On functional equations stemming from actuarial mathematics
Let \(\mathcal X_+\) be the family of all non-negative bounded random variables in a given probability space; let \(w\in [0,\infty)\) be the initial wealth of an insurance company and let \(u:\mathbb R \to \mathbb R\) be its continuous strictly increasing utility function with \(u(0)=0\). Then, for every \(X\in \mathcal X_+\) there exists a unique real number \(H(X)\) such that \[ E[u(w+H(X)-X)]=u(w). \] The functional \(H(X)\) is called the principle of equivalent utility and represents the premium for the risk \(X\). When \(u(x)=cx\), \(c>0\), then we obtain the net premium principle \(H(X)=E[X]\); when \(u(x)=a(1-e^{-cx})\), \(a, c\) positive, we have the exponential premium principle \(H(X)=\frac{1}{c}\ln E[e^{cx}]\). For the rank-dependent utility model, the premium for a risk \(X\) is a solution of the equation \[ E_g[u(w+H(X)-X)]=u(w), \] where \(g:[0,1]\to [0,1]\) is non-decreasing, with \(g(0)=0\), \(g(1)=1\) is called probability distorsion function, and \[ E_g[X]=\int_{-\infty}^0 (g(P(X>t)-1)\;dt+\int_0^{\infty} (g(P(X>t))\;dt. \] The cumulative prospect theory leads to the equation \[ E_{gh}[u(w+H(X)-X)]=u(w), \leqno(1) \] where \(g\) and \(h\) are probability distorsion functions and \[ E_{gh}[X)]=E_g[\max\{X,0\}]-E_h[\max\{-X,0\}], \] for \(X\) bounded random variable. The paper deals with the following problem: determine all functionals of equivalent utility, whose restriction to the family of binary risks, that is the family of random variables taking only two values (\(0\) and a positive value), reduces either to the net premium principle or to exponential premium principle. The paper gives a complete answer to this problem, namely for a given value of \(w\) and \(H\) defined by (1), in the two cases \(H(X)=E[X]\) or \(H(X)=\frac{1}{c}\ln E[e^{cx}]\), a complete description of the functions \(u, g\) and \(h\) is presented.
scientific article; zbMATH DE number 6894803
scientific article