Moreover, the solutions of the optimal control problems are examined and compared to the findings where the jump sizes are assumed to be constant.
📌 3 Key Takeaways
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Objective & Hypothesis:We investigate an insurer's optimal investment and liability problem by maximizing the expected terminal wealth under different utility functions.
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Key Findings & Data:The insurer's aggregate claim payments are modeled by a Lévy risk process.
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Conclusion & Impact:Moreover, the solutions of the optimal control problems are examined and compared to the findings where the jump sizes are assumed to be constant.
🔬Design:Modeling & Simulation
📍Scope:Comprehensive Sample
⚙️Variables:Lévy risk process · Control theory · Pontryagin’s maximum principle
Abstract
We investigate an insurer's optimal investment and liability problem by maximizing the expected terminal wealth under different utility functions. The insurer's aggregate claim payments are modeled by a Lévy risk process. We assume that the financial market consists of a riskless and a risky assets. It is also assumed that the insurer's liability is negatively correlated with the return of the risky asset. The closed-form solution for the optimal investment and liability ratio is obtained using Pontryagin's Maximum Principle. Moreover, the solutions of the optimal control problems are examined and compared to the findings where the jump sizes are assumed to be constant.
ÖZALP, M. A., YILDIRAK, K., OKUR, Y. Y. (2022). Optimal investment strategy and liability ratio for insurer with Lévy risk process. Hacettepe Journal of Mathematics and Statistics, Cilt: 48 - Sayı: 4, 1232–1249. https://ajindex.com/articles/529724
A. Biffis and A. E. Kyprianou. A note on scale functions and the time value of ruin for Lévy insurance risk processes. Insurance: Mathematics and Economics, 46(1), 85-91, 2010
S, Browne. Optimal Investment Policies for a Firm With a Random Risk Process: Exponential Utility and Minimizing the Probability of Ruin. Mathematics of Operations Research, 20(4),937-458, 1995
A. Castañer and M. Mercé Claramunt. Optimal Stop-loss Reinsurance: a Dependence Analysis. Hacettepe Journal of Mathematics and Statistics, 45(2),497-519, 2016
R. Cont and P. Tankov. Financial Modelling with Jump Processes. Chapman & Hall/CRC, 2004
W. H. Fleming and H. M. Soner. Controlled Markov Processes and Viscosity Solutions (Vol. 25). New York: Springer Science & Business Media, 2006
N. C. Framstad, B. Øksendal and A. Sulem. Sufficient stochastic maximum principle for the optimal control of jump diffusions and applications to finance. Journal of Optimization Theory and Applications, 121(1), 77-98, 2004
J. M. Harrison. and D. M. Kreps. Martingales and arbitrage in multiperiod securities markets. Journal of Economic Theory, 20(3), 381-408, 1979
C. Hipp and M. Taksar. Stochastic control for optimal new business. Insurance: Mathematics and Economics, 26(2), 185-192, 2000
I. Karatzas, J. P. Lehoczky, S. E. Shreve and G. Xu. Martingale and duality methods for utility maximization in an incomplete market. SIAM Journal on Control and optimization, 29(3), 702-730, 1991
A. E. Kyprianou. Introductory Lectures on Fluctuations of Lévy Processes with Applications. New York: Springer Science & Business Media, 2006
B. K. Øksendal and A. Sulem. Applied Stochastic Control of Jump Diffusions (Vol. 498). Berlin: Springer, 2005
S. D. Promislow and V. R. Young. Unifying framework for optimal insurance. Insurance: Mathematics and Economics, 36(3), 347-364, 2005
H. Schmidli. Optimal proportional reinsurance policies in a dynamic setting. Scandinavian Actuarial Journal, 2001(1), 55-68, 2001
J. L. Stein. Stochastic optimal control and the US financial debt crisis. New York: Springer, 2012
M. Taksar. Optimal risk and dividend distribution control models for an insurance company. Mathematical Methods of Operations Research, 51(1), 1-42, 2000
N. Wang. Optimal investment for an insurer with exponential utility preference. Insurance: Mathematics and Economics, 40(1), 77-84, 2007
Z.Wang, J. Xia and L. Zhang. Optimal investment fo