In this paper we study backward stochastic differential equations with general terminal value and general random generator. In particular, we do not require the terminal value be given by a forward diffusion equation. The randomness of the generator does not need to be from a forward equation, either. Motivated from applications to numerical simulations, first we obtain the L p -Hölder continuity of the solution. Then we construct several numerical approximation schemes for backward stochastic differential equations and obtain the rate of convergence of the schemes based on the obtained L p -Hölder continuity results. The main tool is the Malliavin calculus.

In this paper we study a singular stochastic differential equation driven by an additive fractional Brownian motion with Hurst parameter H > 1 2 . Under some assumptions on the drift, we show that there is a unique solution, which has moments of all orders. We also apply the techniques of Malliavin calculus to prove that the solution has an absolutely continuous law at any time t > 0.

In this paper, an adaptive nonsingular terminal sliding mode control (ANTSMC) is investigated for attitude tracking of spacecraft with actuator faults. First, a nonsingular fast terminal sliding mode surface is designed to avoid the singularity. Finite-time attitude control is developed using the nonsingular terminal sliding mode technique, which can make the attitude and angular velocity tracking errors converge to zero in finite time in the presence of uncertainties and external disturbances. Second, the total uncertainty is deduced to be bounded. The adaptive laws are incorporated to develop the ANTSMC, removing the restriction on the upper bound of the lumped uncertainty. The finite-time convergence of the closed-loop system with ANTSMC is proved using the Lyapunov stability theory. Finally, the simulation results are presented to demonstrate the performance of the proposed controllers. INDEX TERMS Spacecraft, nonsingular terminal sliding mode, adaptive, finite time, attitude control, actuator faults.

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In this paper, we study spatial averages for the parabolic Anderson model in the Skorohod sense driven by rough Gaussian noise, which is colored in space and time. We include the case of a fractional noise with Hurst parameters H 0 in time and H 1 in space, satisfying H 0 ∈ (1/2, 1), H 1 ∈ (0, 1/2) and H 0 + H 1 > 3/4. Our main result is a functional central limit theorem for the spatial averages. As an important ingredient of our analysis, we present a Feynman-Kac formula that is new for these values of the Hurst parameters.