Narrow Results

In this note, we consider an optimal control problem associated to a differential equation driven by a Hölder continuous function g of index β > 1/2. We split our study into two cases. If the coefficient of dg_t does not depend on the control process, we prove an existence theorem for a slightly generalized control problem, that is we obtain a literal extension of the corresponding situation for ordinary differential equations. If the coefficient of dg_t depends on the control process, we also prove an existence theorem but here we are obliged to restrict the set of controls to sufficiently regular functions.

We consider a least square-type method to estimate the drift parameters for the mean-reverting Ornstein–Uhlenbeck process of the second kind $\{X_t,t\ge 0\}$ defined as $d{X}_t=𝜃(\mu +X_t)dt+d{Y}_{t,G}^{(1)},t\ge 0$, with unknown parameters $𝜃>0$ and $\mu \in ℝ$, where $Y_{t,G}^{(1)}:={\int }_0^t{e}^{-s}d{G}_{a_s}$ with $a_t={\mathrm{\text{He}}}^{\frac{t}{H}}$, and $\{G_t,t\ge 0\}$ is a Gaussian process. In order to establish the consistency and the asymptotic distribution of least square-type estimators of $𝜃$ and $\mu$ based on the continuous-time observations $\{X_t,t\in [0,T]\}$ as $T\to \infty$, we impose some technical conditions on the process $G$, which are satisfied, for instance, if $G$ is a fractional Brownian motion with Hurst parameter $H\in (0,1)$, $G$ is a subfractional Brownian motion with Hurst parameter $H\in (0,1)$ or $G$ is a bifractional Brownian motion with Hurst parameters $(H,K)\in (0,1)\times (0,1]$. Our method is based on pathwise properties of $\{X_t,t\ge 0\}$ and $\{Y_{t,G}^{(1)},t\ge 0\}$ proved in the sequel.

In this paper, we use the techniques of fractional calculus to study the existence of a unique solution to semilinear fractional differential equation driven by a $\gamma$-Hölder continuous function $𝜃$ with $\gamma \in (\frac{2}{3},1)$. Here, the initial condition is a function that may not be defined at zero and the involved integral with respect to $𝜃$ is the extension of the Young integral [An inequality of the Hölder type, connected with Stieltjes integration, Acta Math. 67 (1936) 251–282] given by Zähle [Integration with respect to fractal functions and stochastic calculus I, Probab. Theory Related Fields 111 (1998) 333–374].