The law of iterated logarithm for subordinated Gaussian sequences: uniform Wasserstein bounds

in ALEA: Latin American Journal of Probability and Mathematical Statistics (2016), 13

Parameter estimation based on discrete observations of fractional Ornstein-Uhlenbeck process of the second kind

in Statistical Inference for Stochastic Processes (2015), 18(3), 205227

Optimal Berry-Esseen bounds on the Poisson space.

E-print/Working paper (2015)

Necessary and sufficient conditions for Hölder continuity of Gaussian processes

in Statistics and Probability Letters (2014), 94

The continuity of Gaussian processes is an extensively studied topic and it culminates in Talagrand’s notion of majorizing measures that gives complicated necessary and sufficient conditions. In this note ... [more ▼]

The continuity of Gaussian processes is an extensively studied topic and it culminates in Talagrand’s notion of majorizing measures that gives complicated necessary and sufficient conditions. In this note we study the Hölder continuity of Gaussian processes. It turns out that necessary and sufficient conditions can be stated in a simple form that is a variant of the celebrated Kolmogorov–Čentsov condition. [less ▲]

Asymptotic normality of randomized periodogram for estimating quadratic variation in mixed Brownian-fractional Brownian model

E-print/Working paper (2014)

We study asymptotic normality of the randomized periodogram estimator of quadratic variation in the mixed Brownian--fractional Brownian model. In the semimartingale case, that is, where the Hurst ... [more ▼]

We study asymptotic normality of the randomized periodogram estimator of quadratic variation in the mixed Brownian--fractional Brownian model. In the semimartingale case, that is, where the Hurst parameter H of the fractional part satisfies H∈(3/4,1), the central limit theorem holds. In the nonsemimartingale case, that is, where H∈(1/2,3/4], the convergence toward the normal distribution with a nonzero mean still holds if H=3/4, whereas for the other values, that is, H∈(1/2,3/4), the central convergence does not take place. We also provide Berry--Esseen estimates for the estimator. [less ▲]

Spectral characterization of the quadratic variation of mixed Brownian–fractional Brownian motion

in Statistical Inference for Stochastic Processes (2013), 16(2), 97-112

Dzhaparidze and Spreij (Stoch Process Appl, 54:165–174, 1994) showed that the quadratic variation of a semimartingale can be approximated using a randomized periodogram. We show that the same ... [more ▼]

Dzhaparidze and Spreij (Stoch Process Appl, 54:165–174, 1994) showed that the quadratic variation of a semimartingale can be approximated using a randomized periodogram. We show that the same approximation is valid for a special class of continuous stochastic processes. This class contains both semimartingales and non-semimartingales. The motivation comes partially from the recent work by Bender et al. (Finance Stoch, 12:441–468, 2008), where it is shown that the quadratic variation of the log-returns determines the hedging strategy. [less ▲]

Rate of Convergence for Discretization of Integrals with Respect to Fractional Brownian Motion

in Journal of Theoretical Probability (2013)

In this article, an uniform discretization of stochastic integrals $\int_{0}^{1} f'_-(B_t)\ud B_t$, where $B_t$ denotes the fractional Brownian motion with Hurst parameter $H \in (\frac{1}{2},1)$, for a ... [more ▼]

In this article, an uniform discretization of stochastic integrals $\int_{0}^{1} f'_-(B_t)\ud B_t$, where $B_t$ denotes the fractional Brownian motion with Hurst parameter $H \in (\frac{1}{2},1)$, for a large class of convex functions $f$ is considered. In $\big[$\cite{a-m-v}, Statistics \& Decisions, \textbf{27}, 129-143$\big]$, for any convex function $f$, the almost sure convergence of uniform discretization to such stochastic integral is proved. Here we prove $L^r$- convergence of uniform discretization to stochastic integral. In addition, we obtain a rate of convergence. It turns out that the rate of convergence can be brought arbitrary close to $H - \frac{1}{2}$. [less ▲]

On the fractional Black-Scholes market with transaction costs

in Communications in Mathematical Finance (2013), 2(3), 21-40

We consider fractional Black-Scholes market with proportional transaction costs. When transaction costs are present, one trades periodically i.e. we have the discrete trading with equidistance $n^{-1 ... [more ▼]

We consider fractional Black-Scholes market with proportional transaction costs. When transaction costs are present, one trades periodically i.e. we have the discrete trading with equidistance $n^{-1}$ between trading times. We derive a non trivial hedging error for a class of European options with convex payoff in the case when the transaction costs coefficients decrease as $n^{-(1-H)}$. We study the expected hedging error and asymptotic behavior of the hedge as Hurst parameter $H$ approaches $\frac{1}{2}$. [less ▲]