tricomi–Weeks expansion which is slightly more general than Equation (35) Weideman (1999)—see also Section 9. We now introduce a different method.

**Remark**

**6.**

**Remark**

**7.**

#### 5.2. Combining the Tijms–Padé and Laguerre–Tricomi–Weeks Approximations

## 6. Mixed Exponential Claims

**Example**

**2.**

**Example**

**3.**

**Example**

**4.**

## 7. Padé Approximation of Heavy-Tailed Claims

## 8. The Wiener–Hopf Factorization and Risk Theory with Rational Roots

#### A Computer Program “Rat” That Outputs a Spectrally Negative Lévy Process with Given Roots and Poles of Its Symbol

- a vector of length N containing the negatives ${\gamma}_{1},\cdots ,{\gamma}_{N}$ of the Cramér–Lundberg roots with negative real part, and
- a vector ${\beta}_{1},..,{\beta}_{n},$ of length $n=\left\{\begin{array}{cc}N\hfill & \sigma =0\hfill \\ N-1\hfill & \sigma >0\hfill \end{array}\right.$, containing the diagonal of the triangular matrix B representing the Coxian distribution (the rest of the parameters of this law are only provided indirectly, via the Cramér–Lundberg roots).

**Remark**

**8.**

**Remark**

**9.**

**Remark**

**10.**

- Eventual ruin probabilities
- Cumulants generating function (Laplace exponent) of the Levy process $\kappa \left(s\right)$
- Homogeneous scale function ${W}^{\left(q\right)}\left(y\right)$, for one fixed q

## 9. Further Background on the Laguerre–Tricomi–Weeks Method

## Author Contributions

## Funding

## Conflicts of Interest

## References

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1 | Even when barrier strategies do not achieve the optimum, and multi-band policies must be used instead, constructing the solution must start by determining the global maximum of the barrier function Avram et al. (2015); Azcue and Muler (2005); Schmidli (2007). |

2 | Note that only moments starting from 1 are required, so this may be applied to processes whose subordinator part has infinite activity as well. |

3 | This equation is important in establishing the nonnegativity of the optimal dividends barrier. |

4 | Equation (25) is easy to check by taking the Laplace transform, but quite important, numerically, since ${W}_{0}^{\left({\mathsf{\Phi}}_{q}\right)}\left(x\right)$ is a monotone bounded function (with values in the interval (${lim}_{s\to \infty}\frac{s}{\kappa \left(s\right)},\frac{1}{{\kappa}^{\prime}\left({\mathsf{\Phi}}_{q}\right)})$). |

5 | Since ${W}^{\left({\mathsf{\Phi}}_{q}\right)}\left(x\right)$ still converges to a constant when $x\to \infty $, this non-zero limit at ∞ must be removed first. |

6 | Higher-order rational approximations may be considered as well. |

7 | C can also be included in the coefficients ${B}_{n}$, but introducing it does render the computation of Equation (40) more convenient. |

8 | Beyond $n=40$, the precision needs to be changed to obtain better results. |

9 | Nowadays, the simplest way to obtain them is by solving linear systems with Mathematica, Maple, Sage, etc. |

**Figure 1.**Graphs of the Loeffen example for $\kappa \left(s\right)=\frac{{\sigma}^{2}{s}^{2}}{2}+c\phantom{\rule{0.277778em}{0ex}}s+\lambda \left(\frac{1}{{(s+1)}^{2}}-1\right),c=\frac{107}{5},\lambda =10,q=\frac{1}{10}$, ${\sigma}^{2}/2\in \{{1}{/}{2}{,}\textcolor[rgb]{}{1}\textcolor[rgb]{}{,}{3}{/}{2}{,}{2}\}$.

**Figure 2.**The series expansion of $\kappa \left(s\right)$ has multiple contact with $\kappa \left(s\right)$ at $s=0$, but increases asymptotically at a smaller rate. Despite that, it yields quite reasonable approximations of ${\mathsf{\Phi}}_{q}$, for q small enough.

**Figure 4.**Relative errors of the Laguerre–Tricomi–Weeks inversion with mixed exponential claims of order 2.

**Figure 5.**Relative errors of the Laguerre–Tricomi–Weeks inversion with mixed exponential claims of order 3.

**Figure 6.**The scale function ${W}_{q}\left(x\right)$ (on the left) and its derivative (on the right) in the case of the perturbed model.

**Figure 7.**On the left the expected value of total discounted dividends, ${V}^{b]}\left(x\right)$, as function of b with $x=1$ in the case of the perturbed model. On the right the absolute error of approximations of ${V}^{b]}\left(x\right)$ of order 4 (blue), 6 (orange) and 8 (green).

**Figure 8.**The scale function ${W}_{q}\left(x\right)$ (on the left) and its derivative (on the right) in the case of the unperturbed model.

**Figure 9.**On the left the expected value of total discounted dividends, ${V}^{b]}\left(x\right)$, as function of b with $x=1$ in the case of the unperturbed model. On the right the absolute error of approximations of ${V}^{b]}\left(x\right)$ of order 4 (blue), 6 (orange) and 8 (green).

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**MDPI and ACS Style**

Avram, F.; Horváth, A.; Provost, S.; Solon, U. On the Padé and Laguerre–Tricomi–Weeks Moments Based Approximations of the Scale Function *W* and of the Optimal Dividends Barrier for Spectrally Negative Lévy Risk Processes. *Risks* **2019**, *7*, 121.
https://doi.org/10.3390/risks7040121

**AMA Style**

Avram F, Horváth A, Provost S, Solon U. On the Padé and Laguerre–Tricomi–Weeks Moments Based Approximations of the Scale Function *W* and of the Optimal Dividends Barrier for Spectrally Negative Lévy Risk Processes. *Risks*. 2019; 7(4):121.
https://doi.org/10.3390/risks7040121

**Chicago/Turabian Style**

Avram, Florin, Andras Horváth, Serge Provost, and Ulyses Solon. 2019. "On the Padé and Laguerre–Tricomi–Weeks Moments Based Approximations of the Scale Function *W* and of the Optimal Dividends Barrier for Spectrally Negative Lévy Risk Processes" *Risks* 7, no. 4: 121.
https://doi.org/10.3390/risks7040121