On solutions to Ito stochastic differential equations

Application of DJ method to Ito stochastic differential equations

‎This paper develops iterative method described by [V‎. ‎Daftardar-Gejji‎, ‎H‎. ‎Jafari‎, ‎An iterative method for solving nonlinear functional equations‎, ‎J‎. ‎Math‎. ‎Anal‎. ‎Appl‎. ‎316 (2006) 753-763] to solve Ito stochastic differential equations‎. ‎The convergence of the method for Ito stochastic differential equations is assessed‎. ‎To verify efficiency of method‎, ‎some examples are ex...

Some Estimates on Exponentials of Solutions to Stochastic Differential Equations

On representations of solutions to certain stochastic differential equations

In this paper we obtain general conditions under which stochastic differential equations possess a strong solution representable in an explicit form as a functional of the Wiener process. Particular interest bears the problem of determining conditions that guarantee nonexplosion of the solution. The necessary as well as sufficient condition is derived.

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...