# Asymptotic Expected Utility of Dividend Payments in a Classical Collective Risk Process

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## Abstract

**:**

## 1. Introduction

## 2. Hamilton–Jacobi–Bellman Equation

**Theorem 1.**

**Corollary 1.**

**Proof.**

**Lemma 1.**

**Proof.**

## 3. Asymptotic Analysis

#### 3.1. Classical Risk Process (1) and Power Utility Function

**Lemma 2.**

**Proof.**

**Theorem 2.**

**Remark 1.**

#### 3.2. Classical Risk Process (1) and Logarithmic Utility Function

**Theorem 3.**

## 4. Numerical Analysis

**Remark 2.**

- Set initial value ${v}_{x}\left(0\right)=:b$,
- From the equality (13) derive initial value $v\left(0\right)=a$;
- Solve numerically the differential Equation (12) with the initial condition $v\left(0\right)=a$;
- Calculate $c\left(x\right)$ using $c\left(x\right)={v}_{x}{\left(x\right)}^{-\frac{1}{1-\alpha}}$;
- Using the least squares method, approximate $c\left(x\right)$ be the linear function $\widehat{c}\left(x\right)={a}_{1}x+{b}_{1}$. Because of our results from Theorem 2, we assume that $\widehat{c}\left(x\right)$ is a linear function;
- Let $x\left(t\right)$ be a trajectory of the regulated process starting from 0 until the first time claim arrival T. Hence$$\mu -\widehat{c}\left(x\left(t\right)\right)={x}^{\prime}\left(t\right),\phantom{\rule{1.em}{0ex}}x\left(0\right)=0,$$$$x\left(t\right)=\frac{\mu -{b}_{1}}{{a}_{1}}-\frac{\mu -{b}_{1}}{{a}_{1}}{e}^{-{a}_{1}t};$$
- Using the least squares method, approximate $v\left(x\right)$ by a function of the form $\widehat{v}\left(x\right)={a}_{2}{x}^{\alpha}+{b}_{2}$. Because of our results from Theorem 2, we assume that $\widehat{v}\left(x\right)$ is a power function;
- Calculate$$A=\mathbb{E}\left[{e}^{-\beta T}\widehat{v}(X\left(T\right)-S)\right]+\mathbb{E}\left[{\int}_{0}^{T}{e}^{-\beta t}U(\widehat{c}\left(X\left(t\right)\right)dt\right],$$
- Calculate the value $a-A$;
- Repeat until $|a-A|<\u03f5$ for fixed $\u03f5>0$.

## Author Contributions

## Funding

## Data Availability Statement

## Conflicts of Interest

## Correction Statement

## Appendix A. Proof of Theorem 2

**Proof.**

- (a)
- $\xi \beta v{z}^{2q-2p+1}$ and $\xi \frac{q-p}{p}{z}^{2q-p+1}$;
- (b)
- $\xi \frac{q-p}{p}{z}^{2q-p+1}$ and $\left(p-q\right){z}^{q}{z}_{v}$;
- (c)
- $\xi \beta v{z}^{2q-2p+1}$ and $\left(q-p\right){z}^{q}{z}_{v}$.

**Lemma**

**A1.**

**Proof.**

- I.
- If $q\ne 2p$, then, via the separation of variables, we have$$z\left(v\right)\sim {\left(\frac{l\xi \beta (2p-q)}{q-p}\right)}^{\frac{1}{2p-q}}{\left(\frac{{v}^{2}}{2}+c\right)}^{\frac{1}{2p-q}}.$$
- II.
- If $q=2p$, then a simple integration leads to$$z\left(v\right)\sim {e}^{\frac{l\xi \beta}{p}\left(\frac{{v}^{2}}{2}+c\right)}.$$

## Appendix B. Proof of Theorem 3

**Proof.**

- (a)
- ${y}_{v}$ and $\xi \beta v+\xi $;
- (b)
- ${y}_{v}$ and $\xi lny$;
- (c)
- $\xi \beta v+\xi $ and $\xi lny$.

