# Solution of Ruin Probability for Continuous Time Model Based on Block Trigonometric Exponential Neural Network

^{1}

^{2}

^{3}

^{4}

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Extreme Learning Machine Algorithm

- Step 1. Randomly initialized parameters in the hidden layer;
- Step 2. Calculate the output matrix of the hidden layer;
- Step 3. Obtain the output weight $\beta $ by least square method.

## 3. Block Trigonometric Exponential Neural Network

**Remark**

**1.**

**Convergence theorem**

**Proof.**

## 4. The BTENN Model for the Ruin Probability Equation

#### 4.1. Classical Risk Model

#### 4.2. BTENN for Classical Risk Model

#### 4.3. Erlang(2) Risk Model

#### 4.4. BTENN for Erlang(2) Risk Model

_{1}< u

_{2}⋯< um = b, the above equation can be represented as follows:

## 5. Numerical Results and Analysis

#### 5.1. Numerical Example 1

#### 5.2. Numerical Example 2

#### 5.3. Numerical Example 3

## 6. Conclusion and Prospects

## Author Contributions

## Funding

## Conflicts of Interest

## References

- Drugdova, B. The issue of the commercial insurance, commercial insurance market and insurance of non-life risks. In Financial Management of Firms and Financial Institutions: 10th International Scientific Conference, Pts I-Iv; Culik, M., Ed.; Vsb-Tech Univ. Ostrava: Feecs, Czech Republic, 2015; pp. 202–208. [Google Scholar]
- Opeshko, N.S.; Ivashura, K.A. Improvement of stress testing of insurance companies in view of european requirements. Financ. Credit Act. Probl. Theory Pract.
**2017**, 1, 112–119. [Google Scholar] [CrossRef] [Green Version] - Jia, F. Analysis of State-owned holding Insurance Companies’ Risk Management on the Basis of Equity Structure, 2nd China International Conference on Insurance and Risk Management (CICIRM); Tsinghua University Press: Beijing, China, 2011; pp. 60–63. [Google Scholar]
- Cejkova, V.; Fabus, M. Management and Criteria for Selecting Commercial Insurance Company for Small and Medium-Sized Enterprises; Masarykova Univerzita: Brno, Czech Republic, 2014; pp. 105–110. [Google Scholar]
- Belkina, T.A.; Konyukhova, N.B.; Slavko, B.V. Solvency of an Insurance Company in a Dual Risk Model with Investment: Analysis and Numerical Study of Singular Boundary Value Problems. Comput. Math. Math. Phys.
**2019**, 59, 1904–1927. [Google Scholar] [CrossRef] - Jin, B.; Yan, Q. Diversification, Performance and Risk Taking of Insurance Company; Tsinghua University Press: Beijing, China, 2013; pp. 178–188. [Google Scholar]
- Wang, Y.; Yu, W.; Huang, Y.; Yu, X.; Fan, H. Estimating the Expected Discounted Penalty Function in a Compound Poisson Insurance Risk Model with Mixed Premium Income. Mathematics
**2019**, 7, 305. [Google Scholar] [CrossRef] [Green Version] - Song, Y.; Li, X.Y.; Li, Y.; Hong, X. Risk investment decisions within the deterministic equivalent income model. Kybernetes
**2020**. [Google Scholar] [CrossRef] - Stellian, R.; Danna-Buitrago, J.P. Financial distress, free cash flow, and interfirm payment network: Evidence from an agent-based model. Int. J. Financ. Econ.
**2019**. [Google Scholar] [CrossRef] - Emms, P.; Haberman, S. Asymptotic and numerical analysis of the optimal investment strategy for an insurer. Insur. Math. Econ.
