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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.

Gärtner–Ellis condition for squared asymptotically stationary Gaussian processes

Marina Kleptsyna Alain Le Breton Bernard Ycart

https://doi.org/10.15559/15-MSTA38CNF

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

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

Editorial

K. Kubilius Yu. Mishura L. Sakhno

https://doi.org/10.15559/15-MSTA23EDI

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.

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

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