Modern Stochastics: Theory and Applications

Entropic Value-at-Risk (EVaR) measure is a convenient coherent risk measure. Due to certain difficulties in finding its analytical representation, it was previously calculated explicitly only for the normal distribution. We succeeded to overcome these difficulties and to calculate Entropic Value-at-Risk (EVaR) measure for Poisson, compound Poisson, Gamma, Laplace, exponential, chi-squared, inverse Gaussian distribution and normal inverse Gaussian distribution with the help of Lambert function that is a special function, generally speaking, with two branches.

Despite the fact that the theory of stability of continuous-time difference equations has a long history, is well developed and very popular in research, there is a simple and clearly formulated problem about the stability of stochastic difference equations with continuous time and distributed delay, which has not been solved for more than 13 years. This paper offers to readers some generalization on this unsolved problem in the hope that it will help move closer to its solution.

The Gaussian-Volterra process with a linear kernel is considered, its properties are established and projection coefficients are explicitly calculated, i.e. one of possible prediction problems related to Gaussian processes is solved.

The integral with respect to a multidimensional stochastic measure, assuming only its σ-additivity in probability, is studied. The continuity and differentiability of realizations of the integral are established.

Models of generalized counting processes time-changed by a general inverse subordinator are considered, their distributions are characterized, and governing equations for them are presented. The equations are given in terms of the generalized fractional derivatives, namely, convolution-type derivatives with respect to Bernštein functions. Some particular examples are presented.

The theory of the so-called ${\mathcal{W}_{q}}$ and ${\mathcal{Z}_{q}}$ scale functions is developped for the fluctuations of right-continuous discrete time and space killed random walks. Explicit expressions are derived for the resolvents and two-sided exit problem when killing depends on the present level of the process. Similar results in the reflected case are also considered. All the expressions are given in terms of new generalisations of the scale functions, which are obtained using arguments different from the continuous case (spectrally negative Lévy processes). Hence, the connections between the two cases are spelled out. For a specific form of the killing function, the probability of bankruptcy is obtained for the model known as omega model in the actuarial literature.

Consistent estimators of the baseline hazard rate and the regression parameter are constructed in the Cox proportional hazards model with heteroscedastic measurement errors, assuming that the baseline hazard function belongs to a certain class of functions with bounded Lipschitz constants.