Approximation of SDE solutions using local asymptotic expansions

We develop an asymptotic expansion for small time of the density of the solution of a non-degenerate system of stochastic differential equations with smooth coefficients, and apply this to the stepwise approximation of solutions. The asymptotic expansion, which takes the form of a multivariate Edgeworth-type expansion, is obtained from the Kolmogorov forward equation using some standard PDE results. To generate one step of the approximation to the solution, we use a Cornish–Fisher type expansion derived from the Edgeworth expansion. To interpret the approximation generated in this way as a strong approximation we use couplings between the (normal) random variables used and the Brownian path driving the SDE. These couplings are constructed using techniques from optimal transport and Vaserstein metrics. The type of approximation so obtained may be regarded as intermediate between a conventional strong approximation and a weak approximation. In principle approximations of any order can be obtained, though for higher orders the algebra becomes very heavy. In order 1/2 the method gives the usual Euler approximation; in order 1 it gives a variant of the Milstein method, but which needs only normal variables to be generated. However the method is somewhat limited by the non-degeneracy requirement.

Linear subdiffusion in weighted fractional Hölder spaces

<p style='text-indent:20px;'>For <inline-formula><tex-math id="M1">\begin{document}$ \nu\in(0,1) $\end{document}</tex-math></inline-formula>, we investigate the nonautonomous subdiffusion equation:</p><p style='text-indent:20px;'><disp-formula> <label/> <tex-math id="FE1"> \begin{document}$ \mathbf{D}_{t}^{\nu}u-\mathcal{L}u = f(x,t), $\end{document} </tex-math></disp-formula></p><p style='text-indent:20px;'>where <inline-formula><tex-math id="M2">\begin{document}$ \mathbf{D}_{t}^{\nu} $\end{document}</tex-math></inline-formula> is the Caputofractional derivative and <inline-formula><tex-math id="M3">\begin{document}$ \mathcal{L} $\end{document}</tex-math></inline-formula> is a uniformly ellipticoperator with smooth coefficients depending on time. Undersuitable conditions on the given data and a minimal number (i.e.the necessary number) of compatibility conditions, the globalclassical solvability to the related initial-boundary valueproblems are established in the weighted fractional Hölderspaces.</p>

Stability results for an elastic–viscoelastic wave equation with localized Kelvin–Voigt damping and with an internal or boundary time delay

We investigate the stability of a one-dimensional wave equation with non smooth localized internal viscoelastic damping of Kelvin–Voigt type and with boundary or localized internal delay feedback. The main novelty in this paper is that the Kelvin–Voigt and the delay damping are both localized via non smooth coefficients. Under sufficient assumptions, in the case that the Kelvin–Voigt damping is localized faraway from the tip and the wave is subjected to a boundary delay feedback, we prove that the energy of the system decays polynomially of type t − 4 . However, an exponential decay of the energy of the system is established provided that the Kelvin–Voigt damping is localized near a part of the boundary and a time delay damping acts on the second boundary. While, when the Kelvin–Voigt and the internal delay damping are both localized via non smooth coefficients near the boundary, under sufficient assumptions, using frequency domain arguments combined with piecewise multiplier techniques, we prove that the energy of the system decays polynomially of type t − 4 . Otherwise, if the above assumptions are not true, we establish instability results.

Boundary value problems with nonlocal conditions for hyperbolic systems of equations with two independent variables

Nonlocal boundary value problems for arbitrary order hyperbolic systems with one spatial variable are considered. A priori estimates for general nonlocal mixed problems for systems with smooth and piecewise smooth coefficients are obtained. The correct solvability of such problems is proved.Examples of additional conditions necessity are provided.