**6. Mixed Exponential Claims**

We will now consider examples with rational roots. The idea for obtaining "rational first passage probabilities", which appeared essentially in Dufresne and Gerber (1991a), is to replace the additive parametrization of Equation (8) with that provided by the so-called Wiener–Hopf factorization of Equation (44). For Cramér–Lundberg processes, it is enough to specify the negative roots *γi*, *i* ≥ 1 and poles *βi* (for example, by specifying the Wiener–Hopf factor, Equation (44), and also the positive root <sup>Φ</sup>*q* of the symbol. The additive decomposition of the model's symbol, Equation (8), may be recovered using its behavior when *s* → <sup>∞</sup>, and partial fractions. This is automatized in the Mathematica program RatC which is available upon request from the authors.

This package is useful for providing testing cases for our approximations, since in such examples the case of mixed exponential claims is reduced to exact rational root-solving and partial fractions (included automatically in Mathematica's command InverseLaplaceTransform).

**Example 2.** *A Cramér–Lundberg model with exponential mixture jumps of order two. This example illustrates the computational steps of the previous section, and the fact that the second order Tijms approximation for mixed exponential claims of order* 2 *is exact (and so is in particular the optimal barrier).*

*We chose* <sup>Φ</sup>*q* = 1 3, *c* = 1/2 *and a negative Wiener–Hopf factor*

$$\phi\_{-}(s) = \frac{\left(\frac{s}{2} + 1\right)(s+1)}{\left(\frac{2s}{3} + 1\right)(2s+1)}.$$

*which is input into our Rat program—see Section 8—by specifying the interspersed roots and poles exr = (1/2, 3/2); exc = (1, 2).*

*This corresponds—see Section 8—to a Cramér–Lundberg process with cumulant generating function*

$$\kappa(s) = cs + \frac{8}{48}(\frac{1}{s+1} - 1) + \frac{21}{48}(\frac{2}{s+2} - 1), c = \frac{1}{2}s$$

*where we used λ* = 2948 *, and claim density*

$$b(\mathbf{x}) = \frac{8}{29}e^{-\mathbf{x}} + \frac{21}{29}2e^{-2\mathbf{x}},$$

*with mean m*1 = 3758 *and ρ* = *λ<sup>m</sup>*1 *c* = 3748 *. The resulting scale function with q* = 1/16 *is*

$$\mathcal{W}\_{\mathfrak{q}}(\mathfrak{x}) = -\frac{3}{11}\mathfrak{e}^{-3\mathfrak{x}/2} - \frac{9\mathfrak{e}^{-\mathfrak{x}/2}}{5} + \frac{224\mathfrak{e}^{\mathfrak{x}/3}}{55}.$$

*see Figure 3, and the optimal barrier is b*∗ = 0.642265*. Recall from Remark 5 that the Tijms–Padé approximation is exact at order 2.*

**Figure 3.** *<sup>W</sup>*(*x*).

*After an Esscher shift of* 13*, the scale function transform becomes*

$$\widehat{W^{(\Phi\_q)}}(\mathbf{s}) = \frac{8(\mathbf{4} + \mathbf{3s})(7 + \mathbf{3s})}{\mathbf{s}(\mathbf{5} + \mathbf{6s})(11 + \mathbf{6s})},$$

*with dominant non-zero pole* −56. *After the removal of the pole* 0*, this becomes*

$$
\widehat{G}(\mathbf{s}) = \frac{72(57\mathbf{\hat{s}} + 97)}{55(6\mathbf{\hat{s}} + 5)(6\mathbf{\hat{s}} + 11)}.
$$

*The Padé approximation of order* (0, 1) *is* 677448 <sup>55</sup>(<sup>6177</sup>*s*+<sup>5335</sup>)*, the Laguerre exponent is α*/2 = 5335/6177 = 0.863688*, and the largest error with n* = 30 *is* 6 × 10−16*—see Figure 4.*

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

**Example 3.** *A perturbed Cramér–Lundberg model with exponential mixtures jumps of order three. Our next example is produced by taking* <sup>Φ</sup>*q* = 13*and a negative Wiener–Hopf factor*

$$\phi\_{-}(s) = \frac{\left(\frac{s}{2} + 1\right)(s+1)}{\left(\frac{2s}{5} + 1\right)\left(\frac{2s}{3} + 1\right)(2s+1)},\tag{42}$$

*which corresponds to the cumulant generating function:*

$$\kappa(s) = s^2 + \frac{7s}{6} + \frac{1}{2(s+1)} + \frac{7}{8(s+2)} - \lambda\_r \lambda = \frac{15}{16}.$$

