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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.

Tempered Hermite process

Farzad Sabzikar

https://doi.org/10.15559/15-MSTA34

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

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.

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-MSTA36CNF

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

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.

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

Taras Shalaiko Georgiy Shevchenko

https://doi.org/10.15559/15-MSTA35CNF

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

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.

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

Ievgen Turchyn

https://doi.org/10.15559/15-MSTA33

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

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.

Weak approximation rates for integral functionals of Markov processes

Iurii Ganychenko Alexei Kulik

https://doi.org/10.15559/15-MSTA37CNF

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

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