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We study the asymptotic of the ruin probability for a process which is thesolution of linear SDE defined by a pair of independent L\'evy processes. Ourmain interest is the model describing the evolution of the capital reserve ofan insurance company selling annuities and investing in a risky asset. Let$\beta>0$ be the root of the cumulant-generating function $H$ of the incrementof the log price process $V$. We show that the ruin probability admits theexact asymptotic $Cu^{-\beta}$ as the initial capital $u\to\infty$ assumingonly that the law of $V_T$ is non-arithmetic without any further assumptions onthe price process.

The ruin problem for L\'evy-driven linear stochastic equations with applications to actuarial models with negative risk sums