*Note that the only impact of the presence of Brownian motion with σ* > 0 *is that the degree of the numerator in Equation* (42) *equals the degree of the denominator* <sup>−</sup>1*.*

*The resulting scale function with q* = 516*is*

$$\mathcal{W}\_{\mathfrak{q}}(\mathbf{x}) = -\frac{9}{68}\mathfrak{e}^{-5\mathbf{x}/2} - \frac{3}{22}\mathfrak{e}^{-3\mathbf{x}/2} - \frac{9\mathfrak{e}^{-\mathbf{x}/2}}{20} + \frac{672\mathfrak{e}^{\mathbf{x}/3}}{935}.$$

*The input to the Laguerre–Tricomi–Weeks inversion is*

$$\widehat{G}(s) = \frac{9}{11(6s+11)} + \frac{27}{34(6s+17)} + \frac{27}{10(6s+5)}\sqrt{$$

*its Padé approximation of order* (0, 1) *is* 4639156488 <sup>935</sup>(<sup>8004369</sup>*s*+<sup>7505245</sup>)*, the Laguerre exponent is α*/2 = 0.937644*, and the largest error with n* = 40 *is* 4 × 10−14*.*<sup>8</sup> *Other exponents do better however. For example, the larger exponent α*/2 = <sup>6</sup>*κ*(<sup>Φ</sup>*q*)*κ*(<sup>Φ</sup>*q*) <sup>3</sup>(*κ*(<sup>Φ</sup>*q*))<sup>2</sup>−*κ*(<sup>Φ</sup>*q*)*κ*(<sup>Φ</sup>*q*) = 1.00688*, where we have erased the* 2 *in the denominator, has a smaller larger error when n* = 40 *of* 4 × 10−15*.*

**Example 4.** *A Cramér–Lundberg model with exponential mixtures jumps of order three. Our next example is produced by taking* <sup>Φ</sup>*q* = 13 , *c* = 1 *and a negative Wiener–Hopf factor*

$$\phi\_{-}(s) = \frac{\left(\frac{s}{3} + 1\right)\left(\frac{s}{2} + 1\right)(s + 1)}{\left(\frac{2s}{5} + 1\right)\left(\frac{2s}{3} + 1\right)(2s + 1)}$$

<sup>8</sup> Beyond *n* = 40, the precision needs to be changed to obtain better results.

*with poles* −12, −32, −52*. This corresponds to a Cramér–Lundberg process with cumulant generating function:*

$$\kappa(\mathbf{s}) = \mathbf{c}\mathbf{s} + \frac{1}{4(s+1)} + \frac{7}{8(s+2)} + \frac{25}{8(s+3)} - \lambda, \mathbf{c} = 1, \lambda = \frac{83}{48}.$$

*The scale function with q* = 548*is*

$$\mathcal{W}\_q(\mathbf{x}) = -\frac{9}{136} \mathfrak{e}^{-5\mathbf{x}/2} - \frac{9}{44} \mathfrak{e}^{-3\mathbf{x}/2} - \frac{9 \mathfrak{e}^{-\mathbf{x}/2}}{8} + \frac{448 \mathfrak{e}^{\mathbf{x}/3}}{187}$$

*and the optimal barrier is b*∗ = 0.866289*. The Padé Tijms approximation is b*∗ = 0.876898*. The input to the Laguerre–Tricomi–Weeks inversion is*

$$\hat{G}(\mathbf{s}) = \frac{216\left(261\mathbf{s}^2 + 1155\mathbf{s} + 120\mathbf{2}\right)}{187(6\mathbf{s} + 5)(6\mathbf{s} + 11)(6\mathbf{s} + 17)}.$$

*the Padé approximation of order* (0, 1) *is* 312077664 <sup>187</sup>(<sup>1278399</sup>*s*+<sup>1123870</sup>)*, the Laguerre exponent is α*/2 = 0.879123*, and the largest error with n* = 40 *is* 4 × 10−14*—see Figure 5.*

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

*Again, the exponent α*/2 = <sup>6</sup>*κ*(<sup>Φ</sup>*q*)*κ*(<sup>Φ</sup>*q*) <sup>3</sup>(*κ*(<sup>Φ</sup>*q*))<sup>2</sup>−*κ*(<sup>Φ</sup>*q*)*κ*(<sup>Φ</sup>*q*) = 1.138 *does better, with the largest error with n* = 40 *of* 6 × 10−16*. This suggests the importance of optimizing α, which is a difficult problem Giunta et al. (1989); Weideman (1999). Recall however our proposal to circumvent it by starting with a higher order Padé approximation of <sup>G</sup>*(*s*)*—see* (41)*.*
