Modern Stochastics: Theory and Applications

We find a multiplicative wavelet-based representation for stochastic processes that can be represented as the exponent of a second-order centered random process. We propose a wavelet-based model for simulation of such a stochastic process and find its rates of convergence to the process in different functional spaces in terms of approximation with given accuracy and reliability. This approach allows us to simulate stochastic processes (including certain classes of processes with heavy tails) with given accuracy and reliability.

A tempered Hermite process modifies the power law kernel in the time domain representation of a Hermite process by multiplying an exponential tempering factor $\lambda >0$ such that the process is well defined for Hurst parameter $H>\frac{1}{2}$. A tempered Hermite process is the weak convergence limit of a certain discrete chaos process.

We consider a finite mixture model with varying mixing probabilities. Linear regression models are assumed for observed variables with coefficients depending on the mixture component the observed subject belongs to. A modification of the least-squares estimator is proposed for estimation of the regression coefficients. Consistency and asymptotic normality of the estimates is demonstrated.

We consider a discrete-time approximation of paths of an Ornstein–Uhlenbeck process as a mean for estimation of a price of European call option in the model of financial market with stochastic volatility. The Euler–Maruyama approximation scheme is implemented. We determine the estimates for the option price for predetermined sets of parameters. The rate of convergence of the price and an average volatility when discretization intervals tighten are determined. Discretization precision is analyzed for the case where the exact value of the price can be derived.

The article is devoted to finding conditions for the packing dimension preservation by distribution functions of random variables with independent $\tilde{Q}$-digits.

We obtain the distance between the exact and approximate distributions of partial maxima of a random sample under power normalization. It is observed that the Hellinger distance and variational distance between the exact and approximate distributions of partial maxima under power normalization is the same as the corresponding distances under linear normalization.

The article is devoted to the estimation of the convergence rate of integral functionals of a Markov process. Under the assumption that the given Markov process admits a transition probability density differentiable in t and the derivative has an integrable upper bound of a certain type, we derive the accuracy rates for strong and weak approximations of the functionals by Riemannian sums. We also develop a version of the parametrix method, which provides the required upper bound for the derivative of the transition probability density for a solution of an SDE driven by a locally stable process. As an application, we give accuracy bounds for an approximation of the price of an occupation time option.

This paper deals with the discrete-time risk model with nonidentically distributed claims. We suppose that the claims repeat with time periods of three units, that is, claim distributions coincide at times $\{1,4,7,\dots \}$, at times $\{2,5,8,\dots \}$, and at times $\{3,6,9,\dots \}$. We present the recursive formulas to calculate the finite-time and ultimate ruin probabilities. We illustrate the theoretical results by several numerical examples.