The object of investigation is the mixed fractional Brownian motion of the form Xt=κBH1t+σBH2t, driven by two independent fractional Brownian motions BH1 and BH2 with Hurst parameters H1<H2. Strongly consistent estimators of unknown model parameters (H1,H2,κ2,σ2)⊤ are constructed based on the equidistant observations of a trajectory. Joint asymptotic normality of these estimators is proved for 0<H1<H2<34.
The paper is devoted to a stochastic heat equation with a mixed fractional Brownian noise. We investigate the covariance structure, stationarity, upper bounds and asymptotic behavior of the solution. Based on its discrete-time observations, we construct a strongly consistent estimator for the Hurst index H and prove the asymptotic normality for H<3/4. Then assuming the parameter H to be known, we deal with joint estimation of the coefficients at the Wiener process and at the fractional Brownian motion. The quality of estimators is illustrated by simulation experiments.
A problem of drift parameter estimation is studied for a nonergodic weighted fractional Vasicek model defined as dXt=θ(μ+Xt)dt+dBa,bt, t≥0, with unknown parameters θ>0, μ∈R and α:=θμ, whereas Ba,b:={Ba,bt,t≥0} is a weighted fractional Brownian motion with parameters a>−1, |b|<1, |b|<a+1. Least square-type estimators (˜θT,˜μT) and (˜θT,˜αT) are provided, respectively, for (θ,μ) and (θ,α) based on a continuous-time observation of {Xt,t∈[0,T]} as T→∞. The strong consistency and the joint asymptotic distribution of (˜θT,˜μT) and (˜θT,˜αT) are studied. Moreover, it is obtained that the limit distribution of ˜θT is a Cauchy-type distribution, and ˜μT and ˜αT are asymptotically normal.
The paper deals with a stochastic heat equation driven by an additive fractional Brownian space-only noise. We prove that a solution to this equation is a stationary and ergodic Gaussian process. These results enable us to construct a strongly consistent estimator of the diffusion parameter.
We consider a stochastic differential equation of the form
dXt=θa(t,Xt)dt+σ1(t,Xt)σ2(t,Yt)dWt
with multiplicative stochastic volatility, where Y is some adapted stochastic process. We prove existence–uniqueness results for weak and strong solutions of this equation under various conditions on the process Y and the coefficients a, σ1, and σ2. Also, we study the strong consistency of the maximum likelihood estimator for the unknown parameter θ. We suppose that Y is in turn a solution of some diffusion SDE. Several examples of the main equation and of the process Y are provided supplying the strong consistency.