Flows for singular stochastic differential equations with unbounded drifts
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Date
2019-05-12
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Elsevier
Abstract
In this paper, we are interested in the following singular stochastic differential equation (SDE)
dXt = b(t,Xt)dt +dBt, 0 ≤ t ≤ T, X0 = x ∈ Rd,
where the drift coefficient b :[0, T] ×Rd−→Rdis Borel measurable, possibly unbounded and has spatial linear growth. The driving noise Btis a d−dimensional Brownian motion. The main objective of the paper is to establish the existence and uniqueness of a strong solution and a Sobolev differentiable stochastic flow for the above SDE. Malliavin differentiability of the solution is also obtained (cf. [21,23]). Our results constitute significant extensions to those in [31,30,14,21,23]by allowing the drift bto be unbounded. We employ methods from white-noise analysis and the Malliavin calculus. As application, we prove existence of a unique strong Malliavin differentiable solution to the following stochastic delay differential equation.
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Research Article
Keywords
Strong solutions of SDE’s, Irregular drift coefficient, Malliavin calculus, Sobolev flows