Simultaneous Exact Controllability of Mean and Variance of an Insurance Policy

Rajeev Rajaram () and Nathan Ritchey
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Rajeev Rajaram: Department of Mathematical Sciences, Kent State University, Kent, OH 44242, USA
Nathan Ritchey: Department of Mathematical Sciences, Kent State University, Kent, OH 44242, USA

Mathematics, 2023, vol. 11, issue 15, 1-16

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 ? 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 ? 0 , 2 ? T ) represent the mean and variance of future losses at times t = 0 and t = T , respectively. The mean t V and variance 2 ? 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 ? t = b ( t ) , respectively, i.e., at a fine sampling of points in the time interval [ 0 , T ] .

Keywords: Thiele’s equation; Hattendorff equation; exact controllability (search for similar items in EconPapers)
JEL-codes: C (search for similar items in EconPapers)
Date: 2023
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