**2007**, 40, 113–134. [Google Scholar] [CrossRef] [Green Version] - Zhu, S. A Becker-Tomes model with investment risk. Econ. Theory
**2019**, 67, 951–981. [Google Scholar] [CrossRef] - Jiang, W. Two classes of risk model with diffusion and multiple thresholds: The discounted dividends. Hacet. J. Math. Stat.
**2019**, 48, 200–212. [Google Scholar] [CrossRef] - Xie, J.-h.; Zou, W. On the expected discounted penalty function for the compound Poisson risk model with delayed claims. J. Comput. Appl. Math.
**2011**, 235, 2392–2404. [Google Scholar] [CrossRef] - Lundberg, F. Approximerad Framställning Afsannollikhetsfunktionen: II. återförsäkring af Kollektivrisker; Almqvist & Wiksells Boktr: Uppsala, Sweeden, 1903. [Google Scholar]
- Andersen, E.S. On the collective theory of risk in case of contagion between claims. Bull. Inst. Math. Appl.
**1957**, 12, 275–279. [Google Scholar] - Dickson, D.C.M.; Hipp, C. On the time to ruin for Erlang(2) risk processes. Insur. Math. Econ.
**2001**, 29, 333–344. [Google Scholar] [CrossRef] [Green Version] - Li, S.M.; Garrido, J. On a class of renewal risk models with a constant dividend barrier. Insur. Math. Econ.
**2004**, 35, 691–701. [Google Scholar] [CrossRef] - Li, S.M.; Garrido, J. On ruin for the Erlang(n) risk process. Insur. Math. Econ.
**2004**, 34, 391–408. [Google Scholar] [CrossRef] - Gerber, H.U.; Yang, H. Absolute Ruin Probabilities in a Jump Diffusion Risk Model with Investment. N. Am. Actuar. J.
**2007**, 11, 159–169. [Google Scholar] [CrossRef] - Yazici, M.A.; Akar, N. The finite/infinite horizon ruin problem with multi-threshold premiums: A Markov fluid queue approach. Ann. Oper. Res.
**2017**, 252, 85–99. [Google Scholar] [CrossRef] - Lu, Y.; Li, S. The Markovian regime-switching risk model with a threshold dividend strategy. Insur. Math. Econ.
**2009**, 44, 296–303. [Google Scholar] [CrossRef] - Zhu, J.; Yang, H. Ruin theory for a Markov regime-switching model under a threshold dividend strategy. Insur. Math. Econ.
**2008**, 42, 311–318. [Google Scholar] [CrossRef] - Asmussen, S.; Albrecher, H. Ruin Probabilities. Advanced Series on Statistical Science & Applied Probability, 2nd ed.; World Scientific: Singapore, 2010; Volume 14, p. 630. [Google Scholar]
- Wang, G.J.; Wu, R. Some distributions for classical risk process that is perturbed by diffusion. Insur. Math. Econ.
**2000**, 26, 15–24. [Google Scholar] [CrossRef] - Cai, J.; Yang, H.L. Ruin in the perturbed compound Poisson risk process under interest force. Adv. Appl. Probab.
**2005**, 37, 819–835. [Google Scholar] [CrossRef] [Green Version] - Bergel, A.I.; Egidio dos Reis, A.D. Ruin problems in the generalized Erlang(n) risk model. Eur. Actuar. J.
**2016**, 6, 257–275. [Google Scholar] [CrossRef] - Kasumo, C. Minimizing an Insurer’s Ultimate Ruin Probability by Reinsurance and Investments. Math. Comput. Appl.
**2019**, 24, 21. [Google Scholar] [CrossRef] [Green Version] - Xu, L.; Wang, M.; Zhang, B. Minimizing Lundberg inequality for ruin probability under correlated risk model by investment and reinsurance. J. Inequalities Appl.
