Renewal equations on the semi-axis

N. B. Engibaryan

Izvestiya. Mathematics , Tome 63 (1999) no. 1, pp. 57-71

Résumé

We consider the renewal equation $$ \varphi(x)=g(x)+\int_0^x\varphi(x-t)\,dF(t), \qquad g\in L_1(0;\infty), $$ where $F$ is the distribution function of a non-negative random variable. If $F$ has a non-trivial absolutely continuous component or is a distribution of absolutely continuous type, then we prove that the solution of the renewal equation can be written as follows: $$ \varphi=\varphi_1+\varphi_2+\biggl[\int_0^{\infty}x\,dF(x)\biggr]^{-1}\int_0^{\infty}g(x)\,dt, $$ where $\varphi_1\in L_1(0;\infty)$, $\varphi_2\in C[0;\infty)$, and $\varphi_2(+\infty)=0$ If $g$ is bounded and $g(+\infty)=0$, then $\varphi_1(+\infty)=0$. The proof is based on the structural factorization of the renewal equation into absolutely continuous, discrete, and singular components.