We prove the unique solvability of parabolic equations with discontinuous leading coefficients in $W_p^{1,2}((0,T)\times {ℝ}^d)$. Using this result, we establish the uniqueness of diffusion processes with time-dependent discontinuous coefficients.

In this paper, we consider the problem of estimating the covariation of two diffusion processes when observations are subject to non-synchronicity. Building on recent papers [Bernoulli11 (2005) 359–379, Ann. Inst. Statist. Math.60 (2008) 367–406], we derive second-order asymptotic expansions for the distribution of the Hayashi–Yoshida estimator in a fairly general setup including random sampling schemes and non-anticipative random drifts. The key steps leading to our results are a second-order decomposition...

In this paper we solve the basic fractional analogue of the classical linear-quadratic Gaussian regulator problem in continuous-time with partial observation. For a controlled linear system where both the state and observation processes are driven by fractional Brownian motions, we describe explicitly the optimal control policy which minimizes a quadratic performance criterion. Actually, we show that a separation principle holds, i.e., the optimal control separates into two stages based on optimal...

We establish the following sharp local estimate for the family ${R_j}_{j=1}^d$ of Riesz transforms on ${ℝ}^d$. For any Borel subset A of ${ℝ}^d$ and any function $f:{ℝ}^d\to ℝ$, ${\int }_A|R_{j}f\left(x\right)|dx\le C_p{\left|\right|f\left|\right|}_{L^p\left({ℝ}^d\right)}{\left|A\right|}^{1/q}$, 1 < p < ∞. Here q = p/(p-1) is the harmonic conjugate to p, $C_p={[2^{q+2}\Gamma (q+1)/{\pi }^{q+1}{\sum }_{k=0}^{\infty }{(-1)}^k/{(2k+1)}^{q+1}]}^{1/q}$, 1 < p < 2, and $C_p={[4\Gamma (q+1)/{\pi }^q{\sum }_{k=0}^{\infty }1/{(2k+1)}^q]}^{1/q}$, 2 ≤ p < ∞. This enables us to determine the precise values of the weak-type constants for Riesz transforms for 1 < p < ∞. The proof rests on appropriate martingale inequalities, which are of independent interest.

For any locally integrable f on ℝⁿ, we consider the operators S and T which average f over balls of radius |x| and center 0 and x, respectively: $Sf\left(x\right)=1/\left|B\right(0,\left|x\right|\left)\right|{\int }_{B(0,|x\left|\right)}f\left(t\right)dt$, $Tf\left(x\right)=1/\left|B\right(x,\left|x\right|\left)\right|{\int }_{B(x,|x\left|\right)}f\left(t\right)dt$ for x ∈ ℝⁿ. The purpose of the paper is to establish sharp localized LlogL estimates for S and T. The proof rests on a corresponding one-weight estimate for a martingale maximal function, a result which is of independent interest.