Mild Solutions of Quantum Stochastic Differential Equations

Mild Solutions of Neutral Stochastic Partial Functional Differential Equations

$L^p$-existence of mild solutions of fractional differential equations in Banach space

We study the existence of mild solutions for semilinear fractional differential equations with nonlocal initial conditions in $L^p([0,1],E)$, where $E$ is a separable Banach space. The main ingredients used in the proof of our results are measure of noncompactness, Darbo and Schauder fixed point theorems. Finally, an application is proved to illustrate the results of this work. 

Uniqueness of Solutions of Stochastic Differential Equations

It follows from a theorem of Veretennikov [4] that (1) has a unique strong solution, i.e. there is a unique process x(t), adapted to the filtration of the Brownian motion, satisfying (1). Veretennikov in fact proved this for a more general equation. Here we consider a different question, posed by N. V. Krylov [2]: we choose a Brownian path W and ask whether (1) has a unique solution for that pa...

Adaptive Numerical Solutions of Stochastic Differential Equations

In this paper we present an adaptive multi-element generalized polynomial chaos (ME-gPC) method, which can achieve hp-convergence in random space. ME-gPC is based on the decomposition of random space and generalized polynomial chaos (gPC). Using proper numerical schemes to maintain the local orthogonality on-the-fly, we perform gPC locally and adaptively. The key idea is to combine the polynomi...

Existence of Square-mean Almost Periodic Mild Solutions to Some Nonautonomous Stochastic Second-order Differential Equations

In this paper we use the well-known Schauder fixed point principle to obtain the existence of square-mean almost periodic solutions to some classes of nonautonomous second order stochastic differential equations on a Hilbert space.