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Platon Promyslov

Publications and source records attributed to Platon Promyslov.

8 recordsLinked to original sources
arXiv2026

Smoothness of the survival probability in models with a random environment: the annuity and mixed cases

We consider the ruin problem for an insurance company investing its whole reserve in a risky asset whose parameters depend on a Markov random environment. Using the Green's function method we prove $C^2$-smoothness of the survival probability for annuity payments under minimal assumptions-the jump distribution need only be a probability measure on the positive half-line. For two-sided jumps smoothness holds whenever, in each regime, the jump distribution has no atoms on the negative half-line or the survival probability vanishes at the origin. This condition is sharp: an isolated atom combined with a positive value at the origin makes the second derivative discontinuous, and the size of the discontinuity is computed explicitly.

math.PR↗
arXiv2026

Exact asymptotics of the ruin probability in the Sparre Andersen model

For the Sparre Andersen non-life insurance model with investments in an arbitrary L\'evy process, we establish the exact power-law asymptotics of the ruin probability $\Psi(u)\sim C^* u^{-\beta}$ as $u\to\infty$ with a positive finite constant $C^*$; the exponent $\beta$ is the Cram\'er root of the Laplace exponent of the L\'evy process describing the logarithm of the risky asset price. This strengthens previously known two-sided estimates of the order -- the existence of an exact limit had remained an open question. The proof combines a reduction to discrete time, the one-dimensional Kesten-Goldie theorem for the stationary measure of the associated affine recursion, and Goldie's result on the asymptotics of the supremum of a perpetuity.

math.PR↗
arXiv2026

Existence of a classical solution to the integro-differential equation arising in the Cram\'er--Lundberg non-life insurance model with proportional investment

This paper establishes that the survival probability in the non-life Cram\'{e}r--Lundberg insurance model with proportional investment is a classical $C^2$-solution of the associated integro-differential equation under minimal moment conditions: it suffices that the claim size distribution is continuous and possesses a finite moment of some positive order.

math.PR↗
arXiv2023

Ruin Probabilities for a Sparre Andersen Model with Investments: the Case of Annuity Payments

This note is a complement to the paper by Eberlein, Kabanov, and Schmidt on the asymptotic of the ruin probability in a Sparre Andersen non-life insurance model with investments a risky asset whose price follows a geometric L\'evy process. Using the techniques of semi-Markov processes we extend the result of the mentioned paper to the case of annuities and models with two-sided jumps.

math.PR↗