**2018**. [Google Scholar] [CrossRef] [PubMed] - Zou, W.; Xie, J.H. On the probability of ruin in a continuous risk model with delayed claims. J. Korean Math. Soc.
**2013**, 50, 111–125. [Google Scholar] [CrossRef] - Andrulytė, I.M.; Bernackaitė, E.; Kievinaitė, D.; Šiaulys, J. A Lundberg-type inequality for an inhomogeneous renewal risk model. Mod. Stoch. Theory Appl.
**2015**, 2, 173–184. [Google Scholar] [CrossRef] [Green Version] - Fei, W.; Hu, L.; Mao, X.; Xia, D. Advances in the truncated euler-maruyama method for stochastic differential delay equations. Commun. Pure Appl. Anal.
**2020**, 19, 2081–2100. [Google Scholar] [CrossRef] [Green Version] - Li, F.; Cao, Y. Stochastic Differential Equations Numerical Simulation Algorithm for Financial Problems Based on Euler Method. In 2010 International Forum on Information Technology and Applications; IEEE Computer Society: Los Alamitos, CA, USA, 2010; pp. 190–193. [Google Scholar] [CrossRef]
- Zhang, C.; Qin, T. The mixed Runge-Kutta methods for a class of nonlinear functional-integro-differential equations. Appl. Math. Comput.
**2014**, 237, 396–404. [Google Scholar] [CrossRef] - Cardoso, R.M.R.; Waters, H.R. Calculation of finite time ruin probabilities for some risk models. Insur. Math. Econ.
**2005**, 37, 197–215. [Google Scholar] [CrossRef] - Makroglou, A. Computer treatment of the integro-differential equations of collective non-ruin; the finite time case. Math. Comput. Simul.
**2000**, 54, 99–112. [Google Scholar] [CrossRef] - Paulsen, J.; Kasozi, J.; Steigen, A. A numerical method to find the probability of ultimate ruin in the classical risk model with stochastic return on investments. Insur. Math. Econ.
**2005**, 36, 399–420. [Google Scholar] [CrossRef] - Tsitsiashvili, G.S. Computing ruin probability in the classical risk model. Autom. Remote Control
**2009**, 70, 2109–2115. [Google Scholar] [CrossRef] - Zhang, Z. Approximating the density of the time to ruin via fourier-cosine series expansion. Astin Bull.
**2017**, 47, 169–198. [Google Scholar] [CrossRef] - Muzhou Hou, Y.C. Industrial Part Image Segmentation Method Based on Improved Level Set Model. J. Xuzhou Inst. Technol.
**2019**, 40, 10. [Google Scholar] - Wang, Z.; Meng, Y.; Weng, F.; Chen, Y.; Lu, F.; Liu, X.; Hou, M.; Zhang, J. An Effective CNN Method for Fully Automated Segmenting Subcutaneous and Visceral Adipose Tissue on CT Scans. Ann. Biomed. Eng.
**2020**, 48, 312–328. [Google Scholar] [CrossRef] [PubMed] - Hou, M.; Zhang, T.; Weng, F.; Ali, M.; Al-Ansari, N.; Yaseen, Z.M. Global solar radiation prediction using hybrid online sequential extreme learning machine model. Energies
**2018**, 11, 3415. [Google Scholar] [CrossRef] [Green Version] - Muzhou Hou, T.Z.; Yang, Y.; Luo, J. Application of Mec- based ELM algorithm in prediction of PM2.5 in Changsha City. J. Xuzhou Inst. Technol.
**2019**, 34, 1–6. [Google Scholar] - Hahnel, P.; Marecek, J.; Monteil, J.; O’Donncha, F. Using deep learning to extend the range of air pollution monitoring and forecasting. J. Comput. Phys.
**2020**, 408. [Google Scholar] [CrossRef] [Green Version] - Chen, Y.; Xie, X.; Zhang, T.; Bai, J.; Hou, M. A deep residual compensation extreme learning machine and applications. J. Forecast.
