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Equivariant adjusted least squares estimator in two-line fitting model

Sergiy Shklyar

https://doi.org/10.15559/16-VMSTA47

Pub. online: 21 Mar 2016 Type: Research Article

Open Access

Journal: Modern Stochastics: Theory and Applications Volume 3, Issue 1 (2016), pp. 19–45

Abstract

We consider the two-line fitting problem. True points lie on two straight lines and are observed with Gaussian perturbations. For each observed point, it is not known on which line the corresponding true point lies. The parameters of the lines are estimated.

This model is a restriction of the conic section fitting model because a couple of two lines is a degenerate conic section. The following estimators are constructed: two projections of the adjusted least squares estimator in the conic section fitting model, orthogonal regression estimator, parametric maximum likelihood estimator in the Gaussian model, and regular best asymptotically normal moment estimator.

The conditions for the consistency and asymptotic normality of the projections of the adjusted least squares estimator are provided. All the estimators constructed in the paper are equivariant. The estimators are compared numerically.

Functional limit theorems for additive and multiplicative schemes in the Cox–Ingersoll–Ross model

Yuliia Mishura Yevheniia Munchak

https://doi.org/10.15559/16-VMSTA48

Pub. online: 3 Mar 2016 Type: Research Article

Open Access

Journal: Modern Stochastics: Theory and Applications Volume 3, Issue 1 (2016), pp. 1–17

Abstract

In this paper, we consider the Cox–Ingersoll–Ross (CIR) process in the regime where the process does not hit zero. We construct additive and multiplicative discrete approximation schemes for the price of asset that is modeled by the CIR process and geometric CIR process. In order to construct these schemes, we take the Euler approximations of the CIR process itself but replace the increments of the Wiener process with iid bounded vanishing symmetric random variables. We introduce a “truncated” CIR process and apply it to prove the weak convergence of asset prices. We establish the fact that this “truncated” process does not hit zero under the same condition considered for the original nontruncated process.

2010 Mathematics Subject Classification index

https://doi.org/10.15559/15-VMSTA24MI

Pub. online: 31 Dec 2015 Type: 2010 Mathematics Subject Classification Index

Journal: Modern Stochastics: Theory and Applications Volume 2, Issue 4 (2015), pp. 447–450

Subject index

https://doi.org/10.15559/15-VMSTA24SI

Pub. online: 31 Dec 2015 Type: Subject Index

Journal: Modern Stochastics: Theory and Applications Volume 2, Issue 4 (2015), pp. 445–446

Author index

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.

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