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**Figure 1.**Functions $v\left(x\right)$ and ${v}_{x}\left(x\right)$ for $\alpha =0.5$, $\beta =0.05$, $\mu =0.26$, $\xi =0.4$, $\lambda =0.1$ and ${v}_{x}\left(0\right)=2$, $v\left(0\right)=6.8$.

**Figure 2.**Functions $v\left(x\right)$ and ${v}_{x}\left(x\right)$ for $\alpha =0.5$, $\beta =0.05$, $\mu =0.26$, $\xi =0.4$, $\lambda =0.1$ and ${v}_{x}\left(0\right)=1.9$, $v\left(0\right)=6.8021$.

**Figure 3.**Functions $c\left(x\right)$, $\widehat{c}\left(x\right)$ and trajectory $x\left(t\right)$ for $\alpha =0.5$, $\beta =0.05$, $\mu =0.26$, $\xi =0.4$, $\lambda =0.1$ and ${v}_{x}\left(0\right)=1.9$, $v\left(0\right)=6.8021$.

**Figure 4.**Functions $v\left(x\right)$ and $\widehat{v}\left(x\right)$ for $\alpha =0.5$, $\beta =0.05$, $\mu =0.26$, $\xi =0.4$, $\lambda =0.1$ and ${v}_{x}\left(0\right)=1.9$, $v\left(0\right)=6.8021$.

**Table 1.**Functions $v\left(x\right)$ and ${v}_{x}\left(x\right)$ for $\alpha =0.5$, $\beta =0.05$, $\mu =0.26$, $\xi =0.4$, $\lambda =0.1$ and ${v}_{x}\left(0\right)=1.9$, $v\left(0\right)=6.8021$.

x | $\mathit{v}\left(\mathit{x}\right)$ | ${\mathit{v}}_{\mathit{x}}\left(\mathit{x}\right)$ | $\mathit{c}\left(\mathit{x}\right)$ |
---|---|---|---|

0 | 6.8021 | 1.9000 | 0.2770 |

1 | 8.5790 | 1.6929 | 0.3489 |

2 | 10.2022 | 1.5575 | 0.4122 |

3 | 11.7010 | 1.4431 | 0.4802 |

4 | 13.0940 | 1.3454 | 0.5525 |

5 | 14.3963 | 1.2613 | 0.6286 |

6 | 15.6203 | 1.1884 | 0.7081 |

7 | 16.7762 | 1.1247 | 0.7905 |

8 | 17.8723 | 1.0687 | 0.8755 |

9 | 18.9158 | 1.0192 | 0.9626 |

10 | 19.9126 | 0.9752 | 1.0515 |

**Table 2.**Functions $v\left(x\right)$ and ${v}_{x}\left(x\right)$ for $\alpha =0.5$, $\beta =0.05$, $\mu =0.26$, $\xi =0.4$, $\lambda =0.1$ and ${v}_{x}\left(0\right)=2$, $v\left(0\right)=6.8$.

x | $\mathit{v}\left(\mathit{x}\right)$ | ${\mathit{v}}_{\mathit{x}}\left(\mathit{x}\right)$ | $\mathit{c}\left(\mathit{x}\right)$ |
---|---|---|---|

0 | 6.8000 | 2.0000 | 0.2500 |

1 | 9.4022 | 3.1941 | 0.0980 |

2 | 13.3275 | 4.7502 | 0.0443 |

3 | 19.1343 | 7.0039 | 0.0204 |

4 | 27.6771 | 10.2878 | 0.0094 |

5 | 40.2103 | 15.0801 | 0.0044 |

6 | 58.5692 | 22.0787 | 0.0021 |

7 | 85.4378 | 32.3029 | 0.0010 |

8 | 124.7394 | 47.2425 | 0.0004 |

9 | 182.2094 | 69.0750 | 0.0002 |

10 | 266.2320 | 100.9833 | 0.0001 |

**Table 3.**The values of initial conditions obtained from procedure of finding ${v}_{x}\left(0\right)$ (c. = correct, t.b. = too big, t.s. = too small).