**2020**, 1–14. [Google Scholar] [CrossRef] - Weng, F.; Chen, Y.; Wang, Z.; Hou, M.; Luo, J.; Tian, Z. Gold price forecasting research based on an improved online extreme learning machine algorithm. J. Ambient Intell. Humaniz. Comput.
**2020**. [Google Scholar] [CrossRef] - Hou, M.; Liu, T.; Yang, Y.; Zhu, H.; Liu, H.; Yuan, X.; Liu, X. A new hybrid constructive neural network method for impacting and its application on tungsten price prediction. Appl. Intell.
**2017**, 47, 28–43. [Google Scholar] [CrossRef] - Hou, M.; Han, X. Constructive Approximation to Multivariate Function by Decay RBF Neural Network. IEEE Trans. Neural Netw.
**2010**, 21, 1517–1523. [Google Scholar] [CrossRef] - Sun, H.; Hou, M.; Yang, Y.; Zhang, T.; Weng, F.; Han, F. Solving Partial Differential Equation Based on Bernstein Neural Network and Extreme Learning Machine Algorithm. Neural Process. Lett.
**2019**, 50, 1153–1172. [Google Scholar] [CrossRef] - Yang, Y.; Hou, M.; Luo, J. A novel improved extreme learning machine algorithm in solving ordinary differential equations by Legendre neural network methods. Adv. Differ. Equ.
**2018**, 2018, 469. [Google Scholar] [CrossRef] - Yang, Y.; Hou, M.; Luo, J.; Liu, T. Neural Network method for lossless two-conductor transmission line equations based on the IELM algorithm. AIP Adv.
**2018**, 8. [Google Scholar] [CrossRef] - Yang, Y.; Hou, M.; Sun, H.; Zhang, T.; Weng, F.; Luo, J. Neural network algorithm based on Legendre improved extreme learning machine for solving elliptic partial differential equations. Soft Comput.
**2020**, 24, 1083–1096. [Google Scholar] [CrossRef] - Sabir, Z.; Wahab, H.A.; Umar, M.; Sakar, M.G.; Raja, M.A.Z. Novel design of Morlet wavelet neural network for solving second order Lane-Emden equation. Math. Comput. Simul.
**2020**, 172, 1–14. [Google Scholar] [CrossRef] - Hure, C.; Pham, H.; Warin, X. Deep backward schemes for high-dimensional nonlinear pdes. Math. Comput.
**2020**, 89, 1547–1579. [Google Scholar] [CrossRef] [Green Version] - Samaniego, E.; Anitescu, C.; Goswami, S.; Nguyen-Thanh, V.M.; Guo, H.; Hamdia, K.; Zhuang, X.; Rabczuk, T. An energy approach to the solution of partial differential equations in computational mechanics via machine learning: Concepts, implementation and applications. Comput. Methods Appl. Mech. Eng.
**2020**, 362. [Google Scholar] [CrossRef] [Green Version] - Huang, G.; Huang, G.B.; Song, S.; You, K. Trends in extreme learning machines: A review. Neural Netw.
**2015**, 61, 32–48. [Google Scholar] [CrossRef] - Zhou, T.; Liu, X.; Hou, M.; Liu, C. Numerical solution for ruin probability of continuous time model based on neural network algorithm. Neurocomputing
**2019**, 331, 67–76. [Google Scholar] [CrossRef] - Lu, Y.; Chen, G.; Yin, Q.; Sun, H.; Hou, M. Solving the ruin probabilities of some risk models with Legendre neural network algorithm. Digit. Signal Process.
**2020**, 99. [Google Scholar] [CrossRef] - Zhang, X.; Wu, J.; Yu, D. Superconvergence of the composite Simpson’s rule for a certain finite-part integral and its applications. J. Comput. Appl. Math.
**2009**, 223, 598–613. [Google Scholar] [CrossRef] [Green Version]

