Izvestiya of Saratov University.

Mathematics. Mechanics. Informatics

ISSN 1816-9791 (Print)
ISSN 2541-9005 (Online)


For citation:

Belkina T. A., Ogareva A. S. Risky investments and survival probability in the insurance model with two-sided jumps: Problems for integrodifferential equations and ordinary differential equation and their equivalence. Izvestiya of Saratov University. Mathematics. Mechanics. Informatics, 2023, vol. 23, iss. 3, pp. 278-285. DOI: 10.18500/1816-9791-2023-23-3-278-285, EDN: HYOWQI

This is an open access article distributed under the terms of Creative Commons Attribution 4.0 International License (CC-BY 4.0).
Published online: 
31.08.2023
Full text:
(downloads: 582)
Language: 
English
Heading: 
Article type: 
Article
UDC: 
519.624:519.86
EDN: 
HYOWQI

Risky investments and survival probability in the insurance model with two-sided jumps: Problems for integrodifferential equations and ordinary differential equation and their equivalence

Autors: 
Belkina Tatiana Andreevna, Central Economics and Mathematics Institute RAS
Ogareva Anna S., Moscow School of Economics of Lomonosov Moscow State University (MSE MSU)
Abstract: 

We consider a model of an insurance portfolio that includes both non-life and life annuity insurance while assuming  that the surplus (or some of its fraction) is invested in risky assets with the price dynamics given by a geometric Brownian motion. The portfolio  surplus (in the absence of investments)  is described by a stochastic process involving two-sided jumps and a continuous drift. Downward jumps correspond to the claim sizes and upward jumps are interpreted as random gains  that arise at the final moments of the life annuity contracts realizations (i.e. at the moments of the death of policyholders). The drift is determined by the difference between premiums in the non-life insurance contracts and the annuity payments. We study the ruin problem for the model with investment using an approach based on integrodifferential equations (IDE) for the survival probabilities as a function of initial surplus. The main problem in calculating the survival probability as a solution of the IDE is that the initial value of the probability itself or its derivative at a zero initial surplus is priori unknown.  For the case of the exponential distributions of the jumps, we propose a solution to this problem based on the assertion that the problem for an IDE  is equivalent to a problem for an ordinary differential equation (ODE) with some nonlocal condition added. As a result,  a solution to the original problem can be obtained as a solution to the ODE problem with an unknown parameter, which is finally determined using the specified nonlocal condition and a normalization condition.

References: 
  1. Zhang Z., Yang H., Li S. The perturbed compound Poisson risk model with two-sided jumps. Journal of Computational and Applied Mathematics, 2010, vol. 233, iss. 8, pp. 1773–1784. https://doi.org/10.1016/j.cam.2009.09.014
  2. Cheung E. C. K. On a class of stochastic models with two-sided jumps. Queueing Systems, 2011, vol. 69, iss. 1, pp. 1–28. https://doi.org/10.1007/s11134-011-9228-z
  3. Kabanov Yu., Pukhlyakov N. Ruin probabilities with investments: smoothness, integrodifferential and ordinary differential equations, asymptotic behavior. Journal of Applied Probability, 2022, vol. 59, iss. 2, pp. 556–570. https://doi.org/10.1017/jpr.2021.74
  4. Belkina T. Risky investment for insurers and sufficiency theorems for the survival probability. Markov Processes and Related Fields, 2014, vol. 20, iss 3, pp. 505–525. Available at: http://math-mprf.org/journal/articles/id1344/ (accessed November 5, 2022).
  5. Belkina T. A., Konyukhova N. B., Kurochkin S. V. Singular boundary value problem for the integrodifferential equation in an insurance model with stochastic premiums: Analysis and numerical solution. Computational Mathematics and Mathematical Physics, 2012, vol. 52, iss. 10, pp. 1384–1416. https://doi.org/10.1134/S0965542512100077
Received: 
30.11.2022
Accepted: 
25.12.2022
Published: 
31.08.2023