On the integro-differential equation arising in the ruin problem for annuity payment models
Platon Promyslov
TL;DR
This work analyzes ruin probabilities in an annuity model where a fixed fraction $\kappa$ of capital is invested in a risky asset whose price follows a geometric Brownian motion, with a jump component modeling income and expenditures. The main contribution is representing the survival probability $\Phi$ as the solution to a second-order integro-differential equation (IDE) and reducing the equation to integral equations for its derivative $g=\Phi'$, solved via a Banach fixed-point method to obtain a tail solution, followed by a Volterra equation for the near-origin region; this avoids reducing to a higher-order ODE. Under mild jump assumptions with $F(0)=0$ and finite jump moments, the authors prove $\Phi \in C([0,\infty)) \cap C^2((0,\infty))$, $\Phi(0)=0$, $\Phi(\infty)=1$, and establish power-law ruin asymptotics $\Psi(u)\sim C\,u^{-\gamma+1}$ as $u\to\infty$, with $\gamma = \dfrac{2((a-r)\kappa + r)}{\kappa^2 \sigma^2}$ and a computable leading constant $C$. The results relax prior smoothness assumptions on the jump distribution and provide a constructive framework for computing the constant, with implications for risk assessment in investment-based ruin problems with annuity-like cash flows.
Abstract
We study a ruin problem for an annuity model where a fixed fraction of capital is invested in a risky asset. Under weak assumptions on jumps, the ruin probability solves a second-order integro-differential equation and decays as a power function for large initial capital.