Availability of Data and Material: Not applicable. |

Code Availability: All code is executed by Python3.6, mail to [email protected]. |

**Figure 3.**Analytical and numerical solutions obtained by the block trigonometric exponential neural network (BTENN).

**Figure 4.**Comparison of absolute errors of approximate solution by BTENN and legendre neural network (LNN).

**Table 1.**Comparison of Absolute Errors of Approximate Solutions Obtained by BTENN, LNN and Trigonometric Neural Networks (TNN).

$\mathbf{u}$ | Exact | Approximate | Absolute Error by BTENN | Absolute Error by LNN in [57] | Absolute Error by TNN in [56] |
---|---|---|---|---|---|

0.00 | 0.6666666667 | 0.6666666667 | $0.0000\times {10}^{+0}$ | $0.0000\times {10}^{+0}$ | $0.0000\times {10}^{+0}$ |

0.25 | 0.5643211499 | 0.5643211737 | $2.3783\times {10}^{-8}$ | $3.2946\times {10}^{-8}$ | $4.2156\times {10}^{-5}$ |

0.75 | 0.4043537731 | 0.4043537518 | $2.1386\times {10}^{-8}$ | $6.6140\times {10}^{-8}$ | $7.9549\times {10}^{-5}$ |

1.25 | 0.2897321390 | 0.2897321605 | $2.1480\times {10}^{-8}$ | $6.5976\times {10}^{-8}$ | $9.5476\times {10}^{-5}$ |

1.75 | 0.2076021493 | 0.2076021261 | $2.3219\times {10}^{-8}$ | $8.7577\times {10}^{-8}$ | $1.0853\times {10}^{-4}$ |

2.25 | 0.1487534401 | 0.1487534167 | $2.3394\times {10}^{-8}$ | $9.8572\times {10}^{-8}$ | $1.1750\times {10}^{-4}$ |

2.75 | 0.1065864974 | 0.1065865074 | $1.0056\times {10}^{-8}$ | $9.1825\times {10}^{-8}$ | $1.2404\times {10}^{-4}$ |

3.25 | 0.0763725627 | 0.0763725672 | $4.5537\times {10}^{-9}$ | $8.9967\times {10}^{-8}$ | $1.2869\times {10}^{-4}$ |

3.75 | 0.0547233324 | 0.0547233102 | $2.2264\times {10}^{-8}$ | $1.0164\times {10}^{-7}$ | $1.3204\times {10}^{-4}$ |

4.25 | 0.0392109811 | 0.0392109620 | $1.9102\times {10}^{-8}$ | $1.1269\times {10}^{-7}$ | $1.3442\times {10}^{-4}$ |

4.75 | 0.0280958957 | 0.0280959020 | $6.3156\times {10}^{-9}$ | $1.1018\times {10}^{-7}$ | $1.3614\times {10}^{-4}$ |

5.25 | 0.0201315889 | 0.0201315994 | $1.0449\times {10}^{-8}$ | $9.9976\times {10}^{-8}$ | $1.3737\times {10}^{-4}$ |

5.75 | 0.0144249138 | 0.0144249036 | $1.0197\times {10}^{-8}$ | $9.7611\times {10}^{-8}$ | $1.3825\times {10}^{-4}$ |

6.25 | 0.0103359024 | 0.0103358859 | $1.6527\times {10}^{-8}$ | $1.0722\times {10}^{-7}$ | $1.3888\times {10}^{-4}$ |

6.75 | 0.0074059977 | 0.0074060017 | $4.0477\times {10}^{-9}$ | $1.1571\times {10}^{-7}$ | $1.3933\times {10}^{-4}$ |

7.25 | 0.0053066292 | 0.0053066427 | $1.3449\times {10}^{-8}$ | $1.1086\times {10}^{-7}$ | $1.3966\times {10}^{-4}$ |

7.75 | 0.0038023660 | 0.0038023595 | $6.5289\times {10}^{-9}$ | $1.0081\times {10}^{-7}$ | $1.3989\times {10}^{-4}$ |

8.25 | 0.0027245143 | 0.0027245042 | $1.0087\times {10}^{-8}$ | $1.0404\times {10}^{-7}$ | $1.4007\times {10}^{-4}$ |

8.75 | 0.0019521998 | 0.0019522135 | $1.3742\times {10}^{-8}$ | $1.4151\times {10}^{-7}$ | $1.4012\times {10}^{-4}$ |

9.25 | 0.0013988123 | 0.0013988045 | $7.7700\times {10}^{-9}$ | $1.0550\times {10}^{-7}$ | $1.4059\times {10}^{-4}$ |

9.75 | 0.0010022928 | 0.0010023058 | $1.3006\times {10}^{-8}$ | $1.0982\times {10}^{-7}$ | $1.3753\times {10}^{-4}$ |

MAE | $2.3783\times {10}^{-8}$ | $1.4151\times {10}^{-7}$ | $1.4059\times {10}^{-4}$ | ||

