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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.

Editorial

K. Kubilius Yu. Mishura L. Sakhno

https://doi.org/10.15559/15-VMSTA23EDI

Pub. online: 2 Oct 2015 Type: Editorial

Open Access

Journal: Modern Stochastics: Theory and Applications Volume 2, Issue 3 (2015): PRESTO-2015, pp. 199–201

Tempered Hermite process

Farzad Sabzikar

https://doi.org/10.15559/15-VMSTA34

Pub. online: 25 Sep 2015 Type: Research Article

Open Access

Journal: Modern Stochastics: Theory and Applications Volume 2, Issue 4 (2015), pp. 327–341

Abstract

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.

Pricing the European call option in the model with stochastic volatility driven by Ornstein–Uhlenbeck process. Exact formulas

Sergii Kuchuk-Iatsenko Yuliya Mishura

https://doi.org/10.15559/15-VMSTA36CNF

Pub. online: 25 Sep 2015 Type: Research Article

Open Access

Journal: Modern Stochastics: Theory and Applications Volume 2, Issue 3 (2015): PRESTO-2015, pp. 233–249

Abstract

We consider the Black–Scholes model of financial market modified to capture the stochastic nature of volatility observed at real financial markets. For volatility driven by the Ornstein–Uhlenbeck process, we establish the existence of equivalent martingale measure in the market model. The option is priced with respect to the minimal martingale measure for the case of uncorrelated processes of volatility and asset price, and an analytic expression for the price of European call option is derived. We use the inverse Fourier transform of a characteristic function and the Gaussian property of the Ornstein–Uhlenbeck process.

Integral representation with respect to fractional Brownian motion under a log-Hölder assumption

Taras Shalaiko Georgiy Shevchenko

https://doi.org/10.15559/15-VMSTA35CNF

Pub. online: 25 Sep 2015 Type: Research Article

Open Access

Journal: Modern Stochastics: Theory and Applications Volume 2, Issue 3 (2015): PRESTO-2015, pp. 219–232

Abstract

We show that if a random variable is the final value of an adapted log-Hölder continuous process, then it can be represented as a stochastic integral with respect to a fractional Brownian motion with adapted integrand. In order to establish this representation result, we extend the definition of the fractional integral.

A multiplicative wavelet-based model for simulation of a random process

Ievgen Turchyn

https://doi.org/10.15559/15-VMSTA33

Pub. online: 24 Sep 2015 Type: Research Article

Open Access

Journal: Modern Stochastics: Theory and Applications Volume 2, Issue 4 (2015), pp. 309–325

Abstract

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.

Weak approximation rates for integral functionals of Markov processes

Iurii Ganychenko Alexei Kulik

https://doi.org/10.15559/15-VMSTA37CNF

Pub. online: 23 Sep 2015 Type: Research Article

Open Access

Journal: Modern Stochastics: Theory and Applications Volume 2, Issue 3 (2015): PRESTO-2015, pp. 251–266

Abstract

We obtain weak rates for approximation of an integral functional of a Markov process by integral sums. An assumption on the process is formulated only in terms of its transition probability density, and, therefore, our approach is not strongly dependent on the structure of the process. Applications to the estimates of the rates of approximation of the Feynman–Kac semigroup and of the price of “occupation-time options” are provided.

Convergence of hitting times for jump-diffusion processes

Georgiy Shevchenko

https://doi.org/10.15559/15-VMSTA32

Pub. online: 23 Sep 2015 Type: Research Article

Open Access

Journal: Modern Stochastics: Theory and Applications Volume 2, Issue 3 (2015): PRESTO-2015, pp. 203–218

Abstract

We investigate the convergence of hitting times for jump-diffusion processes. Specifically, we study a sequence of stochastic differential equations with jumps. Under reasonable assumptions, we establish the convergence of solutions to the equations and of the moments when the solutions hit certain sets.

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

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