The asymptotic behaviour of probabilities of small deviations has been investigated for various classes of stochastic processes. Asymptotics of probabilities of small deviations for sums of independent random variables and homogeneous processes with independent increments were found in Mogul’skii [1], Borovkov and Mogul’skii [2] and references therein. Note that the last class of processes incl...

We consider risk processes with reinsurance. A general family of reinsurance contracts is allowed, including proportional and excess-of-loss policies. Claim occurrence is regulated by a classical compound Poisson process or by aMarkov-modulated compound Poisson process. We provide some large deviation results concerning these two risk processes in the small-claim case. Finally, we derive the so...

We introduce an algorithm for the segmentation of a class of regime switching processes. The segmentation algorithm is a non parametric statistical method able to identify the regimes (patches) of a time series. The process is composed of consecutive patches of variable length. In each patch the process is described by a stationary compound Poisson process, i.e. a Poisson process where each cou...

The problem of disorder seeks to determine a stopping time which is as close as possible to the unknown time of ’disorder’ when the observed process changes its probability characteristics. We give a partial answer to this question for some special cases of Lévy processes and present a complete solution of the Bayesian and variational problem for a compound Poisson process with exponential jump...

Let (Nt : t > 0) be the simple counting process associated with the number of jumps in (0, t]. Then, Sn and Xn are the respectively the arrival instant and the size of the nth jump, and we can write Zt = ∑t i=1 Xi. Let Fs = σ(Zu : u ∈ (0,s]), and F• = (Fs : s> 0) be the natural filtration associated with the process Z. Clearly, jump times (Sn : n∈N) are stopping times with respect to filtration...

We overview the results on topic of compound Poisson approximation to distribution a sum Sn=X1+⋯+Xn (possibly dependent) random variables. indicate number open problems and discuss directions further research.