Random Recurrence Equations and Ruin in a Markov-dependent Stochastic

Random Recurrence Equations

We develop sharp large deviation asymptotics for the probability of ruin in a Markov-dependent stochastic economic environment and study the extremes for some related Markovian processes which arise in financial and insurance mathematics, related to perpetuities and the ARCH(1) and GARCH(1,1) time series models. Our results build upon work of Goldie [Ann. Appl. Probab. 1 (1991) 126–166], who ha...

Dividend Payments and Related Problems in a Markov-Dependent Insurance Risk Model under Absolute Ruin

In this paper, we study the dividend payments prior to absolute ruin in a Markov-dependent risk process in which the claim occurrence and the claim amount are regulated by an external discrete time Markov chain. A system of integrodifferential equations with boundary conditions satisfied by the moment-generating function, the nth moment of the discounted dividend payments prior to absolute ruin...

A Stochastic algorithm to solve multiple dimensional Fredholm integral equations of the second kind

In the present work‎, ‎a new stochastic algorithm is proposed to solve multiple dimensional Fredholm integral equations of the second kind‎. ‎The solution of the‎ integral equation is described by the Neumann series expansion‎. ‎Each term of this expansion can be considered as an expectation which is approximated by a continuous Markov chain Monte Carlo method‎. ‎An algorithm is proposed to sim...

Central and Local Limit Theories in Markov Dependent Random Variables

Large Deviations for Solutions to Stochastic Recurrence Equations under Kesten’s Condition

In this paper we prove large deviations results for partial sums constructed from the solution to a stochastic recurrence equation. We assume Kesten’s condition [17] under which the solution of the stochastic recurrence equation has a marginal distribution with power law tails, while the noise sequence of the equations can have light tails. The results of the paper are analogs of those obtained...