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https://doi.org/10.15559/15-VMSTA24AI

Pub. online: 31 Dec 2015 Type: Author Index

Journal: Modern Stochastics: Theory and Applications Volume 2, Issue 4 (2015), p. 443

Ruin probability in the three-seasonal discrete-time risk model

Andrius Grigutis Agneška Korvel Jonas Šiaulys

https://doi.org/10.15559/15-VMSTA45

Pub. online: 30 Dec 2015 Type: Research Article

Open Access

Journal: Modern Stochastics: Theory and Applications Volume 2, Issue 4 (2015), pp. 421–441

Abstract

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.

Accuracy of discrete approximation for integral functionals of Markov processes

Iurii Ganychenko Victoria Knopova Alexei Kulik

https://doi.org/10.15559/15-VMSTA46

Pub. online: 30 Dec 2015 Type: Research Article

Open Access

Journal: Modern Stochastics: Theory and Applications Volume 2, Issue 4 (2015), pp. 401–420

Abstract

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.

Distance between exact and approximate distributions of partial maxima under power normalization

Attahalli Shivanarayanaprasad Praveena Sreenivasan Ravi

https://doi.org/10.15559/15-VMSTA42

Pub. online: 23 Dec 2015 Type: Research Article

Open Access

Journal: Modern Stochastics: Theory and Applications Volume 2, Issue 4 (2015), pp. 391–399

Abstract

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.

On packing dimension preservation by distribution functions of random variables with independent

\tilde{Q}

-digits

Oleksandr Slutskyi

https://doi.org/10.15559/15-VMSTA44

Pub. online: 23 Dec 2015 Type: Research Article

Open Access

Journal: Modern Stochastics: Theory and Applications Volume 2, Issue 4 (2015), pp. 371–389

Abstract

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

The notion of “faithfulness of fine packing systems for packing dimension calculation” is introduced, and connections between this notion and packing dimension preservation are found.

Option pricing in the model with stochastic volatility driven by Ornstein–Uhlenbeck process. Simulation

Sergii Kuchuk-Iatsenko Yuliya Mishura

https://doi.org/10.15559/15-VMSTA43

Pub. online: 17 Dec 2015 Type: Research Article

Open Access

Journal: Modern Stochastics: Theory and Applications Volume 2, Issue 4 (2015), pp. 355–369

Abstract

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.

Linear regression by observations from mixture with varying concentrations

Daryna Liubashenko Rostyslav Maiboroda

https://doi.org/10.15559/15-VMSTA41

Pub. online: 4 Dec 2015 Type: Research Article

Open Access

Journal: Modern Stochastics: Theory and Applications Volume 2, Issue 4 (2015), pp. 343–353

Abstract

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.

Extreme residuals in regression model. Minimax approach

Aleksander Ivanov Ivan Matsak Sergiy Polotskiy

https://doi.org/10.15559/15-VMSTA40CNF

Pub. online: 5 Oct 2015 Type: Research Article

Open Access

Journal: Modern Stochastics: Theory and Applications Volume 2, Issue 3 (2015): PRESTO-2015, pp. 297–308

Abstract

We obtain limit theorems for extreme residuals in linear regression model in the case of minimax estimation of parameters.

Fredholm representation of multiparameter Gaussian processes with applications to equivalence in law and series expansions

Tommi Sottinen Lauri Viitasaari

https://doi.org/10.15559/15-VMSTA39CNF

Pub. online: 2 Oct 2015 Type: Research Article

Open Access

Journal: Modern Stochastics: Theory and Applications Volume 2, Issue 3 (2015): PRESTO-2015, pp. 287–295

Abstract

We show that every multiparameter Gaussian process with integrable variance function admits a Wiener integral representation of Fredholm type with respect to the Brownian sheet. The Fredholm kernel in the representation can be constructed as the unique symmetric square root of the covariance. We analyze the equivalence of multiparameter Gaussian processes by using the Fredholm representation and show how to construct series expansions for multiparameter Gaussian processes by using the Fredholm kernel.

Gärtner–Ellis condition for squared asymptotically stationary Gaussian processes

Marina Kleptsyna Alain Le Breton Bernard Ycart

https://doi.org/10.15559/15-VMSTA38CNF

Pub. online: 2 Oct 2015 Type: Research Article

Open Access

Journal: Modern Stochastics: Theory and Applications Volume 2, Issue 3 (2015): PRESTO-2015, pp. 267–286

Abstract

We establish the Gärtner–Ellis condition for the square of an asymptotically stationary Gaussian process. The same limit holds for the conditional distribution given any fixed initial point, which entails weak multiplicative ergodicity. The limit is shown to be the Laplace transform of a convolution of gamma distributions with Poisson compound of exponentials. A proof based on the Wiener–Hopf factorization induces a probabilistic interpretation of the limit in terms of a regression problem.

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Online ISSN: 2351-6054
Print ISSN: 2351-6046

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