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Simultaneous Exact Controllability of Mean and Variance of an Insurance Policy

Author

Listed:
  • Rajeev Rajaram

    (Department of Mathematical Sciences, Kent State University, Kent, OH 44242, USA
    These authors contributed equally to this work.)

  • Nathan Ritchey

    (Department of Mathematical Sciences, Kent State University, Kent, OH 44242, USA
    These authors contributed equally to this work.)

Abstract

We explore the simultaneous exact controllability of mean and variance of an insurance policy by utilizing the benefit S t and premium P t as control inputs to manage the policy value t V and the variance 2 σ t of future losses. The goal is to determine whether there exist control inputs that can steer the mean and variance from a prescribed initial state at t = 0 to a prescribed final state at t = T , where the initial–terminal pair of states ( 0 V , T V ) and ( 2 σ 0 , 2 σ T ) represent the mean and variance of future losses at times t = 0 and t = T , respectively. The mean t V and variance 2 σ t are governed by Thiele’s and Hattendorff’s differential equations in continuous time and recursive equations in discrete time. Our study focuses on solving the problem of exact controllability in both continuous and discrete time. We show that our result can be used to devise control inputs S t , P t in the interval [ 0 , T ] so that the mean and variance partially track a specified curve t V = a ( t ) and 2 σ t = b ( t ) , respectively, i.e., at a fine sampling of points in the time interval [ 0 , T ] .

Suggested Citation

  • Rajeev Rajaram & Nathan Ritchey, 2023. "Simultaneous Exact Controllability of Mean and Variance of an Insurance Policy," Mathematics, MDPI, vol. 11(15), pages 1-16, July.
    • Handle: RePEc:gam:jmathe:v:11:y:2023:i:15:p:3296-:d:1203452
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    References listed on IDEAS

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    1. Viktor Stojkoski & Trifce Sandev & Ljupco Kocarev & Arnab Pal, 2021. "Geometric Brownian Motion under Stochastic Resetting: A Stationary yet Non-ergodic Process," Papers 2104.01571, arXiv.org, revised Aug 2021.
    2. Dalila Guerdouh & Nabil Khelfallah & Josep Vives, 2022. "Optimal Control Strategies for the Premium Policy of an Insurance Firm with Jump Diffusion Assets and Stochastic Interest Rate," JRFM, MDPI, vol. 15(3), pages 1-19, March.
    3. Stein, Elias M & Stein, Jeremy C, 1991. "Stock Price Distributions with Stochastic Volatility: An Analytic Approach," The Review of Financial Studies, Society for Financial Studies, vol. 4(4), pages 727-752.
    4. Rajeev Rajaram & Nathan Ritchey, 2021. "Hattendorff Differential Equation for Multi-State Markov Insurance Models," Risks, MDPI, vol. 9(9), pages 1-19, September.
    5. Dickson,David C. M. & Hardy,Mary R. & Waters,Howard R., 2020. "Solutions Manual for Actuarial Mathematics for Life Contingent Risks," Cambridge Books, Cambridge University Press, number 9781108747615.
    6. Viktor Stojkoski & Trifce Sandev & Lasko Basnarkov & Ljupco Kocarev & Ralf Metzler, 2020. "Generalised geometric Brownian motion: Theory and applications to option pricing," Papers 2011.00312, arXiv.org.
    7. Roman V. Ivanov, 2023. "On the Stochastic Volatility in the Generalized Black-Scholes-Merton Model," Risks, MDPI, vol. 11(6), pages 1-23, June.
    8. Emms, Paul & Haberman, Steven, 2005. "Pricing General Insurance Using Optimal Control Theory," ASTIN Bulletin, Cambridge University Press, vol. 35(2), pages 427-453, November.
    9. Black, Fischer & Scholes, Myron S, 1973. "The Pricing of Options and Corporate Liabilities," Journal of Political Economy, University of Chicago Press, vol. 81(3), pages 637-654, May-June.
    10. Cox, John C. & Ross, Stephen A. & Rubinstein, Mark, 1979. "Option pricing: A simplified approach," Journal of Financial Economics, Elsevier, vol. 7(3), pages 229-263, September.
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