Well-posedness and stability for a class of stochastic delay differential equations with singular drift

Abstract

In this paper we prove well-posedness and stability of a class of stochastic delay differential equations with singular drift. Moreover, we show local wellposedness under localized assumptions.

On the strong Feller property for stochastic delay differential equations with singular drift

“…such that M x is a global weak solution for equation (1). By Lemma 3.3, Lemma 3.4, condition (C2) and condition (C3), it holds…”

“…At first, we show the existence of a weak solution. The additional statement about the pathwise uniqueness has been shown in [1], Theorem 1.5, and the existence of a strong solution then follows from the Theorem of Yamada and Watanabe [19]. The strong solution M x is by definition (F t ) t≥0 -adapted where (F t ) t≥0 is the augmented filtration generated by W .…”

“…The following stability result has essentially been shown in [1], where σ was supposed to be weakly differentiable with…”

“…In order to deal with discontinuous drift coefficients, we consider a certain convergence for topological spaces, see Theorem 1.8. As a result, we gain a simple method to derive the strong Feller property of equation (1) from the well-posedness and the strong Feller property of the simpler special case B ≡ 0 and b ≡ 0. equipped with the supremum norm · ∞ . For a process X defined on [t − r, t] with t ≥ 0, we write X t (s) := X(t + s), s ∈ [−r, 0].…”

Convergence Rate of Euler–Maruyama Scheme for FSDEs

“…So, in this work, our goal is to discuss convergence rate of EM method for SDEs with rough coefficients, which may allow SDEs involved to be degenerate. For wellposedness of (path-dependent) SDEs with singular coefficients, we refer to, e.g., [1,19,20,21] for more details.…”