Modern Stochastics: Theory and Applications

For a class of non-autonomous parabolic stochastic partial differential equations defined on a bounded open subset $D\subset {\mathbb{R}^{d}}$ and driven by an ${L^{2}}(D)$-valued fractional Brownian motion with the Hurst index $H>1/2$, a new result on existence and uniqueness of a mild solution is established. Compared to the existing results, the uniqueness in a fully nonlinear case is shown, not assuming the coefficient in front of the noise to be affine. Additionally, the existence of moments for the solution is established.

Our aim in this paper is to establish some strong stability properties of a solution of a stochastic differential equation driven by a fractional Brownian motion for which the pathwise uniqueness holds. The results are obtained using Skorokhod’s selection theorem.

We consider the simulation of sample paths of a fractional Brownian motion with small values of the Hurst index and estimate the behavior of the expected maximum. We prove that, for each fixed N, the error of approximation $\mathbf{E}\max _{t\in [0,1]}{B}^{H}(t)-\mathbf{E}\max _{i=\overline{1,N}}{B}^{H}(i/N)$ grows rapidly to ∞ as the Hurst index tends to 0.

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.

We construct an estimator of the unknown drift parameter $\theta \in \mathbb{R}$ in the linear model where ${B}^{H_{1}}$ and ${B}^{H_{2}}$ are two independent fractional Brownian motions with Hurst indices $H_{1}$ and $H_{2}$ satisfying the condition $\frac{1}{2}\le H_{1}<H_{2}<1$. Actually, we reduce the problem to the solution of the integral Fredholm equation of the 2nd kind with a specific weakly singular kernel depending on two power exponents. It is proved that the kernel can be presented as the product of a bounded continuous multiplier and weak singular one, and this representation allows us to prove the compactness of the corresponding integral operator. This, in turn, allows us to establish an existence–uniqueness result for the sequence of the equations on the increasing intervals, to construct accordingly a sequence of statistical estimators, and to establish asymptotic consistency.

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 $H\in (3/4,1)$, the central limit theorem holds. In the nonsemimartingale case, that is, where $H\in (1/2,3/4]$, the convergence toward the normal distribution with a nonzero mean still holds if $H=3/4$, whereas for the other values, that is, $H\in (1/2,3/4)$, the central convergence does not take place. We also provide Berry–Esseen estimates for the estimator.