b | ||||
---|---|---|---|---|

Correctness | Value | a | A | $\mathit{a}-\mathit{A}$ |

t.b. | ≥1.97 | - | - | - |

c. | 1.96 | 6.798693877 | 6.783185889 | 0.015507988 |

c. | 1.95 | 6.798803418 | 6.784849201 | 0.013954217 |

c. | 1.94 | 6.799092783 | 6.786580941 | 0.012511842 |

c. | 1.93 | 6.799564767 | 6.788388955 | 0.011175812 |

c. | 1.92 | 6.800222221 | 6.790283409 | 0.009938812 |

c. | 1.91 | 6.801068062 | 6.792277924 | 0.008790138 |

c. | 1.90 | 6.802105263 | 6.794392618 | 0.007712645 |

c. | 1.89 | 6.803336861 | 6.796662198 | 0.006674663 |

t.s. | 1.88 | - | - | - |

t.s. | 1.881 | - | - | - |

c. | 1.882 | 6.804464186 | 6.798652236 | 0.005811950 |

c. | 1.8819 | 6.804479085 | 6.798679195 | 0.005799890 |

t.s. | 1.8818 | - | - | - |

t.s. | ⋮ | - | - | - |

t.s. | 1.88185 | - | - | - |

c. | 1.88186 | 6.804485051 | 6.798690050 | 0.005795001 |

c. | 1.881859 | 6.804485199 | 6.798690322 | 0.005794877 |

c. | 1.881858 | 6.804485348 | 6.798690594 | 0.005794754 |

c. | 1.881857 | 6.804485498 | 6.798690867 | 0.005794631 |

c. | 1.881856 | 6.804485647 | 6.798691139 | 0.005794508 |

c. | 1.881855 | 6.804485795 | 6.798691412 | 0.005794383 |

c. | 1.881854 | 6.804485945 | 6.798691685 | 0.005794260 |

c. | 1.881853 | 6.804486095 | 6.798691958 | 0.005794137 |

c. | 1.881852 | 6.804486243 | 6.798692231 | 0.005794012 |

c. | 1.881851 | 6.804486392 | 6.798692504 | 0.005793888 |

t.s. | 1.881850 | - | - | - |

t.s. | ⋮ | - | - | - |

t.s. | 1.8818503 | - | - | - |

c. | 1.8818504 | 6.804486482 | 6.798692667 | 0.005793815 |

c. | 1.88185039 | 6.804486484 | 6.798692671 | 0.005793813 |

c. | 1.88185038 | 6.804486485 | 6.798692673 | 0.005793812 |

c. | 1.88185037 | 6.804486486 | 6.798692675 | 0.005793811 |

c. | 1.88185036 | 6.804486488 | 6.798692679 | 0.005793809 |

c. | 1.88185035 | 6.804486489 | 6.798692681 | 0.005793808 |

t.s. | 1.88185034 | - | - | - |

t.s. | 1.881850341 | - | - | - |

c. | 1.881850342 | 6.804486491 | 6.798692684 | 0.005793807 |

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**MDPI and ACS Style**

Baran, S.; Constantinescu, C.; Palmowski, Z.
Asymptotic Expected Utility of Dividend Payments in a Classical Collective Risk Process. *Risks* **2023**, *11*, 64.
https://doi.org/10.3390/risks11040064

**AMA Style**

Baran S, Constantinescu C, Palmowski Z.
Asymptotic Expected Utility of Dividend Payments in a Classical Collective Risk Process. *Risks*. 2023; 11(4):64.
https://doi.org/10.3390/risks11040064

**Chicago/Turabian Style**

Baran, Sebastian, Corina Constantinescu, and Zbigniew Palmowski.
2023. "Asymptotic Expected Utility of Dividend Payments in a Classical Collective Risk Process" *Risks* 11, no. 4: 64.
https://doi.org/10.3390/risks11040064