MSE | $2.2044\times {10}^{-16}$ | $1.0124\times {10}^{-14}$ | $1.6124\times {10}^{-8}$ |

Training Points | MSE | MAE |
---|---|---|

21 | $2.2044\times {10}^{-16}$ | $2.3783\times {10}^{-8}$ |

30 | $9.8427\times {10}^{-17}$ | $1.0836\times {10}^{-8}$ |

50 | $8.2830\times {10}^{-17}$ | $9.2712\times {10}^{-9}$ |

100 | $4.6419\times {10}^{-17}$ | $8.1964\times {10}^{-9}$ |

Hidden Neurons | MSE | MAE |
---|---|---|

12 | $2.2044\times {10}^{-16}$ | $2.3783\times {10}^{-8}$ |

18 | $8.0732\times {10}^{-17}$ | $6.1296\times {10}^{-9}$ |

20 | $1.0714\times {10}^{-18}$ | $8.3127\times {10}^{-10}$ |

50 | $9.1445\times {10}^{-22}$ | $7.1498\times {10}^{-12}$ |

$\mathit{u}.$ | Exact | BTENN Solution | Absolute Error by BTENN | Absolute Error by LNN in [57] | Absolute Error by TNN in [56] |
---|---|---|---|---|---|

0.0 | 0.7822293562 | 0.7822293562 | $0.0000\times {10}^{+0}$ | $0.0000\times {10}^{+0}$ | $0.0000\times {10}^{+0}$ |

0.5 | 0.7015293026 | 0.7015292999 | $2.7167\times {10}^{-9}$ | $2.4647\times {10}^{-7}$ | $2.8123\times {10}^{-4}$ |

1.0 | 0.6291548105 | 0.6291548124 | $2.0193\times {10}^{-9}$ | $4.5465\times {10}^{-7}$ | $2.8636\times {10}^{-4}$ |

1.5 | 0.5642469590 | 0.5642469609 | $1.8784\times {10}^{-9}$ | $1.0367\times {10}^{-7}$ | $1.5895\times {10}^{-4}$ |

2.0 | 0.5060354389 | 0.5060354357 | $3.3062\times {10}^{-9}$ | $3.4282\times {10}^{-7}$ | $2.6196\times {10}^{-6}$ |

2.5 | 0.4538294117 | 0.4538294097 | $2.1078\times {10}^{-9}$ | $2.8821\times {10}^{-7}$ | $1.3678\times {10}^{-4}$ |

3.0 | 0.4070093102 | 0.4070093132 | $3.0732\times {10}^{-9}$ | $8.5184\times {10}^{-8}$ | $2.1145\times {10}^{-4}$ |

3.5 | 0.3650194860 | 0.3650194894 | $3.4673\times {10}^{-9}$ | $3.3444\times {10}^{-7}$ | $2.1746\times {10}^{-4}$ |

4.0 | 0.3273616151 | 0.3273616137 | $1.4722\times {10}^{-9}$ | $2.3922\times {10}^{-7}$ | $1.6260\times {10}^{-4}$ |

4.5 | 0.2935887841 | 0.2935887798 | $4.3992\times {10}^{-9}$ | $7.1659\times {10}^{-8}$ | $6.6239\times {10}^{-5}$ |

5.0 | 0.2633001860 | 0.2633001850 | $1.0998\times {10}^{-9}$ | $2.9814\times {10}^{-7}$ | $4.5630\times {10}^{-5}$ |

5.5 | 0.2361363638 | 0.2361363676 | $3.7978\times {10}^{-9}$ | $2.4169\times {10}^{-7}$ | $1.4506\times {10}^{-4}$ |

6.0 | 0.2117749447 | 0.2117749479 | $3.3257\times {10}^{-9}$ | $3.5002\times {10}^{-8}$ | $2.0657\times {10}^{-4}$ |

6.5 | 0.1899268137 | 0.1899268117 | $1.9615\times {10}^{-9}$ | $2.6648\times {10}^{-7}$ | $2.1149\times {10}^{-4}$ |

