Parameter estimation for Ornstein–Uhlenbeck processes driven by fractional Lévy process

Abstract

Abstract We study the minimum Skorohod distance estimation θε∗ $\theta _{\varepsilon}^{\ast }$ and minimum L1 $L_{1}$-norm estimation θε˜ $\widetilde {\theta _{\varepsilon}}$ of the drift parameter θ of a stochastic differential equation dXt=θXtdt+εdLtd $dX_{t}=\theta X_{t}\,dt+\varepsilon \,dL^{d}_{t}$, X0=x0 $X_{0}=x_{0}$, where {Ltd,0≤t≤T} $\{L^{d}_{t},0\leq t\leq T\}$ is a fractional Lévy process, ε∈(0,1] $\varepsilon \in (0,1]$. We obtain their consistency and limit distribution for fixed T, when ε→0 $\varepsilon \rightarrow 0$. Moreover, we also study the asymptotic laws of their limit distributions for T→∞ $T\rightarrow \infty $.

Keywords

• Fractional Lévy process
• Minimum Skorohod distance estimation
• Minimum L 1 $L_{1}$ -norm estimation
• Consistency
• Limit distribution
• Asymptotic law