Asymptotic Expected Utility of Dividend Payments in a Classical Collective Risk Process
Abstract
1. Introduction
2. Hamilton–Jacobi–Bellman Equation
3. Asymptotic Analysis
3.1. Classical Risk Process (1) and Power Utility Function
3.2. Classical Risk Process (1) and Logarithmic Utility Function
4. Numerical Analysis
- Set initial value ,
- From the equality (13) derive initial value ;
- Solve numerically the differential Equation (12) with the initial condition ;
- Calculate using ;
- Using the least squares method, approximate be the linear function . Because of our results from Theorem 2, we assume that is a linear function;
- Let be a trajectory of the regulated process starting from 0 until the first time claim arrival T. Hence
- Using the least squares method, approximate by a function of the form . Because of our results from Theorem 2, we assume that is a power function;
- Calculate
- Calculate the value ;
- Repeat until for fixed .
Author Contributions
Funding
Data Availability Statement
Conflicts of Interest
Correction Statement
Appendix A. Proof of Theorem 2
- (a)
- and ;
- (b)
- and ;
- (c)
- and .
- I.
- If , then, via the separation of variables, we have
- II.
- If , then a simple integration leads to
Appendix B. Proof of Theorem 3
- (a)
- and ;
- (b)
- and ;
- (c)
- and .
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x | |||
---|---|---|---|
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 |
x | |||
---|---|---|---|
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 |
b | ||||
---|---|---|---|---|
Correctness | Value | a | 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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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
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 StyleBaran, 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
APA StyleBaran, S., Constantinescu, C., & Palmowski, Z. (2023). Asymptotic Expected Utility of Dividend Payments in a Classical Collective Risk Process. Risks, 11(4), 64. https://doi.org/10.3390/risks11040064