7.0 | 0.1703326833 | 0.1703326792 | $4.2003\times {10}^{-9}$ | $2.1959\times {10}^{-7}$ | $1.5223\times {10}^{-4}$ |

7.5 | 0.1527600154 | 0.1527600159 | $3.2072\times {10}^{-10}$ | $6.6730\times {10}^{-8}$ | $3.6624\times {10}^{-5}$ |

8.0 | 0.1370002624 | 0.1370002664 | $4.0026\times {10}^{-9}$ | $2.6165\times {10}^{-7}$ | $1.0754\times {10}^{-4}$ |

8.5 | 0.1228663918 | 0.1228663912 | $3.5653\times {10}^{-10}$ | $7.4952\times {10}^{-8}$ | $2.2779\times {10}^{-4}$ |

9.0 | 0.1101906665 | 0.1101906634 | $3.1964\times {10}^{-9}$. | $2.3406\times {10}^{-7}$ | $2.4184\times {10}^{-4}$ |

9.5 | 0.0988226546 | 0.0988226577 | $2.6166\times {10}^{-9}$ | $4.8428\times {10}^{-8}$ | $3.2086\times {10}^{-5}$ |

10.0 | 0.0886274433 | 0.0886274378 | $9.7183\times {10}^{-10}$ | $3.7546\times {10}^{-7}$ | $5.6048\times {10}^{-4}$ |

MAE | $4.3992\times {10}^{-9}$ | $4.5465\times {10}^{-7}$ | $5.6048\times {10}^{-4}$ | ||

MSE | $8.7029\times {10}^{-18}$ | $6.0264\times {10}^{-7}$ | $4.4036\times {10}^{-8}$ |

$\mathbf{u}$ | BTENN Solution | LNN Solution [57] | Absolute Difference |
---|---|---|---|

230 | 0.7771674726 | 0.7771671970 | $2.7562\times {10}^{-7}$ |

1162 | 0.6262266038 | 0.6262251570 | $1.4468\times {10}^{-6}$ |

2094 | 0.5177610722 | 0.5177563420 | $4.7302\times {10}^{-6}$ |

3026 | 0.4367401507 | 0.4367394040 | $7.4668\times {10}^{-7}$ |

3958 | 0.3719139416 | 0.3719008950 | $1.3047\times {10}^{-5}$ |

4890 | 0.3188717066 | 0.3188620670 | $9.6396\times {10}^{-6}$ |

5822 | 0.2748805781 | 0.2748686660 | $1.1912\times {10}^{-5}$ |

6754 | 0.2380806874 | 0.2380170420 | $6.3645\times {10}^{-5}$ |

7686 | 0.2069896444 | 0.2069120930 | $7.7551\times {10}^{-5}$ |

8618 | 0.1805589985 | 0.1805008800 | $5.8118\times {10}^{-5}$ |

9550 | 0.1580081081 | 0.1579619740 | $4.6134\times {10}^{-5}$ |

© 2020 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (http://creativecommons.org/licenses/by/4.0/).

## Share and Cite

**MDPI and ACS Style**

Chen, Y.; Yi, C.; Xie, X.; Hou, M.; Cheng, Y.
Solution of Ruin Probability for Continuous Time Model Based on Block Trigonometric Exponential Neural Network. *Symmetry* **2020**, *12*, 876.
https://doi.org/10.3390/sym12060876

**AMA Style**

Chen Y, Yi C, Xie X, Hou M, Cheng Y.
Solution of Ruin Probability for Continuous Time Model Based on Block Trigonometric Exponential Neural Network. *Symmetry*. 2020; 12(6):876.
https://doi.org/10.3390/sym12060876

**Chicago/Turabian Style**

Chen, Yinghao, Chun Yi, Xiaoliang Xie, Muzhou Hou, and Yangjin Cheng.
2020. "Solution of Ruin Probability for Continuous Time Model Based on Block Trigonometric Exponential Neural Network" *Symmetry* 12, no. 6: 876.
https://doi.org/10.3390/sym12060876