IIARD International Institute of Academic Research and Development

INTERNATIONAL JOURNAL OF APPLIED SCIENCES AND MATHEMATICAL THEORY (IJASMT )

E- ISSN 2489-009X
P- ISSN 2695-1908
VOL. 10 NO. 3 2024
DOI: 10.56201/ijasmt.v10.no3.2024.pg68.82

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Analytical Solutions of Insurer and Reinsurer Strategy with Environmental Noise under Logarithm Utility

Njoku Kevin Ndubuisi Chikezie. Edikan E. Akpanibah

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Abstract

In this paper, the insurer and reinsurer’s strategy and the reinsurer’s surplus were studied in the presence of some random environmental noise on the risky asset under logarithm utility function. A portfolio with one risky and risk free asset was considered such that the risky asset follows the geometric Brownian motion (GBM). It was also assume that the claim process of the insurer is a stochastic differential equation, and the reinsurer can buy proportional reinsurance policy as a backup for their investment. The maximum principle theory and Ito’s lemma were used to derive our optimization problem. The Legendre transformation and dual theory with variable separation technique were used to solve the optimization problem under logarithm utility to obtain the optimal reinsurer strategy (ORS), optimal reinsurer policy (ORP) and the reinsurer’s surplus. More so, some numerical analyses were presented to discuss the effectof somesensitive parameters on the ORS and ORP. The ORS was observed to be a decreasing function of the instantaneous volatility, risk free interest rate but an increasing function of the appreciation rate of the of the risky asset. Furthermore, the relationship between the surplus process and time, risky asset and environmental noise was also given.

keywords:

Surplus process, Insurer and reinsurance strategy, Legendre transforms,Logarithm

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