Convergence in Skorokhod J-topology for compositions of stochastic processes

A survey on functional limit theorems for compositions of stochastic processes is presented. Applications to stochastic processes with random scaling of time, random sums, extremes with random sample size, generalised exceeding processes, sum- and max-processes with renewal stopping, and shock processes are discussed

Asymptotic expansions for distributions of the surplus prior and at the time of ruin

Asymptotic expansions for the distribution of the surplus prior to and at the time of a ruin are given for nonlinearly perturbed risk processes

Asymptotic expansions for quasi-stationary distributions of nonlinearly perturbed semi-Markov processes

Asymptotic expansions are given for quasi-stationary distributions of nonlinearly perturbed semi-Markov processes

Necessary and sufficient conditions for weak convergence of first-rare-event times for semi-Markov processes. I

Necessary and sufficient conditions for weak convergence of first-rareevent times for semi-Markov processes with finite set of states are obtained. These results are applied to risk processes and give necessary and sufficient conditions for stable approximation of ruin probabilities including the case of diffusion approximation

Necessary and sufficient conditions for weak convergence of first-rare-event times for semi-Markov processes. II

Necessary and suffcient conditions for weak convergence of first-rareevent times for semi-Markov processes, obtained in the first part of this paper [66], are applied to counting processes generating by flows of rare events controlled by semi-Markov processes, random geometric sums, and risk processes. In particular, necessary and sufficient conditions for stable approximation of ruin probabilities including the case of diffusion approximation are given

Asymptotic Expansions for Stationary Distributions of Perturbed Semi-Markov Processes

New algorithms for computing of asymptotic expansions for stationary distributions of nonlinearly perturbed semi-Markov processes are presented. The algorithms are based on special techniques of sequential phase space reduction, which can be applied to processes with asymptotically coupled and uncoupled finite phase spaces.Comment: 83 page

Convergence of option rewards for Markov type price processes modulated by stochastic indices

A general price process represented by a two-component Markov process is considered. Its first component is interpreted as a price process and the second one as an index process modulating the price component. American type options with pay-off functions, which admit power type upper bounds, are studied. Both the transition characteristics of the price processes and the pay- off functions are assumed to depend on a perturbation parameter d= 0 and to converge to the corresponding limit characteristics as d¿ 0. In the first part of the paper, asymptotically uniform skeleton approximations connecting reward functionals for continuous and discrete time models are given. In the second part of the paper, these skeleton approximations are used for getting results about the convergence of reward functionals for American type options for perturbed price processes in discrete and continuous time. Examples related to modulated exponential price processes with independent increments are given

Convergence of option rewards for Markov type price processes modulated by stochastic indices. II

A general price process represented by a two-component Markov process is considered. Its first component is interpreted as a price process and the second one as an index process modulating the price component. American type options with pay-off functions, which admit power type upper bounds, are studied. Both the transition characteristics of the price processes and the pay- off functions are assumed to depend on a perturbation parameter d= 0 and to converge to the corresponding limit characteristics as d¿ 0. In the first part of the paper, asymptotically uniform skeleton approximations connecting reward functionals for continuous and discrete time models are given. In the second part of the paper, these skeleton approximations are used for getting results about the convergence of reward functionals for American type options for perturbed price processes in discrete and continuous time. Examples related to modulated exponential price processes with independent increments are given